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+\chapter{Brownian Motion Monte Carlo} + +\section{Brownian Motion Preliminaries} +We will present here some standard invariants of Brownian motions. The proofs are standard and can be found in for instance \cite{durrett2019probability} and \cite{karatzas1991brownian}. + + +\begin{lemma}[Markov property of Brownian motions] + Let $T \in \R$, $t \in [0,T]$, and $d \in \N$. Let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a probability space. Let $\mathcal{W}_t: \lb 0, T \rb \times \Omega \rightarrow \R^d$ be a standard Brownian motion. Fix $s\in [0,\infty)$. Let $\mathfrak{W}_t = \mathcal{W}_{s+t}-\mathcal{W}_s$. Then $\mathfrak{W} = \left\{ \mathfrak{W}_t : t\in [0,\infty) \right\}$ is also a standard Brownian motion independent of $\mathcal{W}$. +\end{lemma} + +\begin{proof} + We check against the Brownian motion axioms. First note that $\mathfrak{W}_0 = \mathcal{W}_{s+0} - \mathcal{W}_s = 0$ with $\mathbb{P}$-a.s. + + Note that $t\mapsto \mathcal{W}_{s+t} - \mathcal{W}_s$ is $\mathbb{P}$-a.s. continuous as it is the difference of two functions that are also $\mathbb{P}$-a.s. continuous. + + Note next that for $h\in \lp 0,\infty\rp$ it is the case that: + \begin{align} + \E\lb \mathfrak{W}_{t+h} -\mathfrak{W}_t\rb &= \E \lb \mathcal{W}_{s+t+h}-\mathcal{W}_{s+h} -\mathcal{W}_{s+t}+\mathcal{W}_s\rb \nonumber \\ + &= \E \lb \mathcal{W}_{s+t+h}-\mathcal{W}_{s+t}\rb -\E \lb \mathcal{W}_{s+h}-\mathcal{W}_s\rb \nonumber \\ + &=0-0 =0 + \end{align} + + We note finally that: + \begin{align} + \var \lb \mathfrak{W}_{t+h} -\mathfrak{M}_t\rb &= \var \lb \mathcal{W}_{s+t+h}-\mathcal{W}_{s} -\mathcal{W}_{s+t}+\mathcal{W}_s\rb \nonumber \\ + &= \var \lb \mathcal{W}_{s+t+h}-\mathcal{W}_{s+t}\rb -\var \lb \mathcal{W}_{s}-\mathcal{W}_s\rb + \cancel{\cov \lp \mathcal{W}_{s+t+h}-\mathcal{W}_{s+t}, \mathcal{W}_{s+h}-\mathcal{W}_s\rp} \nonumber \\ + &=h-0=h \nonumber + \end{align} + Finally note that two stochastic processes $\mathcal{W}$, $\mathcal{X}$ are independent whenever given a set of sample points $t_1,t_2,\hdots, t_n \in \lb 0,T\rb$ it is the case that the vectors $\lb \mathcal{W}_{t_1}, \mathcal{W}_{t_2},\hdots, \mathcal{W}_{t_n}\rb^\intercal$ and $\lb \mathcal{X}_{t_1},\mathcal{X}_{t_2},\hdots, \mathcal{X}_{t_n}\rb^\intercal$ are independent vectors. + + That being the case note that the independent increments property of Brownian motions yields that, $\mathcal{W}_{s+t_1} - \mathcal{W}_s$, $\mathcal{W}_{s+t_2}-\mathcal{W}_s, \hdots, \mathcal{W}_{s+t_n}-\mathcal{W}_s$ is independent of $\mathcal{W}_{t_1},\mathcal{W}_{t_2},\hdots, \mathcal{W}_{t_n}$, i.e. $\mathfrak{W}$ and $\mathcal{W}$ are independent. +\end{proof} +\begin{lemma}[Independence of Brownian Motion]\label{iobm} + Let $T \in \lp 0,\infty\rp$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $\mathcal{X}, \mathcal{Y}: \lb 0,T\rb \times \Omega \rightarrow \R^d$ be standard Brownian motions. It is then the case that they are independent of each other. +\end{lemma} +\begin{proof} + We say that two Brownian motions are independent of each of each other if given a sampling vector of times $\lp t_1,t_2,\hdots,t_n\rp$, the vectors $\lp \mathcal{X}_{t_1}, \mathcal{X}_{t_2},\hdots \mathcal{X}_{t_n}\rp$ and $\lp \mathcal{Y}_{t_1}, \mathcal{Y}_{t_2},\hdots, \mathcal{Y}_{t_n}\rp$ are independent. As such let $n\in \N$ and let $\lp t_1,t_2,\hdots t_n \rp$ be a vector or times with samples as given above. Consider now a new Brownian motion $\mathcal{X} - \mathcal{Y}$, wherein our samples are now $\lp \mathcal{X}_{t_1} - \mathcal{Y}_{t_1}, \mathcal{X}_{t_2}-\mathcal{Y}_{t_2}, \hdots, \mathcal{X}_{t_n} - \mathcal{Y}_{t_n} \rp$. By the independence property of Brownian motions, these increments must be independent of each other. Whence it is the case that the vectors $\lp \mathcal{X}_{t_1}, \mathcal{X}_{t_2},\hdots, \mathcal{X}_{t_n}\rp$ and $\lp \mathcal{Y}_{t_1}, \mathcal{Y}_{t_2},\hdots, \mathcal{Y}_{t_n}\rp$ are independent. +\end{proof} +\begin{lemma}[Scaling Invariance] + Let $T \in \R$, $t \in [0,T]$, and $d \in \N$. Let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a probability space. Let $\mathcal{W}_t: \lb 0, T \rb \times \Omega \rightarrow \R^d$ be a standard Brownian motion. Let $a \in \R \setminus \{ 0\}$. It is then the case that $\mathcal{X}_t \coloneqq \frac{1}{a} \mathcal{W}_{a^2\cdot t}$ is also a standard Brownian motion. +\end{lemma} +\begin{proof} + We check against the Brownian motion axioms. Note for instance that the function $t \mapsto \mathcal{X}_t$ is a product of a constant with a function that is $\mathbb{P}$-a.s. continuous yielding a function that is also $\mathbb{P}$-a.s. continuous. + + Note also for instance that $\mathcal{X}_0 = \frac{1}{a} \cdot \mathcal{W}_{a^2 \cdot 0} = 0$ with $\mathbb{P}$-a.s. + + Note that for all $h \in \lp 0,\infty\rp$, and $t\in \lb 0,T\rb$ it is the case that: + \begin{align} + \E \lb \mathcal{X}_{t+h} - \mathcal{X}_t\rb &= \E \lb \frac{1}{a}\mathcal{W}_{a^2 \cdot \lp t+h\rp} - \frac{1}{a}\mathcal{W}_{a^2 \cdot t}\rb \nonumber \\ + &=\frac{1}{a} \E \lb \mathcal{W}_{a^2\cdot \lp t+h\rp} - \mathcal{W}_{a^2\cdot t}\rb \nonumber \\ + &=0\nonumber + \end{align} + + Note that for all $h \in \lp 0,\infty\rp$, and $t\in \lb 0,T\rb$ it is the case that: + + \begin{align} + \var\lb \mathcal{X}_{t+h } - \mathcal{X}_t\rb &= \var\lb \frac{1}{a}\mathcal{W}_{a^2\cdot \lp t+h\rp} - \frac{1}{a}\mathcal{W}_{a^2\cdot t}\rb \nonumber \\ + &=\frac{1}{a^2}\var\lb\mathcal{W}_{a^2\cdot \lp t+h\rp} - \mathcal{W}_{a^2\cdot t}\rb \nonumber\\ + &= \frac{1}{\cancel{a^2}}\cancel{a^2} \lp \cancel{t}+h-\cancel{t}\rp \nonumber\\ + &=h + \end{align} + Finally note that for $t \in \lb 0,T\rb$ and $s \in \lb 0,t\rp$ it is the case that $\mathcal{W}_{a^2 \cdot t} - \mathcal{W}_{a^2 \cdot s}$ is independent of $\mathcal{W}_{a^2\cdot s}$. Whence it is also the case that $\mathcal{X}_t-\mathcal{X}_s$ is independent of $\mathcal{X}_s$. +\end{proof} + +\begin{lemma}[Summation of Brownian Motions] + Let $T \in \R$, $t \in [0,T]$ and $d \in \N$. Let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a probability space. Let $\mathcal{W}_t, \mathcal{X}_t: \lb 0,T \rb \times \Omega \rightarrow \R^d$ be a standard independent Brownian motions. It is then the case that the process $\mathcal{Y}_t$ defined as $\mathcal{Y}_t = \frac{1}{\sqrt{2}}\lp \mathcal{W}_t + \mathcal{X}_t \rp$ is also a standard Brownian motion. +\end{lemma} +\begin{proof} + Note that $t \mapsto \frac{1}{\sqrt{2}}\lp \mathcal{W}_t+\mathcal{X}_t\rp$ is $\mathbb{P}$-a.s. continuous as it is the linear combination of two functions that are also $\mathbb{R}$-a.s. continuous. + + Note also that $\mathcal{Y}_0 = \frac{1}{\sqrt{2}}\lp \mathcal{W}_0+\mathcal{X}_0\rp = 0+0=0$ with $\mathbb{P}$-a.s. + + Note that for all $h \in \lp 0,\infty\rp$ and $t \in \lb t,T\rb$ it is the case that: + \begin{align} + \E\lb \frac{1}{\sqrt{2}}\lp \mathcal{Y}_{t+h} - \mathcal{Y}_t\rp \rb &= \E \lb \frac{1}{\sqrt{2}} \lp\mathcal{W}_{t+h}+\mathcal{X}_{t+h} - \mathcal{W}_t-\mathcal{X}_t \rp\rb \nonumber \\ + &= \frac{1}{\sqrt{2}}\E \lb \mathcal{W}_{t+h}-\mathcal{W}_t\rb + \frac{1}{\sqrt{2}}\E \lb \mathcal{X}_{t+h}-\mathcal{X}_t\rb \nonumber \\ + &=0 \nonumber + \end{align} + + Note that for all $h \in \lp 0,\infty\rp$, and $t\in \lb 0,T\rb$ it is the case that: + \begin{align} + \var \lb \frac{1}{\sqrt{2}}\lp\mathcal{Y}_{t+h} - \mathcal{Y}_{t}\rp\rb &= \var \lb \frac{1}{\sqrt{2}}\lp \mathcal{W}_{t+h}+\mathcal{X}_{t+h} - \mathcal{W}_t-\mathcal{X}_t\rp \rb \nonumber \\ + &=\var \lb \frac{1}{\sqrt{2}}\lp \mathcal{W}_{t+h} - \mathcal{W}_t\rp + \frac{1}{\sqrt{2}}\lp \mathcal{X}_{t+h}-\mathcal{X}_t\rp\rb \nonumber\\ + &= \frac{1}{2}\var \lb \mathcal{W}_{t+h}-\mathcal{W}_t\rb +\frac{1}{2}\var\lb \mathcal{X}_{t+h}-\mathcal{X}_t\rb + \cancel{\cov \lp \mathcal{W},\mathcal{X}\rp} \nonumber \\ + &= h \nonumber + \end{align} +\end{proof} + + + +\begin{definition}[Of $\mathfrak{k}$]\label{def:1.17} + Let $p \in [2,\infty)$. We denote by $\mathfrak{k}_p \in \R$ the real number given by $\mathfrak{k}:=\inf \{ c\in \R \}$ where it holds that for every probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and every random variable $\mathcal{X}: \Omega \rightarrow \R$ with $\E[|\mathcal{X}|] < \infty$ that $\lp \E \lb \lv \mathcal{X} - \E \lb \mathcal{X} \rb \rp^p \rb \rp ^{\frac{1}{p}} \leqslant c \lp \E \lb \lv \mathcal{X} \rv^p \rb \rp ^{\frac{1}{p}}.$ +\end{definition} + +\begin{definition}[Primary Setting]\label{primarysetting} Let $d,m \in \mathbb{N}$, $T, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \mathbb{Z}$, $g \in C(\mathbb{R}^d,\mathbb{R})$, assume for all $t \in [0,T],x\in \mathbb{R}^d$ that: +\begin{align}\label{(2.1.2)} + \max\{|g(x)|\} \leqslant \mathfrak{L} \lp 1+\|x\|_E^p \rp +\end{align} + +and let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space. Let $\mathcal{W}^{\theta}: [0,T] \times \Omega \rightarrow \mathbb{R}^d$, $\theta \in \Theta$ be independent standard Brownian motions, let $u \in C([0,T] \times \mathbb{R}^d,\mathbb{R})$ satisfy for all $t \in [0,T]$, $x\in \mathbb{R}^d$, that $\mathbb{E}[|g(x+\mathcal{W}^0_{T-t})|] < \infty$ and: +\begin{align}\label{(1.12)} + u(t,x) &= \mathbb{E} \lb g \lp x+\mathcal{W}^0_{T-t} \rp \rb +\end{align} +and let let $U^\theta:[0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$, $\theta \in \Theta$ satisfy, $\theta \in \Theta$, $t \in [0,T]$, $x\in \mathbb{R}^d$, that: +\begin{align}\label{(2.1.4)} + U^\theta_m(t,x) = \frac{1}{m}\left[\sum^{m}_{k=1}g\left(x+\mathcal{W}^{(\theta,0,-k)}_{T-t}\right)\right] +\end{align} +\end{definition} +\begin{lemma} \label{lemma1.1} + +Assume Setting \ref{primarysetting} then: +\begin{enumerate}[label = (\roman*)] + \item it holds for all $n\in \N_0$, $\theta \in \Theta$ that $U^\theta:[0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$ is a continuous random field. + \item it holds that for all $\theta \in \Theta$ that $\sigma \lp U^\theta \rp \subseteq \sigma \lp \lp \mathcal{W}^{(\theta, \mathcal{V})}\rp_{\mathcal{V} \in \Theta}\rp $. + \item it holds that $\lp U^\theta \rp_{\theta \in\Theta}$,$\lp \mathcal{W}^\theta \rp_{\theta \in \Theta}$, are independent. + \item it holds for all $n,m \in $, $i,k,\mathfrak{i},\mathfrak{k}\in \mathbb{Z}$, with $(i,k) \neq (\mathfrak{i},\mathfrak{k})$ that $(U^{(\theta,i,k)})_{\theta \in \Theta}$ and $\left(U^{(\theta,\mathfrak{i},\mathfrak{k})}\right)_{\theta \in \Theta}$ are independent and, + \item it holds that $\lp U^\theta \rp_{\theta \in \Theta}$ are identically distributed random variables. +\end{enumerate} +\end{lemma} + +\begin{proof} For (i) Consider that $\mathcal{W}^{(\theta,0,-k)}_{T-t}$ are continuous random fields and that $g\in C(\mathbb{R}^d,\mathbb{R})$, we have that $U^\theta(t,x)$ is the composition of continuous functions with $m > 0$ by hypothesis, ensuring no singularities. Thus $U^\theta: [0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$. + +\medskip + +For (ii) observe that for all $\theta \in \Theta$ it holds that $\mathcal{W}^\theta$ is $\mathcal{B} \lp \lb 0, T \rb \otimes \sigma \lp W^\theta \rp \rp /\mathcal{B}\lp \mathbb{R}^d \rp$-measurable, this, and induction on prove item (ii). + +\medskip +Moreover observe that item (ii) and the fact that for all $\theta \in \Theta$ it holds that $\lp\mathcal{W}^{\lp \theta, \vartheta\rp}_{\vartheta \in \Theta}\rp$, $\mathcal{W}^\theta$ are independently establish item (iii). + +\medskip +Furthermore, note that (ii) and the fact that for all $i,k,\mathfrak{i},\mathfrak{k} \in \mathbb{Z}$, $\theta \in \Theta$, with $(i,k) \neq (\mathfrak{i},\mathfrak{k})$ it holds that $\lp\mathcal{W}^{\lp\theta, i,k,\vartheta\rp}\rp_{\vartheta \in \Theta}$ and $\lp\mathcal{W}^{\lp\theta,\mathfrak{i},\mathfrak{k},\vartheta\rp}\rp_{\vartheta \in \Theta}$ are independent establish item (iv). + +\medskip +Hutzenhaler \cite[Corollary~2.5 ]{hutzenthaler_overcoming_2020} establish item (v). This completes the proof of Lemma 1.1. +\end{proof} + +\begin{lemma}\label{lem:1.20} Assume Setting \ref{primarysetting}. Then it holds for $\theta \in \Theta$, $s \in [0,T]$, $t\in [s,T]$, $x\in \mathbb{R}^d$ that: +\begin{align} + \mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s}\rp \rv \rb +\mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}\rp \rv \rb + \int^T_s \E \lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s} \rp \rv \rb dr < \infty +\end{align} +\end{lemma} + +\begin{proof} +Note that (\ref{(2.1.2)}), the fact that for all $r,a,b \in [0,\infty)$ it holds that $(a+b)^r \leqslant 2^{\max\{r-1,0\}}(a^r+b^r)$, and the fact that for all $\theta \in \Theta$ it holds that $\mathbb{E}\lb \|\mathcal{W}^\theta_T\|\rb < \infty$, assure that for all $s \in [0,T]$, $t\in[s,T]$, $\theta \in \Theta$ it holds that: +\begin{align}\label{(2.1.6)} + \mathbb{E}\lb \lv g(x+\mathcal{W}^\theta_{t-s})\rv \rb &\leqslant \mathbb{E}\lb\mathfrak{L}\lp 1+\|x+\mathcal{W}^\theta_{t-s}\|_E^p\rp\rb \nonumber\\ + &\leqslant \mathfrak{L}\lb 1+2^{\max\{p-1,0\}}\lp \|x\|_E^p+\mathbb{E} \lb \left\|\mathcal{W}^\theta_T\right\|_E^p \rb \rp\rb<\infty +\end{align} +\label{eq:1.4} +We next claim that for all $s\in [0,T]$, $t\in[s,T]$, $\theta \in \Theta$ it holds that: +\begin{align}\label{(1.17)} + \mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s} \rp \rv \rb+ \int^T_s \mathbb{E}\lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s}\rp \rv \rb dr < \infty +\end{align} + +To prove this claim observe the triangle inequality and (\ref{(2.1.4)}), demonstrate that for all $s\in[0,T]$, $t\in[s,T]$, $\theta \in \Theta$, it holds that: +\begin{align}\label{(1.18)} + \mathbb{E}\lb \lv U^\theta \lp t,x+\mathcal{W}^\theta_{t-s}\rp \rv \rb \leqslant \frac{1}{m}\left[ \sum^{m}_{i=1}\mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}+\mathcal{W}^{(\theta,0,-i)}_{T-t} \rp \rv \rb \rb +\end{align} + +Now observe that (\ref{(2.1.6)}) and the fact that $(W^\theta)_{\theta \in \Theta}$ are independent imply that for all $s \in [0,T]$, $t\in [s,T]$, $\theta \in \Theta$, $i\in \mathbb{Z}$ it holds that: +\begin{align}\label{(1.19)} + \mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{t-s}+\mathcal{W}^{(\theta,0,i)}_{T-t} \rp \rv \rb = \mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{(t-s)+(T-t)}\rp \rv \rb = \mathbb{E}\lb \lv g \lp x+\mathcal{W}^\theta_{T-s}\rp \rv \rb <\infty +\end{align} +\medskip +Combining (\ref{(1.18)}) and (\ref{(1.19)}) demonstrate that for all $s \in [0,T]$, $t\in[s,T]$, $\theta \in \Theta$ it holds that: +\begin{align}\label{(1.20)} + \mathbb{E}\lb \lv U^\theta(t,x+\mathcal{W}^\theta_{t-s})\rv \rb < \infty +\end{align} + +Finally observe that for all $s\in [0,T]$ $\theta \in \Theta$ it holds that: +\begin{align}\label{(1.21)} +\int^T_s \mathbb{E}\lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s} \rp \rv \rb &\leqslant \lp T-s \rp \sup_{r\in [s,T]} \mathbb{E} \lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s}\rp \rv \rb < \infty +\end{align} + +Combining (\ref{(1.16)}), (\ref{(1.20)}), and (\ref{(1.21)}) completes the proof of Lemma \ref{lem:1.20}. + +\end{proof} +\begin{corollary}\label{cor:1.20.1} Assume Setting \ref{primarysetting}, then we have: +\begin{enumerate}[label = (\roman*)] + \item it holds that $t \in [0,T],x\in \mathbb{R}^d$ that: + \begin{align} + \mathbb{E}\lb \lv U^0 \lp t,x \rp \rv \rb + \mathbb{E}\lb \lv g \lp x+\mathcal{W}^{(0,0,-1)}_{T-t} \rp \rv \rb < \infty + \end{align} + \item it holds that $t\in [0,T],x\in \mathbb{R}^d$ that: + \begin{align} + \mathbb{E}\lb U^0\lp t,x \rp \rb = \mathbb{E} \lb g \lp x+\mathcal{W}^{(0,0,-1)}_{T-t}\rp\rb + \end{align} +\end{enumerate} +\end{corollary} + +\begin{proof} +(i) is a restatement of Lemma \ref{lem:1.20} in that for all $t\in [0,T]$: +\begin{align} + &\mathbb{E}\left[ \left| U^0\left( t,x \right) \right| \right] + \mathbb{E} \left[ \left|g \left(x+\mathcal{W}^{(0,0,-1)}_{T-t}\right)\right|\right] \nonumber\\ + &<\mathbb{E} \left[ \left|U^\theta \lp t,x+\mathcal{W}^\theta_{t-s} \rp \right| \right] +\mathbb{E}\left[ \left|g \left(x+\mathcal{W}^\theta_{t-s}\right) \right| \right]+ \int^T_s \mathbb{E}\lb \lv U^\theta \lp r,x+\mathcal{W}^\theta_{r-s} \rp \rv \rb dr \nonumber\\ + &< \infty +\end{align} + +Furthermore (ii) is a restatement of (\ref{(1.14)}) with $\theta = 0$, $m=1$, and $k=1$. This completes the proof of Corollary \ref{cor:1.20.1}. +\end{proof} + +\section{Monte Carlo Approximations} + + +\begin{lemma}\label{lem:1.21}Let $p \in (2,\infty)$,$n\in \mathbb{N}$, let $(\Omega, \mathcal{F}, \mathbb{P})$, be a probability space and let $\mathcal{X}_i: \Omega \rightarrow \mathbb{R}$, $i \in \{1,2,...,n\}$ be i.i.d. random variables with $\mathbb{E}[|\mathcal{X}_1|]<\infty$. Then it holds that: +\begin{align} + \lp\E \lb \lv \E \lb \mathcal{X}_1 \rb-\frac{1}{n} \lp \sum^n_{i=1} \mathcal{X}_i \rp \rv^p \rb \rp^{\frac{1}{p}} \leqslant \lb \frac{p-1}{n}\rb ^{\frac{1}{2}}\left(\E\lb \lv \mathcal{X}_1-\E \lb \mathcal{X}_1 \rb \rv^p \rp \rb^{\frac{1}{p}} +\end{align} +\end{lemma} + +\begin{proof} +The hypothesis that for all $i \in \{1,2,...,n\}$ it holds that $\mathcal{X}_i:\Omega \rightarrow \mathbb{R}$ are i.i.d. random variables ensures that: +\begin{align} + \E \lb \lv \E \lb \mathcal{X}_1\rb - \frac{1}{n} \lp \sum^n_{i=1} \mathcal{X}_i \rp \rv ^p \rb + = \E \lb \lv \frac{1}{n} \lp \sum^n_{i=1} \lp \E \lb \mathcal{X}_1 \rb - \mathcal{X}_i \rp \rp \rv ^p \rb = \frac{1}{n^p} \E \lb \lv \sum^n_{i=1} \lp \E \lb \mathcal{X}_i \rb - \mathcal{X}_i \rp \rv^p \rb +\end{align} + +This combined with the fact that for all $i \in \{1,2,...,n\}$ it is the case that $\mathcal{X}_i: \Omega \rightarrow \R$ are i.i.d. random variables and e.g. \cite[Theorem~2.1]{rio_moment_2009} (with $p \curvearrowleft p$, $ ( S_i )_{i \in \{0,1,...,n\}} \curvearrowleft ( \sum^i_{k=1} ( \E [ X_k ] - X_k))$, $( X_i )_{i \in \{1,2,...,n\}} \curvearrowleft ( \E [ X_i ] - X_i )_{i \in \{1,2,...,n\}}$ in the notation of \cite[Theorem~2.1]{rio_moment_2009} ensures that: + +\begin{align} + \lp \E \lb \lv \E \lb \mathcal{X}_1 \rb - \frac{1}{n} \lp \sum^n_{i=1} \mathcal{X}_i \rp + \rv ^p \rb\rp ^{\frac{2}{p}} &= \frac{1}{n^2} \lp \E \lb \lv + \sum^n_{i=1} \lp \E \lb \mathcal{X}_i \rb - \mathcal{X}_i \rp \rv ^p \rb \rp ^{\frac{2}{p}} + \nonumber\\ + &\leqslant \frac{p-1}{n^2} \lb \sum^n_{i=1} \lp \E \lb \lv \E \lb \mathcal{X}_i \rb -\mathcal{X}_i \rv^p \rb \rp ^{\frac{2}{p}} \rb \nonumber\\ + &= \frac{p-1}{n^2} \lb n \lp \E \lb \lv \E \lb \mathcal{X}_1 \rb - \mathcal{X}_1 \rv ^p \rb \rp^{\frac{2}{p}} \rb \\ + &= \frac{p-1}{n} \lp \E \lb \lv \E \lb \mathcal{X}_1 \rb -\mathcal{X}_1\rv ^p \rb \rp ^{\frac{2}{p}} +\end{align} +This completes the proof of the lemma. +\end{proof} + +\begin{corollary}\label{corollary:1.11.1.} + Let $p\in [2,\infty)$, $n \in \N$, let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a probability space, and let $\mathcal{X}_i: \Omega \rightarrow \R$, $i \in \{1,2,...,n\}$ be i.i.d random variables with $\E\lb \lv \mathcal{X}_1 \rv \rb < \infty$. Then it holds that: + \begin{align}\label{(1.26)} + \lp \E \lb \lv \E \lb \mathcal{X}_1 \rb - \frac{1}{n}\lp \sum^n_{i=1} \mathcal{X}_i \rp \rv ^p \rb \rp^{\frac{1}{p}} \leqslant \lb \frac{p-1}{n} \rb ^{\frac{1}{2}} \lp \E \lb \lv \mathcal{X}_1 - \E \lb \mathcal{X}_1 \rb \rv ^p \rb \rp ^{\frac{1}{p}} + \end{align} +\end{corollary} + +\begin{proof} + Observe that e.g. \cite[Lemma~2.3]{grohsetal} and Lemma \ref{lem:1.21} establish (\ref{(1.26)}). +\end{proof} + +\begin{corollary}\label{cor:1.22.2} + Let $p \in [2,\infty)$, $n\in \N$, let $(\Omega, \mathcal{F}, \mathbb{P})$, be a probability space, and let $\mathcal{X}_i: \Omega \rightarrow \R$, $i \in \{1,2,...,n\}$, be i.i.d. random variables with $\E[|\mathcal{X}_1|] < \infty$, then: + \begin{align} + \lp \E \lb \lv \E \lb \mathcal{X}_1\rb - \frac{1}{n} \lp \sum^n_{i=1} \mathcal{X}_i \rp \rv ^p \rb \rp ^{\frac{1}{p}} \leqslant \frac{\mathfrak{k}_p \sqrt{p-1}}{n^{\frac{1}{2}}} \lp \E \lb \lv \mathcal{X}_1 \rv^p \rb \rp ^{\frac{1}{p}} + \end{align} +\end{corollary} + +\begin{proof} + This a direct consequence of Definition \ref{def:1.17} and Corollary \ref{corollary:1.11.1.}. +\end{proof} +\section{Bounds and Covnvergence} + +\begin{lemma}\label{lem:1.21} Assume Setting \ref{def:1.18}. Then it holds for all $t\in [0,T]$, $x\in \mathbb{R}^d$ +\begin{align} + &\left(\E\left[\left|U^0(t,x+\mathcal{W}^0_t)-\E \left[U^0 \left(t,x+\mathcal{W}^0_t \right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \nonumber\\ + &\leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E\left[ \lv g \lp x+\mathcal{W}^0_T \rp \rv^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}}\right] +\end{align} +\end{lemma} + +\begin{proof} For notational simplicity, let $G_k: [0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$, $k\in \mathbb{Z}$, satisfy for all $k\in \mathbb{Z}$, $t\in[0,T]$, $x\in \mathbb{R}^d$ that: +\begin{align} + G_k(t,x) = g\left(x+\mathcal{W}^{(0,0,-k)}_{T-t}\right) +\end{align} +\medskip +Observe that the hypothesis that $(\mathcal{W}^\theta)_{\theta \in \Theta}$ are independent Brownian motions and the hypothesis that $g \in C(\mathbb{R}^d,\mathbb{R})$ assure that for all $t \in [0,T]$,$x\in \mathbb{R}^d$ it holds that $(G_k(t,x))_{k\in \mathbb{Z}}$ are i.i.d. random variables. This and Corollary \ref{cor:1.22.2} (applied for every $t\in [0,T]$, $x\in \mathbb{R}^d$ with $p \curvearrowleft \mathfrak{p}$, $n \curvearrowleft m$, $(X_k)_{k\in \{1,2,...,m\}} \curvearrowleft (G_k(t,x))_{k\in \{1,2,...,m\}}$), with the notation of Corollary \ref{cor:1.22.2} ensure that for all $t\in [0,T]$, $x \in \mathbb{R}^d$, it holds that: +\begin{align} + \left( \E \left[ \left| \frac{1}{m} \left[ \sum^{m}_{k=1} G_k(t,x) \right] - \E \left[ G_1(t,x) \right] \right| ^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} \leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}}\left(\E \left[|G_1(t,x)|^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} +\end{align} +\medskip +Combining this, with (1.16), (1.17), and item (ii) of Corollary \ref{cor:1.20.1} yields that: +\begin{align} + &\left(\E\left[\left|U^0(t,x) - \E \left[U^0(t,x)\right]\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} \nonumber\\ + &= \left(\E \left[\left|\frac{1}{m}\left[\sum^{m}_{k=1}G_k(t,x)\right]- \E \left[G_1(t,x)\right]\right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} \\ + &\leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}}\left(\E \left[\left| G_1(t,x)\right| ^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \\ + &= \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E \left[\left|g\left(x+\mathcal{W}^1_{T-t}\right)\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}}\right] +\end{align} +This and the fact that $\mathcal{W}^0$ has independent increments ensure that for all $n\in $, $t\in [0,T]$, $x\in \mathbb{R}^d$ it holds that: +\begin{align} + \left(\E \left[\left| U^0 \left(t,x+\mathcal{W}^0_t\right) - \E \left[U^0 \left(t,x+\mathcal{W}^0_t\right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \leqslant \frac{\mathfrak{m}}{m^{\frac{1}{2}}} \left[\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^{\frac{1}{\mathfrak{p}}} \right] +\end{align} +This completes the proof of Lemma \ref{lem:1.21}. +\end{proof} + + +\begin{lemma}\label{lem:1.22} Assume Setting \ref{primarysetting}. Then it holds for all, $t\in [0,T]$, $x\in \mathbb{R}^d$ that: +\begin{align} + \left(\E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t\right) - u \left(t,x+\mathcal{W}^0_t\right) \right|^\mathfrak{p} \right]\right)^\frac{1}{\mathfrak{p}} \leqslant \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right) \left( \E \left[\left| g \left(x+\mathcal{W}^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} +\end{align} +\end{lemma} + +\begin{proof} +Observe that from Corollary \ref{cor:1.20.1} item (ii) we have: +\begin{align} + \E\left[U^0(t,x)\right] = \E \left[ g \left(x+\mathcal{W}^{(0,0,-1)}_{T-t}\right) \right] +\end{align} +This and (\ref{(1.12)}) ensure that: +\begin{align} + u(t,x) - \E \left[U^0(t,x) \right] &= 0 \nonumber \\ + \E \lb U^0(t,x) \rb - u \lp t,x \rp &= 0 +\end{align} + +This, and the fact that $\mathcal{W}^0$ has independent increments, assure that for all, $t\in [0,T]$, $x\in \mathbb{R}^d$, it holds that: +\begin{align} + \left(\E \left[\left| \E \lb U^0 \lp t,x+\mathcal{W}^0_t \rp\right] - u \lp t,x+\mathcal{W}^0_t \rp\right| ^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} = 0 \leqslant \left(\E \left[ \lv u \lp t,x+\mathcal{W}^0_t \rp \rv^\p\right]\right) +\end{align} +This along with (\ref{(1.12)}) ensure that: +\begin{align} + \left(\E \left[\left| \E \left[U^0 \lp t,x+\mathcal{W}^0_t \rp \right] - u \lp t,x+\mathcal{W}^0_t \rp \right| ^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} = 0 \leqslant \left( \E \left[\left| g \left(x+\mathcal{W}^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} +\end{align} +Notice that the triangle inequality gives us: +\begin{align} + \left(\E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t\right) - u \left(t,x+\mathcal{W}^0_t\right) \right|^\mathfrak{p} \right]\right)^\frac{1}{\mathfrak{p}} \leqslant \left(\E \left[\left| U^0(t,x+W^0_t) - \E \left[U^0(t,x+\mathcal{W}^0_t)\right]\right|^\p\right]\right)^{\frac{1}{\p}} \nonumber \\ + +\left(\E \left[\left| \E \left[U^0 \lp t,x+\mathcal{W}^0_t \rp \right]-u \lp t,x+\mathcal{W}^0_t \rp\right|^\p\right]\right)^{\frac{1}{\p}} +\end{align} +This, combined with (1.26), (1.21), the independence of Brownian motions, gives us: +\begin{align} + \left(\E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t\right) - u \left(t,x+\mathcal{W}^0_t\right) \right|^\mathfrak{p} \right]\right)^\frac{1}{\mathfrak{p}} &\leqslant \left(\frac{\mathfrak{m}}{m^{\frac{1}{2}}}\right) \left( \E \left[\left| g \left(x+\mathcal{W}^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} \nonumber \\ + &= \left(\frac{\mathfrak{m}}{m^{\frac{1}{2}}}\right) \left( \E \left[\left| g \left(x+\mathcal{W}^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} +\end{align} +This completes the proof of Lemma \ref{lem:1.22}. +\end{proof} + +\begin{lemma}\label{lem:1.25} Assume Setting \ref{primarysetting}. Then it holds for all $t\in [0,T]$, $x\in \mathbb{R}^d$ that: +\begin{align} + \left( \E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t \right) - u \left( t, x+\mathcal{W}^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +\end{lemma} +\medskip + +\begin{proof} Observe that Lemma \ref{lem:1.22} ensures that: +\begin{align}\label{(1.46)} + \left( \E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t \right) - u \left( t, x+\mathcal{W}^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \leqslant \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} +\end{align} +Observe next that (\ref{(1.12)}) ensures that: +\begin{align}\label{(1.47)} + \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} \leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_T \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +Which in turn yields that: +\begin{align}\label{(1.48)} + \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_T \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} \leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +Combining \ref{(1.46)}, \ref{(1.47)}, and \ref{(1.48)} yields that: +\begin{align} + \left( \E \left[ \left| U^0 \left(t,x+\mathcal{W}^0_t \right) - u \left( t, x+\mathcal{W}^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} &\leqslant \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\E \left[\left| g \left(x+\mathcal{W}^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} \nonumber\\ + &\les\mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s\in[0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +This completes the proof of Lemma \ref{lem:1.25}. +\end{proof} + +\begin{corollary}\label{cor:1.25.1} Assume Setting \ref{primarysetting}. Then it holds for all $t\in[0,T]$, $x\in \R^d$ that: +\begin{align} + \left( \E \left[ \left| U^0 \lp t,x \rp -u(t,x) \rv ^\p \right] \right) ^{\frac{1}{\p}} \leqslant \mathfrak{L} \left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}} \right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \right\| x+\mathcal{W}^0_s \left\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +\end{corollary} + + +\begin{proof} Observe that for all $t \in [0,T-\mft]$ and $\mft \in [0,T]$, and the fact that $W^0$ has independent increments it is the case that: +\begin{align} + u(t+\mft,x) = \E \left[g \left(x+\mathcal{W}^0_{T-(t+\mft)}\right)\right] = \E \left[g \left(x+\mathcal{W}^0_{(T-\mft)-t)}\right)\right] +\end{align} +It is also the case that: +\begin{align*} + U^\theta(t+\mft,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g \left(x+\mathcal{W}^{(\theta,0,-k)}_{T-(t+\mft)}\right)\right] = \frac{1}{m} \left[\sum^{m}_{k=1} g \left(x+\mathcal{W}^{(\theta,0,-k)}_{(T-\mft)-t}\right)\right] +\end{align*} +\medskip +Then, applying Lemma \ref{lem:1.25}, applied for all $\mft \in [0,T]$, with $\mathfrak{L} \curvearrowleft \mathfrak{L}$, $p \curvearrowleft p$, $\mathfrak{p} \curvearrowleft \mathfrak{p}$, $T \curvearrowleft (T-\mft)$ is such that for all $\mft \in [0,T]$, $t \in [0,T-\mft]$, $x \in \R^d$ we have: +\begin{align} + &\left( \E \left[ \left| U^0 \left(t+\mft,x+\mathcal{W}^0_t \right) - u \left( t+\mft, x+\mathcal{W}^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \nonumber \\ + &\leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T-\mft]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} \nonumber \\ + &\leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +Thus we get for all $\mft \in [0,T]$, $x\in \R^d$, $n \in $: +\begin{align} + \left( \E \left[ \left| U^0 \left(\mft,x \right) - u \left(\mft, x \right) \right|^\p \right] \right)^{\frac{1}{\p}} &= \left( \E \left[ \left| U^0 \left(\mft,x+\mathcal{W}^0_0 \right) - u \left(\mft, x+\mathcal{W}^0_0 \right) \right|^\p \right] \right)^{\frac{1}{\p}}\nonumber\\ + &\leqslant \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{1}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left\| x+\mathcal{W}^0_s \right\|_E^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +This completes the proof of Corollary \ref{cor:1.25.1}. +\end{proof} + +\begin{theorem}\label{tentpole_1} Let $T,L,p,q, \mathfrak{d} \in [0,\infty), m \in \mathbb{N}$, $\Theta = \bigcup_{n\in \mathbb{N}} \Z^n$, let $g_d\in C(\R^d,\R)$, and assume that $d\in \N$, $t \in [0,T]$, $x = (x_1,x_2,...,x_d)\in \R^d$, $v,w \in \R$ and that $\max \{ |g_d(x)|\} \leqslant Ld^p \left(1+\Sigma^d_{k=1}\left|x_k \right|\right)$, let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $\mathcal{W}^{d,\theta}: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $\theta \in \Theta$, be independent standard Brownian motions, assume for every $d\in \N$ that $\left(\mathcal{W}^{d,\theta}\right)_{\theta \in \Theta}$ are independent, let $u_d \in C([0,T] \times \R^d,\R)$, $d \in \N$, satisfy for all $d\in \N$, $t\in [0,T]$, $x \in \R^d$ that $\E \left[g_x \left(x+\mathcal{W}^{d,0}_{T-t} \right)\right] < \infty$ and: +\begin{align} + u_d\left(t,x\right) = \E \left[g_d \left(x + \mathcal{W}^{d,0}_{T-t}\right)\right] +\end{align} +Let $U^{d,\theta}_m: [0,T] \times \R^d \times \Omega \rightarrow \R$, $d \in \N$, $m\in \Z$, $\theta \in \Theta$, satisfy for all, $d\in \N$, $m \in \Z$, $\theta \in \Theta$, $t\in [0,T]$, $x\in \R^d$ that: +\begin{align} + U^{d,\theta}_m(t,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g_d \left(x + \mathcal{W}^{d,(\theta, 0,-k)}_{T-t}\right)\right] +\end{align} +and for every $d,n,m \in \N$ let $\mathfrak{C}_{d,n,m} \in \Z$ be the number of function evaluations of $u_d(0,\cdot)$ and the number of realizations of scalar random variables which are used to compute one realization of $U^{d,0}_m(T,0): \Omega \rightarrow \R$. + + There then exists $c \in \R$, and $\mathfrak{N}:\N \times (0,1] \rightarrow \N$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that: +\begin{align}\label{(2.48)} + \sup_{t\in[0,T]} \sup_{x \in [-L,L]^d} \left(\E \left[\left| u_d(t,x) - U^{d,0}_{\mathfrak{N}(d,\epsilon)}\right|^\p\right]\right)^\frac{1}{\p} \leqslant \epsilon +\end{align} + +and: +\begin{align}\label{2.3.27} + \mathfrak{C}_{d,\mathfrak{N}(d,\varepsilon), \mathfrak{N}(d,\varepsilon)} \leqslant cd^c\varepsilon^{-(2+\delta)} +\end{align} +\end{theorem} +\begin{proof} Throughout the proof let $\mathfrak{m}_\mathfrak{p} = \sqrt{\mathfrak{p} -1}$, $\mathfrak{p} \in [2,\infty)$, let $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfy for all $d \in \N$, $t\in [0,T]$ that: +\begin{align}\label{2.3.29} +\mathbb{F}^d_t = \begin{cases} + \bigcap_{s\in[t,T]} \sigma \left(\sigma \left(W^{d,0}_r: r \in [0,s]\right) \cup \{A\in \mathcal{F}: \mathbb{P}(A)=0\}\right) & :t -- 225 +< -- 11 ++ -- 79 +- -- 74 +* -- 465 +:= -- 1011 +add.period$ -- 51 +call.type$ -- 15 +change.case$ -- 108 +chr.to.int$ -- 12 +cite$ -- 16 +duplicate$ -- 228 +empty$ -- 393 +format.name$ -- 95 +if$ -- 1134 +int.to.chr$ -- 4 +int.to.str$ -- 0 +missing$ -- 17 +newline$ -- 80 +num.names$ -- 45 +pop$ -- 93 +preamble$ -- 1 +purify$ -- 112 +quote$ -- 0 +skip$ -- 179 +stack$ -- 0 +substring$ -- 356 +swap$ -- 37 +text.length$ -- 3 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 82 +warning$ -- 1 +while$ -- 41 +width$ -- 0 +write$ -- 200 +(There was 1 error message) diff --git a/Dissertation/Dissertation.fdb_latexmk b/Dissertation/Dissertation.fdb_latexmk new file mode 100644 index 0000000..d53fcb3 --- /dev/null +++ b/Dissertation/Dissertation.fdb_latexmk @@ -0,0 +1,146 @@ +# Fdb version 3 +["bibtex Dissertation"] 1680561453 "Dissertation.aux" "Dissertation.bbl" "Dissertation" 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2370. +LaTeX Info: Redefining \cpagerefrange on input line 2370. +LaTeX Info: Redefining \Cpagerefrange on input line 2370. +LaTeX Info: Redefining \labelcref on input line 2370. +LaTeX Info: Redefining \labelcpageref on input line 2370. +Package cleveref Info: `amsthm' support loaded on input line 3026. +Package cleveref Info: always capitalise cross-reference names on input line 78 +25. +Package cleveref Info: always capitalise cross-reference names on input line 78 +52. +) +(/usr/local/texlive/2021/texmf-dist/tex/latex/natbib/natbib.sty +Package: natbib 2010/09/13 8.31b (PWD, AO) +\bibhang=\skip58 +\bibsep=\skip59 +LaTeX Info: Redefining \cite on input line 694. +\c@NAT@ctr=\count284 +) +\c@theorem=\count285 +\c@corollary=\count286 + +(/usr/local/texlive/2021/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def +File: l3backend-pdftex.def 2021-03-18 L3 backend support: PDF output (pdfTeX) +\l__color_backend_stack_int=\count287 +\l__pdf_internal_box=\box50 +) +(./Dissertation.aux + 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Cross-referencing by name of section + +(/usr/local/texlive/2021/texmf-dist/tex/latex/refcount/refcount.sty +Package: refcount 2019/12/15 v3.6 Data extraction from label references (HO) +) +(/usr/local/texlive/2021/texmf-dist/tex/generic/gettitlestring/gettitlestring.s +ty +Package: gettitlestring 2019/12/15 v1.6 Cleanup title references (HO) +) +\c@section@level=\count294 +) +LaTeX Info: Redefining \ref on input line 60. +LaTeX Info: Redefining \pageref on input line 60. +LaTeX Info: Redefining \nameref on input line 60. + (./Dissertation.out) (./Dissertation.out) +\@outlinefile=\write3 +\openout3 = `Dissertation.out'. + +LaTeX Font Info: Trying to load font information for U+msa on input line 62. + + +(/usr/local/texlive/2021/texmf-dist/tex/latex/amsfonts/umsa.fd +File: umsa.fd 2013/01/14 v3.01 AMS symbols A +) +LaTeX Font Info: Trying to load font information for U+msb on input line 62. + + +(/usr/local/texlive/2021/texmf-dist/tex/latex/amsfonts/umsb.fd +File: umsb.fd 2013/01/14 v3.01 AMS symbols B +) +LaTeX Font Info: Trying to load font information for U+rsfs on input line 62 +. + +(/usr/local/texlive/2021/texmf-dist/tex/latex/jknapltx/ursfs.fd +File: ursfs.fd 1998/03/24 rsfs font definition file (jk) +) [1 + +{/usr/local/texlive/2021/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]pdfTeX w +arning (ext4): destination with the same identifier (name{page.1}) has been alr +eady used, duplicate ignored + + \relax +l.65 \end{abstract} + [1] (./Dissertation.toc) +\tf@toc=\write4 +\openout4 = `Dissertation.toc'. + +pdfTeX warning (ext4): destination with the same identifier (name{page.1}) has +been already used, duplicate ignored + + \relax +l.69 \newpage + [1 + +] +LaTeX Font Info: Trying to load font information for U+euf on input line 80. + + +(/usr/local/texlive/2021/texmf-dist/tex/latex/amsfonts/ueuf.fd +File: ueuf.fd 2013/01/14 v3.01 Euler Fraktur +) [2] [3] +Chapter 1. +[4 + +] [5] +Overfull \hbox (12.0844pt too wide) in paragraph at lines 182--183 +[]\OT1/cmr/m/it/10.95 A strong so-lu-tion to the stochas-tic dif-fer-en-tial eq +ua-tion ([]1.1.5[]) on prob-a-bil-ity space $\OT1/cmr/m/n/10.95 ( +\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 F\OML/cmm/m/it/10.95 ; \U/msb/m/n/10. +95 P\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 (\U/msb/m/n/10.95 F[]\OT1/cmr/m/n/ +10.95 )[])$\OT1/cmr/m/it/10.95 , + [] + +[6] [7] [8] +LaTeX Font Info: Trying to load font information for U+BOONDOX-ds on input l +ine 291. + +(/usr/local/texlive/2021/texmf-dist/tex/latex/boondox/uboondox-ds.fd +File: uboondox-ds.fd 2011/04/26 Fontinst v1.933 font definitions for U/BOONDOX- +ds. +) [9] +Chapter 2. +[10 + +] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] +Chapter 3. + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 749. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 749. + +[24 + +] +LaTeX Font Info: Trying to load font information for U+bbm on input line 832 +. + (/usr/local/texlive/2021/texmf-dist/tex/latex/bbm-macros/ubbm.fd +File: ubbm.fd 1999/03/15 V 1.2 Font definition for bbm font - TH +) [25] +[26] [27] [28] [29] + +LaTeX Warning: Reference `bigsum' on page 30 undefined on input line 959. + +[30] [31] [32] [33] +Overfull \hbox (0.39136pt too wide) in paragraph at lines 1215--1218 +[]\OT1/cmr/m/n/10.95 However note also that since $\OML/cmm/m/it/10.95 G[]$ \OT +1/cmr/m/n/10.95 is up-per semi-continuous, fur-ther the fact that, $\OML/cmm/m/ +it/10.95 ^^^[] \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/n/10.95 , + [] + + +Overfull \hbox (81.7409pt too wide) in paragraph at lines 1215--1218 +\OT1/cmr/m/n/10.95 and then $([]3\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 2\OML/ +cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 27[])$, and $([]3\OML/cmm/m/it/10.95 :\OT1/c +mr/m/n/10.95 2\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 16[])$, im-ply for all $\ +OML/cmm/m/it/10.95 " \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/1 +0.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ we have that: $[][] [] = + [] + +[34] [35] [36] [37] [38] [39] [40] [41] [42] +Overfull \hbox (86.18391pt too wide) in paragraph at lines 1499--1500 +\OT1/cmr/m/it/10.95 ery $\OML/cmm/m/it/10.95 r \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m +/n/10.95 (0\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ \OT +1/cmr/m/it/10.95 sat-isfy the con-di-tion that $\OML/cmm/m/it/10.95 O[] \OMS/cm +sy/m/n/10.95 ^^R O$\OT1/cmr/m/it/10.95 , where $\OML/cmm/m/it/10.95 O[] \OT1/cm +r/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2 +O \OT1/cmr/m/n/10.95 : []\OMS/cmsy/m/n/10.95 g$ + [] + +[43] [44] [45] [46] [47] +Overfull \hbox (2.98135pt too wide) in paragraph at lines 1650--1651 +[]\OT1/cmr/bx/n/10.95 Corollary 3.3.1.1. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/ +it/10.95 T \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OM +S/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$\OT1/cmr/m/it/10.95 , let $[]$ be a prob +-a-bil-ity space, let $\OML/cmm/m/it/10.95 u[] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m +/it/10.95 C[] []$\OT1/cmr/m/it/10.95 , + [] + +[48] [49] +! Argument of \align has an extra }. + + \par +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I've run across a `}' that doesn't seem to match anything. +For example, `\def\a#1{...}' and `\a}' would produce +this error. If you simply proceed now, the `\par' that +I've just inserted will cause me to report a runaway +argument that might be the root of the problem. But if +your `}' was spurious, just type `2' and it will go away. + +Runaway argument? + \| \mu (t,x) - \mu (t,y) \|_E+\|\sigma (t,x) - \sigma (t,y) \|_F &= \ETC. +! Paragraph ended before \align was complete. + + \par +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I suspect you've forgotten a `}', causing me to apply this +control sequence to too much text. How can we recover? +My plan is to forget the whole thing and hope for the best. + +! Missing $ inserted. + + $ +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing \endgroup inserted. + + \endgroup +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \@@par +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +! Extra }, or forgotten \endgroup. +\par ...m \@noitemerr {\@@par }\fi \else {\@@par } + \fi +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Extra }, or forgotten \endgroup. + } + +l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0} + \nonumber\\ +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + + +! LaTeX Error: There's no line here to end. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1755 & + = \|t\nabla_x\alpha(x) - t\nabla_x \alpha(y) \|_E} \nonumber\\ +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +! Misplaced alignment tab character &. + & + +l.1755 & + = \|t\nabla_x\alpha(x) - t\nabla_x \alpha(y) \|_E} \nonumber\\ +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Missing $ inserted. + + $ +l.1755 &= \| + t\nabla_x\alpha(x) - t\nabla_x \alpha(y) \|_E} \nonumber\\ +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +l.1755 ...a_x\alpha(x) - t\nabla_x \alpha(y) \|_E} + \nonumber\\ +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced alignment tab character &. + & + +l.1756 & + \leq \frac{T\|\nabla_x\alpha(x)-\nabla_x\alpha(y)\|_E}{\lp \|x\|_E ... +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. + & + +l.1757 & + \leq 2T\mathfrak{B} \leq +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \cr. +\math@cr@@@ ->\cr + +l.1758 \end{align} + +I can't figure out why you would want to use a tab mark +or \cr or \span just now. If something like a right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced \noalign. +\math@cr@@ ... \iffalse }\fi \math@cr@@@ \noalign + {\vskip #1\relax } +l.1758 \end{align} + +I expect to see \noalign only after the \cr of +an alignment. Proceed, and I'll ignore this case. + +! Missing $ inserted. + + $ +l.1758 \end{align} + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Missing } inserted. + + } +l.1758 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Extra }, or forgotten \endgroup. +\math@cr@@ ...th@cr@@@ \noalign {\vskip #1\relax } + +l.1758 \end{align} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced \noalign. +\black@ #1->\noalign + {\ifdim #1>\displaywidth \dimen@ \prevdepth \nointerlin... +l.1758 \end{align} + +I expect to see \noalign only after the \cr of +an alignment. Proceed, and I'll ignore this case. + + +Overfull \hbox (259.72743pt too wide) detected at line 1758 +[] + [] + +! Extra }, or forgotten \endgroup. +\endalign ->\math@cr \black@ \totwidth@ \egroup + \ifingather@ \restorealignst... +l.1758 \end{align} + +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. + + $ +l.1758 \end{align} + +I've inserted something that you may have forgotten. +(See the above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Display math should end with $$. + + \endgroup +l.1758 \end{align} + +The `$' that I just saw supposedly matches a previous `$$'. +So I shall assume that you typed `$$' both times. + +[50] [51] [52] (./Dissertation.bbl [53 + +]) [54] [55 + + +] [56] (./Dissertation.aux) + +LaTeX Warning: There were undefined references. + + +LaTeX Warning: There were multiply-defined labels. + +Package rerunfilecheck Info: File `Dissertation.out' has not changed. +(rerunfilecheck) Checksum: EA82F68FD5620E41892B53C13EE56636;2174. + ) +(\end occurred inside a group at level 1) + +### simple group (level 1) entered at line 91 ({) +### bottom level +Here is how much of TeX's memory you used: + 12969 strings out of 478994 + 199090 string characters out of 5858184 + 532732 words of memory out of 5000000 + 29889 multiletter control sequences out of 15000+600000 + 421865 words of font info for 102 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 71i,18n,74p,917b,628s stack positions 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They have successfully been used in a wide range of applications, from protein-folding in \cite{tsaban_harnessing_2022}, knot theory in \cite{davies_signature_2021}, and extracting data from gravitational waves in \cite{zhao_space-based_2023}. +\\~\\ +As neural networks become more ubiquitous, we see that the number of parameters required to train them increases, which poses two problems: accessibility on low-power devices and the amount of energy needed to train these models, see for instance \cite{wu2022sustainable} and \cite{strubell2019energy}. Parameter estimates become increasingly crucial in an increasingly climate-challenged world. That we know strict and precise upper bounds on parameter estimates tells us when training becomes wasteful, in some sense, and when, perhaps, different approaches may be needed. +\\~\\ +Our goal in this dissertation is threefold: +\begin{enumerate}[label = (\roman*)] + \item Firstly, we will take something called Multi-Level Picard first introduced in \cite{e_multilevel_2019} and \cite{e_multilevel_2021}, and in particular, the version of Multi-Level Picard that appears in \cite{hutzenthaler_strong_2021}. We show that dropping the drift term and substantially simplifying the process still results in convergence of the method and polynomial bounds for the number of computations required and rather nice properties for the approximations, such as integrability and measurability. + \item We will then go on to realize that the solution to a modified version of the heat equation has a solution represented as a stochastic differential equation by Feynman-Kac and further that a version of this can be realized by the modified multi-level Picard technique mentioned in Item (i), with certain simplifying assumptions since we dropped the drift term. A substantial amount of this is inspired by \cite{bhj20} and much earlier work in \cite{karatzas1991brownian} and \cite{da_prato_zabczyk_2002}. + \item By far, the most significant part of this dissertation is dedicated to expanding and building upon a framework of neural networks as appears in \cite{grohs2019spacetime}. We modify this definition highly and introduce several new neural network architectures to this framework ($\tay, \pwr, \trp, \tun,\etr$, among others) and show, for all these neural networks, that the parameter count grows only polynomially as the accuracy of our model increases, thus beating the curse of dimensionality. This finally paves the way for giving neural network approximations to the techniques realized in Item (ii). We show that it is not too wasteful (defined on the polynomiality of parameter counts) to use neural networks to approximate MLP to approximate a stochastic differential equation equivalent to certain parabolic PDEs as Feynman-Kac necessitates. + \\~\\ + We end this dissertation by proposing two avenues of further research: analytical and algebraic. This framework of understanding neural networks as ordered tuples of ordered pairs may be extended to give neural network approximation of classical PDE approximation techniques such as Runge-Kutta, Adams-Moulton, and Bashforth. We also propose three conjectures about neural networks, as defined in \cite{grohs2019spacetime}. They form a bimodule, and that realization is a functor. +\end{enumerate} +This dissertation is broken down into three parts. At the end of each part, we will encounter tent-pole theorems, which will eventually lead to the final neural network approximation outcome. These tentpole theorems are Theorem \ref{tentpole_1}, Theorem \ref{thm:3.21}, and Theorem. Finally, the culmination of these three theorems is Theorem, the end product of the dissertation. + +\section{Notation, Definitions \& Basic notions.} +We introduce here basic notations that we will be using throughout this dissertation. Large parts are taken from standard literature inspired by \textit{Matrix Computations} by \cite{golub2013matrix}, and \textit{Probability: Theory \& Examples} by Rick \cite{durrett2019probability}. +\subsection{Norms and Inner Products} +\begin{definition}[Euclidean Norm] + Let $\left\|\cdot\right\|_E: \R^d \rightarrow [0,\infty)$ denote the Euclidean norm defined for every $d \in \N_0$ and for all $x= \{x_1,x_2,\cdots, x_d\}\in \R^d$ as: + \begin{align} + \| x\|_E = \lp \sum_{i=1}^d x_i^2 \rp^{\frac{1}{2}} + \end{align} + For the particular case that $d=1$ and where it is clear from context, we will denote $\| \cdot \|_E$ as $|\cdot |$. +\end{definition} +\begin{definition}[Max Norm] + Let $\left\| \cdot \right\|_{\infty}: \R^d \rightarrow [0,\infty )$ denote the max norm defined for every $d \in \N_0$ and for all $x = \left\{ x_1,x_2,\cdots,x_d \right\} \in \R^d$ as: + \begin{align} + \left\| x \right\|_{\infty} = \max_{i \in \{1,2,\cdots,d\}} \left\{\left| x_i \right| \right\} + \end{align} + We will denote the max norm $\left\|\cdot \right\|_{\max}: \R^{m\times n} \rightarrow \lb 0, \infty \rp$ defined for every $m,n \in \N$ and for all $A \in \R^{m\times n}$ as: + \begin{align} + \| A \|_{\max} \coloneqq \max_{\substack {i \in \{1,2,...,m\} \\ j \in \{1,2,...,n\}}} \left| \lb A\rb_{i,j}\right| + \end{align} +\end{definition} + +\begin{definition}[Frobenius Norm] +Let $\|\cdot \|_F: \R^{m\times n} \rightarrow [0,\infty)$ denote the Frobenius norm defined for every $m,n \in \N$ and for all $A \in \R^{m\times n}$ as: +\begin{align} + \|A\|_F = \lp \sum^m_{i=1} \sum^n_{j=1} \lb A \rb^2_{i,j} \rp^{\frac{1}{2}} +\end{align} +\end{definition} + +\begin{definition}[Euclidean Inner Product] + Let $\la \cdot, \cdot \ra: \R^d \times \R^d \rightarrow \R$ denote the Euclidean inner product defined for every $d \in \N$, for all $\R^d \ni x = \{x_1,x_2,...,x_d\}$, and for all $\R^d \ni y = \{y_1,y_2,..., y_d\}$ as: + \begin{align} + \la x, y \ra = \sum^d_{i=1} \lp x_i y_i \rp + \end{align} +\end{definition} + +\subsection{Probability Space and Brownian Motion} +\begin{definition}[Probability Space] + A probability space is a triple $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ where: + \begin{enumerate}[label = (\roman*)] + \item $\Omega$ is a set of outcomes called the \textbf{sample space}. + \item $\mathcal{F}$ is a set of events called the \textbf{event space}, where each event is a set of outcomes from the sample space. More specifically, it is a $\sigma$-algebra on the set $\Omega$. + \item A measurable function $\mathbb{P}: \mathcal{F} \rightarrow [0,1]$ assigning each event in the \textbf{event space} a probability between $0$ and $1$. More specifically, $\mathbb{P}$ is a measure on $\Omega$ with the caveat that the measure of the entire space is $1$, i.e., $\mathbb{P}(\Omega) = 1$. + \end{enumerate} +\end{definition} + +\begin{definition}[Random Variable] + Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, and let $d \in \N_0$. For some $d\in \N_0$ a random variable is a measurable function $\mathcal{X}: \Omega \rightarrow \R^d.$ +\end{definition} + +\begin{definition}[Expectation] + Given a probability space $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$, the expected value of a random variable $X$, denoted $\E \lb X \rb$ is the Lebesgue integral given by: + \begin{align} + \E\lb X \rb=\int_\Omega X d\mathbb{P} + \end{align} +\end{definition} + +\begin{definition}[Stochastic Process] + A stochastic process is a family of random variables over a fixed probability space $(\Omega, \mathcal{F}, \mathbb{R})$, indexed over a set, usually $\lb 0, T\rb$ for $T\in \lp 0,\infty\rp$. +\end{definition} + + +\begin{definition}[Stochastic Basis] + A stochastic basis is a tuple $\lp \Omega, \mathcal{F}, \mathbb{P}, \mathbb{F} \rp$ where: + \begin{enumerate}[label = (\roman*)] + \item $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ is a probability space equipped with a filtration $\mathbb{F}$ where, + \item $\mathbb{F}=(\mathcal{F}_i)_{i \in I}$, is a collection of non-decreasing sets under inclusion where for every $i \in I$, $I$ being equipped in total order, it is the case that $\mathcal{F}_i$ is a sub $\sigma$-algebra of $\mathcal{F}$. + \end{enumerate} +\end{definition} + +\begin{definition}[Brownian Motion Over a Stochastic Basis]\label{def:brown_motion} + Given a stochastic basis $(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F})$ a standard $(\mathbb{F}_t)_{t\in [0,T]}$-Brownian motion $\mathcal{W}_t$ is a mapping $\mathcal{W}_t: [0,T] \times \Omega \rightarrow \R^d$ satisfying: + \begin{enumerate}[label = (\roman*)] + \item $\mathcal{W}_t$ is $\mathcal{F}_t$ measurable for all $t\in [0, \infty)$ + \item $\mathcal{W}_0 = 0$ with $\mathbb{P}$-a.s. + \item $\mathcal{W}_t-\mathcal{W}_s \sim \norm\lp 0,t-s\rp$ when $s\in \lp 0, t \rp $. + \item $\mathcal{W}_t-\mathcal{W}_s$ is independent of $\mathcal{W}_s$ whenever $s \delta$. Thus, by uniform continuity, we have the following: + \begin{align} + |f(x_i)-f(x_j)| < \epsilon \quad \forall i,j \in \{1,2,...,N\} + \end{align} + and thus let $\mathfrak{L}$ be defined as: + \begin{align} + \mathfrak{L} = \max_{\substack{i,j \in \{1,2,...,N\} \\ i \neq j}} \lv \frac{f(x_i)-f(x_j)}{x_i - x_j} \rv + \end{align} + $\mathfrak{L}$ satisfies the Lipschitz property. To see this, let $x_1,x_2$ be two arbitrary points within $\mathcal{K}$. Let $B_\delta(x_i)$ and $B_\delta(x_j)$ be two points such that $x_1 \in B_\delta(x_i)$ and $x_2 \in B_\delta(x_j)$. The triangle inequality then yields that: + \begin{align} +\left|f(x_1)-f(x_2)\right| &\leqslant \left|f(x_1)-f(x_i)\right| + \left|f(x_i)-f(x_j)\right| + \left|f(x_j)-f(x_2)\right| \nonumber\\ +&\leqslant \left|f(x_i)-f(x_j)\right| + 2\epsilon \nonumber\\ +&\leqslant \mathfrak{L}\cdot\left|x_i-x_j\right| + 2\epsilon \nonumber\\ +&\leqslant \mathfrak{L}\cdot\left|x_1-x_2\right| + 2\epsilon \nonumber +\end{align} +for all $\epsilon \in (0,\infty)$. +\end{proof} + + +\begin{definition}[Locally Lipschitz Function]\label{def:1.14} + Let $d \in N_0$. For every $d \in \N_0$ a function $f: \R^d \rightarrow \R^d$ is locally Lipschitz if for all $x_0 \in \R^d$ there exists a compact set $\mathcal{K} \subseteq \domain(f)$ containing $x_0$, and a constant $L \in (0,\infty)$ for that compact set such that + \begin{align} + \sup_{\substack{x,y\in \mathcal{K} \\ x\neq y}} \left\| \frac{f(x)-f(y)}{x-y} \right\|_E \leqslant L + \end{align} + The set of locally Lipschitz functions over set $X$ will be denoted $\lip_L(X)$. +\end{definition} + +\begin{corollary} + A function $f: \R^d \rightarrow \R^d$ that is globally Lipschitz is also locally Lipschitz. More concisely $\lip_G(X) \subsetneq \lip_L(X)$. +\end{corollary} +\begin{proof} + Assume not, that is to say, there exists a point $x\in \domain(f)$, a compact set $\mathcal{K} \subseteq \domain(f)$, and points $x_1,x_2 \in \mathcal{K}$ such that: + \begin{align} + \frac{|f(x_1)-f(x_2)|}{x_1-x_2} \geqslant \mathfrak{L} + \end{align} + This directly contradicts Definition \ref{def:1.13}. +\end{proof} +\subsection{Kolmogorov Equations} +\begin{definition}[Kolmogorov Equation] + We take our definition from \cite[~(7.0.1)]{da_prato_zabczyk_2002} with, $u \curvearrowleft u$, $G \curvearrowleft \sigma$, $F \curvearrowleft \mu$, and $\varphi \curvearrowleft g$, and for our purposes we set $A:\R^d \rightarrow 0$. Given a separable Hilbert space H (in our case $\R^d$), and letting $\mu: [0, T] \times \R^d \rightarrow \R^d$, $\sigma:[0, T] \times \R^d \rightarrow \R^{d\times m}$, and $g:\R^d \rightarrow \R$ be at least Lipschitz, a Kolmogorov Equation is an equation of the form: + \begin{align}\label{(1.7)} + \begin{cases} + \lp \frac{\partial}{\partial t} u \rp \lp t,x \rp = \frac{1}{2} \Trace \lp \sigma \lp t,x \rp \lb \sigma \lp t,x \rp \rb^* \lp \Hess_x u \rp \lp t,x \rp \rp + \la \mu \lp t,x \rp , \lp \nabla_x u \rp \lp t,x \rp \ra \\ + u(0,x) = g(x) + \end{cases} + \end{align} + \end{definition} +\begin{definition}[Strict Solution to Kolmogorov Equation] + Let $d\in \N_0$. For every $d\in \N_0$ a function $u: [0,T] \times \R^d \rightarrow \R$ is a strict solution to (\ref{(1.7)}) if and only if: + \begin{enumerate}[label = (\roman*)] + \item $u \in C^{1,1} \lp \lb 0,T \rb \times \R^d \rp$ and $u(0, \cdot) = g$ + \item $u(t, \cdot) \in UC^{1,2}([0,T] \times \R^d, \R)$ + \item For all $x \in \domain(A)$, $u(\cdot,x)$ is continuously differentiable on $[0,\infty)$ and satisfies (\ref{(1.7)}). + \end{enumerate} +\end{definition} +\begin{definition}[Generalized Solution to Kolmogorov Equation] + A generalized solution to (\ref{(1.7)}) is defined as: + \begin{align} + u(t,x) = \E \lb g \lp \mathcal{X}^{t,x} \rp \rb + \end{align} + Where the stochastic process $\mathcal{X}^{t,x}$ is the solution to the stochastic differential equation, for $x \in \R^d$, $t \in [0,T]$: + \begin{align} + \mathcal{X}^{t,x} = \int^t_0 \mu \lp \mathcal{X}^{t,x}_r \rp dr + \int^t_0 \sigma \lp \mathcal{X}^{t,x}_r \rp dW_r + \end{align} +\end{definition} +\begin{definition}[Laplace Operator w.r.t. $x$] + Let $d \in \N_0$, and $f\in C^2\lp \R^d,\R \rp$. For every $d\in \N_0$, the Laplace operator $\nabla^2_x : C^2(\R^d,\R) \rightarrow \R$ is defined as: + \begin{align} + \Delta_xf = \nabla_x^2f := \nabla \cdot \nabla f = \sum^d_{i=1} \frac{\partial f}{\partial x_i} + \end{align} +\end{definition} +\subsection{Linear Algebra Notation and Definitions} +\begin{definition}[Identity, Zero Matrix, and the 1-matrix] + Let $d \in \N$. We will define the identity matrix for every $d \in \N$ as the matrix $\mathbb{I}_d \in \R^{d\times d}$ given by: + \begin{align} + \mathbb{I}_d = \lb \mathbb{I}_d \rb_{i,j} = \begin{cases} + 1 & i=j \\ + 0 & \text{else} + \end{cases} + \end{align} + Note that $\mathbb{I}_1 =1$. + + Let $m,n,i,j \in \N$. For every $m,n \in \N$, $i \in \left\{1,2,\hdots,m \right\}$, and $j \in \left\{ 1,2,\hdots,n\right\}$ we define the zero matrix $\mymathbb{0}_{m,n} \in \R^{m\times n}$ as: + \begin{align} + \mymathbb{0}_{m,n} =\lb \mymathbb{0}_{m,n} \rb_{i,j} = 0 + \end{align} + Where we only have a column of zeros, it is convenient to denote $\mymathbb{0}_d$ where $d$ is the height of the column. + + Let $m,n,i,j \in \N$. For every $m,n \in \N$, $i \in \left\{ 1,2,\hdots,m\right\}$, and $j \in \left\{1,2,\hdots,n \right\}$ we define matrix of ones $\mymathbb{e}_{m,n} \in \R^{m \times n}$ as: + \begin{align} + \mymathbb{e}_{m,n} = \lb \mymathbb{e} \rb_{i,j} = 1 \quad + \end{align} + Where we only have a column of ones, it is convenient to denote $\mymathbb{e}_d$ where $d$ is the height of the column. +\end{definition} +\begin{definition}[Single-entry matrix] + Let $m,n,k,l \in \N$ and let $c\in \R$. For $k \in \N \cap \lb 1,m\rb$ and $l \in \N \cap \lb 1,n\rb$, we will denote by $\mymathbb{k}^{m,n}_{k,l,c} \in \R^{m \times n}$ as the matrix defined by: + \begin{align} + \mymathbb{k}^{m,n}_{k,l,c} =\lb \mymathbb{k}^{m,n}_{k,l}\rb_{i,j} = \begin{cases} + c &:k=i \land l=j \\ + 0 &:else + \end{cases} + \end{align} +\end{definition} +\begin{definition}[Complex conjugate and transpose] + Let $m,n,i,j \in \N$, and $A \in \mathbb{C}^{m \times n}$. For every $m,n \in \N$, $i \in \left\{1,2,\hdots,m\right\}$ and $j \in \left\{1,2,\hdots, n\right\}$, we denote by $A^* \in \mathbb{C}^{n \times m}$ the matrix: + \begin{align} + A^*\coloneqq \lb A^* \rb _{i,j} = \overline{\lb A \rb _{j,i}} + \end{align} + Where it is clear that we are dealing with real matrices, i.e., $A \in \R^{m\times n}$, we will denote this as $A^\intercal$. +\end{definition} +\begin{definition}[Column and Row Notation]\label{def:1.1.23} + Let $m,n,i,j \in \N$ and let $A \in \R^{m \times n}$. For every $m,n \in N$ and $i \in \left\{ 1,2,\hdots ,m\right\}$ we denote $i$-th row as: + \begin{align} + [A]_{i,*} = \begin{bmatrix} + a_{i,1} & a_{i,2} & \cdots & a_{i,n} + \end{bmatrix} + \end{align} + Similarly for every $m,n \in \N$ and $j \in \left\{ 1,2,\hdots,n\right\}$, we done the $j$-th row as: + \begin{align} + [A]_{*,j} = \begin{bmatrix} + a_{1,j} \\ + a_{2,j} \\ + \vdots \\ + a_{m,j} + \end{bmatrix} + \end{align} +\end{definition} +\begin{definition}[Component-wise notation] + Let $m,n,i,j \in \N$, and let $A \in \R^{m \times n}$. Let $f: \R \rightarrow \R$. For all $m,n \in \N, i \in \left\{1,2,\hdots,m \right\}$, and $j \in \left\{1,2,\hdots,n \right\}$ we will define $f \lp \lb A \rb_{*,*} \rp \in \R^{m \times n}$ as: + \begin{align} + f\lp \lb A\rb_{*,*}\rp \coloneqq \lb f \lp \lb A\rb_{i,j}\rp \rb_{i,j} + \end{align} + Thus under this notation the component-wise square of $A$ is $\lp \lb A \rb_{*,*}\rp^2$, the component-wise $\sin$ is $\sin\lp \lb A \rb_{*,*}\rp$ and the Hadamard product of $A,B \in \R^{m \times n}$ then becomes $ A \odot B = \ \lb A \rb_{*,*} \times \lb B \rb_{*,*}$. +\end{definition} +\begin{remark} + Where we are dealing with a row vector $x \in \R^{d \times 1}$ and it is evident from the context we may choose to write $f\lp \lb x\rb_* \rp$. +\end{remark} +\begin{definition}[The Diagonalization Operator] + Let $m_1,m_2,n_1,n_2 \in \N$. Given $A \in \R^{m_1 \times n_1}$ and $B \in \R^{m_2\times n_2}$, we will denote by $\diag\lp A,B\rp$ the matrix: + \begin{align} + \diag\lp A,B\rp = \begin{bmatrix} + A & \mymathbb{0}_{m_1,n_2}\\ + \mymathbb{0}_{m_2,n_1}& B + \end{bmatrix} + \end{align} +\end{definition} +\begin{remark} + $\diag\lp A_1,A_2,\hdots,A_n\rp$ is defined analogously for a finite set of matrices $A_1,A_2,\hdots,A_n$. +\end{remark} +\begin{definition}[Number of rows and columns notation] + Let $m,n \in \N$. Let $A\in \R^{m \times n}$. Let $\rows:\R^{m \times n} \rightarrow\N$ and $\columns:\R^{m\times n} \rightarrow \N$, be the functions respectively $\rows\lp A \rp = m$ and $\columns\lp A\rp = n$. +\end{definition} + +\subsection{$O$-type Notation and Function Growth} +\begin{definition}[$O$-type notation] +Let $g \in C(\R,\R)$. We say that $f \in C(\R,\R)$ is in $O(g(x))$, denoted $f \in O(g(x))$, if there exists $c\in \lp 0, \infty\rp$ and $x_0 \in \lp 0,\infty\rp$ such that for all $x\in \lb x_0,\infty \rp $ it is the case that: +\begin{align} + 0 \leqslant f(x) \leqslant c \cdot g(x) +\end{align} +We say that $f \in \Omega(g(x))$ if there exists $c\in \lp 0,\infty\rp$ and $x_0 \in \lp 0,\infty \rp$ such that for all $x\in \lb x_0, \infty\rp$ it is the case that: +\begin{align} + 0 \leqslant cg(x) \leqslant f(x) +\end{align} +We say that $f \in \Theta(g(x))$ if there exists $c_1,c_2,x_0 \in \lp 0,\infty\rp$ such that for all $x \in \lb x_0,\infty\rp$ it is the case that: +\begin{align} + 0 \leqslant c_1g(x) \leqslant f \leqslant c_2g(x) +\end{align} +\end{definition} +\begin{corollary}[Bounded functions and $O$-type notation]\label{1.1.20.1} + Let $f(x) \in C(\R,\R)$, then: + \begin{enumerate}[label = (\roman*)] + \item if $f$ is bounded above for all $x\in \R$, it is in $O(1)$ for some constant $c\in \R$. + \item if $f$ is bounded below for all $x \in \R$, it is in $\Omega(1)$ for some constant $c \in \R$. + \item if $f$ is bounded above and below for all $x\in \R$, it is in $\Theta(1)$ for some constant $c\in \R$. + \end{enumerate} +\end{corollary} +\begin{proof} + Assume $f \in C(\R, \R)$, then: + \begin{enumerate}[label = (\roman*)] + \item Assume for all $x \in \R$ it is the case that $f(x) \leqslant M$ for some $M\in \R$, then there exists an $x_0\in \lp 0,\infty \rp$ such that for all $x\in \lp x_0,\infty \rp $ it is also the case that $0 \leqslant f(x) \leqslant M$, whence $f(x) \in O(1)$. + \item Assume for all $x \in \R$ it is the case that $f(x) \geqslant M $ for some $M\in \R$, then there exists an $x_0\in \lp 0,\infty \rp$ such that for all $x\in \lb x_0, \infty \rp$ it is also the case that $f(x) \geqslant M \geqslant 0$, whence $f(x) \in \Omega(1)$. + \item This is a consequence of items (i) and (ii). + \end{enumerate} +\end{proof} +\begin{corollary}\label{1.1.20.2} + Let $n\in \N_0$. For some $n\in \N_0$, let $f \in O(x^n)$. It is then also the case that $f \in O \lp x^{n+1} \rp$. +\end{corollary} +\begin{proof} + Let $f \in O(x^n)$. Then there exists $c_0,x_0 \in \lp 0,\infty\rp$, such that for all $x \in \lb x_0,\infty\rp$ it is the case that: + \begin{align} + f(x) \leqslant c_0\cdot x^n + \end{align} + Note however that for all $n\in \N_0$, there also exists $c_1,x_1 \in \lp 0,\infty\rp$ such that for all $x \in \lp x_1,\infty \rp$ it is the case that: + \begin{align} + x^n \leqslant c_1\cdot x^{n+1} + \end{align} + Thus taken together this implies that for all $x \in \lp \max \left\{ x_0,x_1\right\},\infty\rp$ it is the case that: + \begin{align} + f(x) \leqslant c_0 \cdot x^n \leqslant c_0\cdot c_1 \cdot x^{n+1} + \end{align} +\end{proof} + +\begin{definition}[The floor and ceiling functions] + We denote by $\lfloor\cdot \rfloor: \R \rightarrow \Z$ and $\lceil \cdot \rceil: \R \rightarrow\Z$ the functions satisfying for all $x \in \R$ that $\lfloor x \rfloor = \max \lp \Z \cap \lp -\infty,x \rb \rp $ and $\lceil x \rceil = \min \lp \Z \cap \lp -\infty,x \rb \rp$. +\end{definition} + +\subsection{The Concatenation of Vectors \& Functions} +\begin{definition}[Vertical Vector Concatenation] + Let $m,n \in \N$. Let $x= \lb x_1 \: x_2\: \hdots \: x_m \rb^\intercal \in \R^m$ and $y = \lb y_1,y_2,\hdots,y_n\rb^\intercal \in \R^n$. For every $m,n \in \N$, we will denote by $x \frown y \in \R^m \times \R^n$ the vector given as: + \begin{align} + \begin{bmatrix} + x_1 \\x_2\\ \vdots \\x_m \\y_1 \\y_2\\ \vdots \\y_n + \end{bmatrix} + \end{align} +\end{definition} +\begin{remark} + We will stipulate that when concatenating vectors as $x_1 \frown x_2$, $x_1$ is on top, and $x_2$ is at the bottom. +\end{remark} +\begin{corollary}\label{sum_of_frown_frown_of_sum} + Let $m_1,m_2,n_1,n_2 \in \N$ and let $x \in \R^{m_1}$, $y \in \R^{n_1}$, $\fx\in \R^{m_2}$, and $\fy \in \R^{n_2}$. It is then the case that $\lb x \frown \fx\rb+\lb y \frown \fy\rb = \lb x+y\rb\frown \lb \fx +\fy\rb$. +\end{corollary} +\begin{proof} + This follows straightforwardly from the fact that: + \begin{align} + \lb x \frown \fx \rb + \lb y + \fy\rb = \begin{bmatrix} + x_1 \\ x_2 \\ \vdots \\ x_{m_1} \\ \fx_1 \\ \fx_2 \\ \vdots \\ \fx_{m_2} + \end{bmatrix} + \begin{bmatrix} + y_1 \\ y_2 \\ \vdots \\ y_{n_1} \\ \fy_1\\ \fy_2 \\ \vdots \\ \fy_{n_2} + \end{bmatrix} = \begin{bmatrix} + x_1+y_1 \\ x_2 + y_2 \\ \vdots \\ x_{m_1+n+1} \\ \fx_1+\fy_1 \\ \fx_2 + \fy_2 \\ \vdots \\ \fx_{m_2} + \fy_{n_2} + \end{bmatrix} = \lb x+y\rb\frown \lb \fx +\fy\rb + \end{align} +\end{proof} +\begin{definition}[Function Concatenation] + Let $m_1,n_1,m_2,n_2 \in \N$. Let $f : \R^{m_1} \rightarrow\R^{n_1}$ and $g: \R^{m_2}\rightarrow\R^{n_2}$. We will denote by $f \frown g: \R^{m_1} \times \R^{m_2} \rightarrow \R^{n_1} \times \R^{n_2}$ as the function given for all $x = \{ x_1,x_2,\hdots, x_{m_1}\} \in \R^{m_1}$, $\overline{x} \in \{ \overline{x_1},\overline{x_2},\hdots ,\overline{x}_{m_2}\} \in \R^{m_2}$, and $x \frown \overline{x} =\{x_1,x_2,\hdots,x_{m_1},\overline{x}_1,\overline{x}_2,\hdots,\overline{x}_{m_2}\} \in \R^{m_1} \times \R^{m_2}$ by: + \begin{align} + \begin{bmatrix} + x_1 \\ x_2\\ \vdots \\x_{m_1} \\\overline{x_1} \\\overline{x_2}\\ \vdots \\ \overline{x_{m_2}} + \end{bmatrix} \xrightarrow{\hspace*{1.5cm}} + \begin{bmatrix} + f(x) \\ g(\overline{x}) + \end{bmatrix} + \end{align} +\end{definition} +\begin{corollary}\label{concat_fun_fun_concat} + Let $m,n \in \N$. Let $x_1 \in \R^m$,$x_2 \in \R^n$, and $f\in C\lp \R, \R\rp$. It is then the case that $f\lp x_1 \frown x_2\rp = f \lp x_1\rp \frown f\lp x_2\rp$. +\end{corollary} +\begin{proof} + This follows straightforwardly from the definition of function concatenation. +\end{proof} +\begin{lemma}\label{par_cont} + Let $m_1,m_2,n_1, n_2 \in \N$. Let $f \in C\lp \R^{m_1}, \R^{n_1}\rp$ and $g \in C\lp \R^{m_2}, \R^{n_2}\rp$. It is then also the case that $f \frown g \in C \lp \R^{m_1} \times \R^{n_1}, \R^{m_2} \times \R^{n_2}\rp$. +\end{lemma} + +\begin{proof} + Let $\R^{m_2} \times \R^{n_2}$ be equipped with the usual product topology, i.e., the topology generated by all products $X \times Y$ of open subsets $X \in \R^{m_2}$ and $Y\in \R^{n_2}$. In such a case let $V \subsetneq \R^{m_2} \times \R^{n_2}$ be an open subset. Then let it be that $V_f$ and $V_g$ are the canonical projections to the first and second factors respectively. Since projection under the usual topology is continuous, it is the case that $V_f \subsetneq \R^{m_2}$ and $V_g \subsetneq \R^{n_2}$ are open sets, respectively. As such it is then also the case that $f^{-1}\lp V_f\rp \subsetneq \R^{m_1}$ and $g^{-1}\lp V_g\rp \subsetneq \R^{n_1}$ are open sets as well by continuity of $f$ and $g$. 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One-Dimensional Identity and some associated properties} + +\begin{definition}[One Dimensional Identity Neural Network]\label{7.2.1} + We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that: + \begin{enumerate}[label = (\roman*)] + \item \begin{align} + \id_1 = \lp \lp \begin{bmatrix} + 1 \\ + -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ + 0 + \end{bmatrix}\rp \lp \begin{bmatrix} + 1 \quad -1 + \end{bmatrix},\begin{bmatrix} 0\end{bmatrix}\rp \rp \in \lp \lp \R^{2 \times 1} \times \R^2 \rp \times \lp \R^{1\times 2} \times \R^1 \rp \rp + \end{align} + \item \begin{align}\label{7.2.2} + \id_d = \boxminus^d_{i=1} \id_1 + \end{align} + For $d>1$. + \end{enumerate} +\end{definition} +\begin{lemma}\label{idprop} + Let $d \in \N$, it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\lay(\id_d) = \lp d, 2d, d \rp \in \N^3$. + \item $\real_{\rect} \lp \id_d \rp \in C \lp \R^d, \R^d \rp$. + \item For all $x \in \R^d$ that: + \begin{align} + \lp \real_{\rect} \lp \id_d \rp \rp \lp x \rp = x \nonumber + \end{align} + \item For $d\in \N$ it is the case that $\dep\lp \id_d\rp = 2$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that (\ref{7.2.1}) ensure that $\lay(\id_d) = \lp 1,2,1 \rp$. Furthermore, ($\ref{7.2.2}$) and Remark \ref{5.3.5} prove that $\lay(\id_d) = \lp d,2d,d \rp$ which in turn proves Item (i). Note now that Remark \ref{5.3.5} tells us that: + \begin{align} + \id_d = \boxminus^d_{i=1}\lp \id_1 \rp \in \lp \bigtimes^L_{i=1}\lb \R^{dl_i \times dl_{i-1}} \times \R^{dl_i} \rb \rp = \lp \lp \R^{2d \times d} \times \R^{2d}\rp \times \lp \R^{d \times 2d} \times \R^d\rp \rp + \end{align} + Note that \ref{7.2.1} ensures that for all $x \in \R$ it is the case that: + \begin{align} + \lp \real_{\rect} \lp \id_1 \rp \rp \lp x \rp = \rect(x) - \rect(-x) = \max\{x,0\} - \max\{-x,0\} = x + \end{align} + + And Lemma \ref{5.3.4} shows us that for all $x = \lp x_1,x_2,...,x_d\rp \in \R^d$ it is the case that $\real_{\rect}\lp \id_d \rp \in C \lp \R^d, \R^d \rp $ and that: + \begin{align} + \lp \real_{\act} \lp \id_d \rp \rp \lp x \rp &= \lp \real_{\act} \lp \boxminus_{i=1}^d \lp \id_1\rp \rp \rp \lp x_1,x_2,...,x_d \rp \nonumber \\ + &= \lp \lp \real_{\act} \lp \id_1 \rp \rp \lp x_1 \rp, \lp \real_{\act} \lp \id_1 \rp \rp \lp x_1 \rp,..., \lp \real_{\act} \lp \id_1 \rp \rp \lp x_d \rp \rp \nonumber \\ + &= \lp x_1, x_2,...,x_d \rp = x + \end{align} + This proves Item (ii)\textemdash(iii). Item (iv) follows straightforwardly from Item (i). This establishes the lemma. +\end{proof} +\begin{remark} +Note here the difference between Definition \ref{actnn} and Definition \ref{7.2.1}. +\end{remark} +\begin{lemma}[R\textemdash, 2023]\label{id_param} + Let $d \in \N$. It then the case that for all $d \in \N$ we have that $\param\lp \id_d\rp = 4d^2+3d$ +\end{lemma} +\begin{proof} + By observation we have that $\param \lp \id_1\rp = 4(1)^2+3(1) = 7$. By induction, suppose that this holds for all natural numbers up to and including $n$, i.e., for all naturals up to and including $n$; it is the case that $\param \lp id_n\rp = 4n^2+3n$. Note then that $\id_{n+1} = \id_n \boxminus \id_1$. For $W_1$ and $W_2$ of this new network, this adds a combined extra $8n+4$ parameters. For $b_1$ and $b_2$ of this new network, this adds a combined extra $3$ parameters. Thus, we have the following: + \begin{align} + 4n^2+3n + 8n+4 + 3 &= 4(n+1)^2+3(n+1) + \end{align} + This completes the induction and hence proves the Lemma. +\end{proof} +\begin{lemma}\label{7.2.3} + Let $\nu \in \neu$ with end-widths $d$. It is then the case that $ \real_{\rect} \lp \id_d \bullet \nu \rp \lp x \rp = \real_{\rect} \lp \nu \bullet \id_d \rp = \real_{\rect} \lp \nu\rp $, i.e. $\id_d$ acts as a compositional identity. +\end{lemma} +\begin{proof} From (\ref{5.2.1}) and Definition \ref{7.2.1} we have eight cases. + +Case 1 where $d=1$ and subcases: +\begin{enumerate}[label = (1.\roman*)] + \item $\id_d \bullet \nu$ where $\dep(\nu) = 1$ + \item $\id_d \bullet \nu$ where $\dep(\nu) > 1$ + \item $\nu \bullet \id_d$ where $\dep(\nu) =1$ + \item $\nu \bullet \id_d$ where $\dep(\nu) > 1$ +\end{enumerate} + +Case 2 where $d>1$ and subcases: +\begin{enumerate}[label = (2.\roman*)] + \item $\id_d \bullet \nu$ where $\dep(\nu) = 1$ + \item $\id_d \bullet \nu$ where $\dep(\nu) > 1$ + \item $\nu \bullet \id_d$ where $\dep(\nu) =1$ + \item $\nu \bullet \id_d$ where $\dep(\nu) > 1$ +\end{enumerate} + +\textit{Case 1.i:} Let $\nu = \lp \lp W_1,b_1 \rp \rp$. Deriving from Definitions \ref{7.2.1} and \ref{5.2.1} we have that: +\begin{align} + \id_1 \bullet \nu &=\lp \lp \begin{bmatrix} + 1 \\ + -1 + \end{bmatrix} W_1, \begin{bmatrix} + 1 \\ + -1 + \end{bmatrix}b_1 + \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 \quad -1, + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix} \rp \rp \\ + &= \lp \lp \begin{bmatrix} + W_1 \\-W_{1} + \end{bmatrix}, \begin{bmatrix} + b_1 \\ -b_1 + \end{bmatrix} \rp,\lp \begin{bmatrix} + 1 \quad -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix} \rp \rp +\end{align} +Let $x \in \R$. Upon instantiation with $\rect$ and $d=1$ we have: +\begin{align} + \lp \real_{\rect}\lp \id_1\bullet \nu \rp \rp \lp x \rp &= \rect(W_1x+b_1)-\rect(-W_1x - b_1) \nonumber\\ + &= \max\{W_1x+b_1,0\}-\max\{-W_1x-b_1,0\} \nonumber \\ + &= W_1x+b_1 \nonumber\\ + &= \real_{\rect}(\nu) \nonumber +\end{align} +\textit{Case 1.ii:} Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp, ..., \lp W_L, b_L \rp \rp $. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have that: +\begin{align} + \id_1\bullet \nu &= \lp \lp W_1,b_1\rp,\lp W_2,b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix} + 1 \\-1 + \end{bmatrix} W_L, \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}b_L + \begin{bmatrix} + 0 \\ 0 + \end{bmatrix} \rp, \lp \begin{bmatrix} + 1 \quad -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix} \rp \rp \nonumber \\ + &= \lp \lp W_1,b_1\rp, \lp W_2, b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix} + W_L \\ -W_L + \end{bmatrix} ,\begin{bmatrix} + b_L \\ -b_L + \end{bmatrix} \rp ,\lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix} \rp \rp \nonumber + \end{align} + Let $x \in \R$. Note that upon instantiation with $\rect$, the last two layers are: + \begin{align} + &\rect(W_Lx+b_L)-\rect(-W_Lx - b_L,0) \nonumber\\ + &=\max\{W_Lx+b_L,0\}-\max\{-W_Lx-b_L,0\} \nonumber \\ + &= W_Lx+b_L \label{7.2.8} + \end{align} +This, along with Case 1. i, implies that the uninstantiated last layer is equivalent to $(W_L,b_L)$ whence $\id_1\bullet \nu = \nu$. + +\textit{Case 1.iii:} Let $\nu = \lp \lp W_1,b_1\rp \rp$. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have: +\begin{align} + \nu \bullet \id_1 &= \lp \lp \begin{bmatrix} + 1 \\-1 + \end{bmatrix}, \begin{bmatrix} + 0 \\0 + \end{bmatrix}\rp, \lp W_1\begin{bmatrix} + 1 \quad -1 + \end{bmatrix},W_1 \begin{bmatrix} + 0 + \end{bmatrix} + b_1\rp \rp \nonumber \\ + &= \lp \lp \begin{bmatrix} + 1 \\-1 + \end{bmatrix}, \begin{bmatrix} + 0 \\0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + W_1 \quad -W_1 + \end{bmatrix}, b_1\rp \rp \nonumber +\end{align} +Let $x \in \R$. Upon instantiation with $\rect$ we have that: +\begin{align} + \lp \real_{\rect} \lp \nu \bullet \id_1 \rp \rp \lp x \rp &= \begin{bmatrix} + W_1 \quad -W_1 + \end{bmatrix} \rect \lp \begin{bmatrix} + x \\ -x + \end{bmatrix} \rp +b_1 \nonumber \\ + &= W_1\rect(x)-W_1\rect(-x) + b_1 \nonumber \\ + &=W_1 \lp \rect(x) - \rect(-x) \rp +b_1 \nonumber \\ + &=W_x+b_1 = \real_{\rect} \lp \nu \rp +\end{align} +\textit{Case 1.iv:} Let $\nu = \lp \lp W_1,b_1\rp , \lp W_2,b_2 \rp,...,\lp W_L, b_L \rp \rp $. Deriving from Definitions \ref{7.2.1} and \ref{5.2.1} we have that: +\begin{align} + \nu \bullet \id_1 = \lp \lp \begin{bmatrix} + 1 \\-1 + \end{bmatrix}, \begin{bmatrix} + 0 \\0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + W_1 \quad -W_1 + \end{bmatrix}, b_1\rp, \lp W_2,b_2 \rp ,...,\lp W_L,b_L \rp \rp +\end{align} +Let $x \in \R$. Upon instantiation with $\rect$, we have that the first two layers are: +\begin{align} + &\begin{bmatrix} + W_1 \quad -W_1 + \end{bmatrix} \rect \lp \begin{bmatrix} + x \\ -x + \end{bmatrix} \rp +b_1 \nonumber \\ + &= W_1\rect(x)-W_1\rect(-x) + b_1 \nonumber \\ + &=W_1 \lp \rect(x) - \rect(-x) \rp + b_1 \nonumber \\ + &= W_1x+b_1 = \real_{\rect} \lp \nu \rp +\end{align} +This, along with Case 1. iii, implies that the uninstantiated first layer is equivalent $(W_1,b_1)$ whence we have that $\nu \bullet \id_1 = \nu$. + +Observe that Definitions \ref{5.2.5} and \ref{7.2.1} tells us that: + +\begin{align} + \boxminus^d_{i=1} \id_i = \lp \lp \overbrace{\begin{bmatrix} + \we_{\id_1,1} \\ + &&\ddots \\ + &&& \we_{\id_1,1} + \end{bmatrix}}^{d-many} , \mymathbb{0}_{2d}\rp, \lp \overbrace{\begin{bmatrix} + \we_{\id_1,2} \\ + && \ddots \\ + &&& \we_{\id_1,2} + \end{bmatrix}}^{d-many}, \mymathbb{0}_d\rp \rp \nonumber +\end{align} + +\textit{Case 2.i} Let $d \in \N \cap [1,\infty)$. Let $\nu \in \neu$ be $\nu = \lp W_1,b_1 \rp$ with end-widths $d$. Deriving from Definitions \ref{5.2.1} and \ref{7.2.1} we have: +\begin{align} + \id_d \bullet \nu = \lp \lp \begin{bmatrix} + \we_{\id_1,1} \\ + &&\ddots \\ + &&& \we_{\id_1,1} + \end{bmatrix}W_1 , \begin{bmatrix} + \we_{\id_1,1} \\ + &&\ddots \\ + &&& \we_{\id_1,1} + \end{bmatrix} b_1\rp, \right. \nonumber\\ \left. \lp \begin{bmatrix} + \we_{\id_1,2} \\ + && \ddots \\ + &&& \we_{\id_1,2} + \end{bmatrix}, \mymathbb{0}_d\rp \rp \nonumber \\ + = \lp \lp \begin{bmatrix} + [W_1]_{1,*} \\ + -[W_1]_{1,*} \\ + \vdots \\ + [W_1]_{d,*}\\ + -[W_1]_{d,*} + \end{bmatrix}, \begin{bmatrix} + [b_1]_1\\ + -[b_1]_1 \\ + \vdots \\ + [b_1]_d \\ + -[b_1]_d + \end{bmatrix} \rp, \lp \begin{bmatrix} + \we_{\id_1,2} \\ + && \ddots \\ + &&& \we_{\id_1,2} + \end{bmatrix}, \mymathbb{0}_d\rp \rp \nonumber +\end{align} +Let $x \in \R^d$. Upon instantiation with $\rect$ we have that: +\begin{align} + &\lp \real_{\rect} \lp \id_d \bullet \nu \rp \rp \lp x \rp \nonumber \\ &= \rect([W_1]_{1,*} \cdot x + [b_1]_1)-\rect(-[W_1]_{1,*}\cdot x -[b_1]_1)+\cdots \nonumber\\& +\rect([W_1]_{d,*}\cdot x+[b_1]_d)-\rect (-[W_1]_{d,*}\cdot x-[b_1]_d) \nonumber \\ + &= [W_1]_{1,*}\cdot x + [b_1]_1 + \cdots + [W_1]_{d,*}\cdot x + [b_1]_d \nonumber \\ + &= W_1x + b_1 = \real_{\rect} \lp \nu \rp \nonumber +\end{align} +\textit{Case 2.ii:} Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp, ..., \lp W_L, b_L \rp \rp $. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have that: +\begin{align} + \id_d \bullet \nu =\lp \lp W_1,b_1\rp, \lp W_2, b_2 \rp,...,\lp W_{L-1},b_{L-1} \rp, \lp \begin{bmatrix} + [W_L]_{1,*} \\ + -[W_L]_{1,*}\\ + \vdots \\ + [W_L]_{d,*} \\ + -[W_L]_{d,*} + \end{bmatrix} ,\begin{bmatrix} + [b_L]_1 \\ + -[b_L]_1 \\ + \vdots \\ + [b_L]_d \\ + -[b_L]_d + \end{bmatrix} \rp ,\lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix} \rp \rp \nonumber +\end{align} +Note that upon instantiation with $\rect$, the last two layers become: +\begin{align} + &\rect([W_L]_{1,*} \cdot x + [b_L]_1)-\rect(-[W_L]_{1,*}\cdot x -[b_L]_1)+\cdots \nonumber\\& +\rect([W_L]_{d,*}\cdot x+[b_L]_d)-\rect (-[W_L]_{d,*}\cdot x-[b_L]_d) \nonumber \\ + &=[W_L]_{1,*}\cdot x + [b_L]_1 + \cdots + [W_L]_{d,*}\cdot x + [b_L]_d \nonumber \\ + &= W_Lx + b_L +\end{align} +This, along with Case 2.i implies that the uninstantiated last layer is equivalent to $(W_L,b_L)$ whence $\id_d\bullet \nu = \nu$. + +\textit{Case 2.iii:} Let $\nu = \lp \lp W_1,b_1\rp \rp$. Deriving from Definition \ref{7.2.1} and \ref{5.2.1} we have: +\begin{align} + &\nu \bullet \id_d \nonumber\\ &= \lp \lp \begin{bmatrix} + \we_{\id_1,1} \\ + &&\ddots \\ + &&& \we_{\id_1,1} + \end{bmatrix}, \mymathbb{0}_{2d}\rp, \lp W_1\begin{bmatrix} + \we_{\id_1,2} \\ + &&\ddots \\ + &&& \we_{\id_1,2} + \end{bmatrix}, b_1\rp \rp \nonumber +\end{align} +Upon instantiation with $\rect$ we have that: +\begin{align} + &\lp \real_{\rect} \lp \nu \rp \rp \lp x \rp \\ &= \begin{bmatrix} + [W_1]_{*,1} \ -[W_1]_{*,1} \ \cdots \ [W_1]_{*,d} \ -[W_1]_{*,d} + \end{bmatrix}\rect \lp \begin{bmatrix} + [x]_1 \\ + -[x]_1 \\ + \vdots \\ + [x]_d \\ + -[x]_d + \end{bmatrix}\rp + b_1 \nonumber \\ + &= [W_1]_{*,1} \rect([x]_1) - [W_1]_{*,1} \rect(-[x]_1)+ \cdots +[W_1]_{*,d}\rect([x]_d)-[W_1]_{*,d}\rect(-[x]_d) + b_1 \nonumber \\ + &= [W_1]_{*,1}\cdot [x]_1 + \cdots + [W_1]_{*,d} \cdot [x]_d \nonumber \\ + &= W_1x+b_1 = \real_{\rect}(\nu) +\end{align} + +\textit{Case 2.iv:} Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp ,...,\lp W_L,b_L \rp \rp $. Deriving from Definitions \ref{7.2.1} and \ref{5.2.1} we have: +\begin{align} + &\nu \bullet \id_d \nonumber \\ = &\lp \lp \begin{bmatrix} + \we_{\id_1,1} \\ + &&\ddots \\ + &&& \we_{\id_1,1} + \end{bmatrix}, \mymathbb{0}_{2d}\rp, \lp \begin{bmatrix} + [W_1]_{*,1} \ -[W_1]_{*,1} \ \cdots \ [W_1]_{*,d} \ -[W_1]_{*,d} + \end{bmatrix}, b_1\rp,... \right. \nonumber \\ &\left.\lp W_2,b_2 \rp ,...,\lp W_L,b_L \rp \rp \nonumber +\end{align} +Upon instantiation with $\rect$, we have that the first two layers are: +\begin{align} + &\lp \real_{\rect} \lp \nu \rp \rp \lp x \rp \\ &= \begin{bmatrix} + [W_1]_{*,1} \ -[W_1]_{*,1} \ \cdots \ [W_1]_{*,d} \ -[W_1]_{*,d} + \end{bmatrix}\rect \lp \begin{bmatrix} + [x]_1 \\ + -[x]_1 \\ + \vdots \\ + [x]_d \\ + -[x]_d + \end{bmatrix}\rp + b_1 \nonumber \\ + &= [W_1]_{*,1} \rect([x]_1) - [W_1]_{*,1} \rect(-[x]_1)+ \cdots +[W_1]_{*,d}\rect([x]_d)-[W_1]_{*,d}\rect(-[x]_d) + b_1 \nonumber \\ + &= [W_1]_{*,1}\cdot [x]_1 + \cdots + [W_1]_{*,d} \cdot [x]_d \nonumber \\ + &= W_1x+b_1 +\end{align} +This, along with Case 2. iii, implies that the uninstantiated first layer is equivalent to $(W_L,b_L)$ whence $\id_d\bullet \nu = \nu$. + +This completes the proof. +\end{proof} +\begin{definition}[Monoid] + Given a set $X$ with binary operation $*$, we say that $X$ is a monoid under the operation $*$ if: + \begin{enumerate}[label = (\roman*)] + \item For all $x,y \in X$ it is the case that $x*y \in X$ + \item For all $x,y,z \in X$ it is the case that $(x *y)*z = x*(y*z)$ + \item The exists a unique element $e \in X$ such that $e*x=x*e = x$ + \end{enumerate} +\end{definition} +\begin{theorem} + Let $d\in \N$. For a fixed $d$, the set of all neural networks $\nu \in \neu$ with instantiations in $\rect$ and end-widths $d$ form a monoid under the operation of $\bullet$. +\end{theorem} +\begin{proof} + This is a consequence of Lemma \ref{7.2.3} and Lemma \ref{5.2.3}. +\end{proof} + +\begin{remark} + By analogy with matrices, we may find it helpful to refer to neural networks of end-widths $d$ as ``square neural networks of size $d$''. +\end{remark} +%\section{Modulus of Continuity} +%\begin{definition} +% Let $A\subseteq \R$ and let $f:A \rightarrow \R$. We denote the modulus of continuity $\omega_f: \lb 0,\infty \rb \rightarrow \lb 0,\infty \rb$ as the function given for all $h \in \lb 0,\infty \rb$ as: +% \begin{align}\label{9.3.1} +% \omega_f \lp h \rp = \sup \lp \left\{\left| f(x) - f(y)\right| \in \lb 0 ,\infty \rp : \lp x,y \in A, \left| x-y\right| \les h\rp \right\} \cup \left\{ 0\right\} \rp +% \end{align} +%\end{definition} +%\begin{lemma} +% Let $\alpha \in \lb -\infty, \infty \rb$, $b \in \lb a, \infty \rb$, and let $f: \lb a,b \rb \cap \R \rightarrow \R$ be a function. It is then the case that for all all $x,y \in \lb a,b\rb \cap \R$ that $\left| f(x) -f(y)\right| \les \omega_f \lp \left| x-y \right| \rp$. +%\end{lemma} +%\begin{proof} +% Note that (\ref{9.3.1}) implies the lemma. +%\end{proof} +%\begin{lemma}\label{lem:9.3.3} +% Let $A\subseteq \R$, $L \in \lb 0,\infty \rp$, and let $f:A \rightarrow \R$ satisfy for all $x,y \in A$ that $\left| f(x) - f(y)\right| \les L \left|x-y \right|$. It is then the case for all $h \in \lb 0,\infty \rp$ that $\omega_f(h) \les Lh$. +%\end{lemma} +%\begin{proof} +% Since it holds for all $x,y \in \R$ that $\left| f(x) - f(y)\right| \les L \left|x-y \right|$, it then, with (\ref{9.3.1}) imply for all $h \in \lb 0,\infty \rp$ that: +% \begin{align} +% \omega_f \lp h \rp &= \sup \lp \left\{\left| f(x) - f(y)\right| \in \lb 0 ,\infty \rp : \lp x,y \in A, \left| x-y\right| \les h\rp \right\} \cup \left\{ 0\right\} \rp \nonumber\\ +% &\les \sup \lp \left\{L\left|x -y\right| \in \lb 0 ,\infty \rp : \lp x,y \in A, \left| x-y\right| \les h\rp \right\} \cup \left\{ 0\right\} \rp \nonumber \\ +% &\les \sup \lp \left\{Lh,0 \right\} \rp = Lh +% \end{align} +% This completes the proof of the lemma. +%\end{proof} +%\section{Linear Interpolation of Real-Valued Functions} +%Note that we need a framework for approximating generic 1-dimensional continuous functions to approximate more complex functions. We introduce the linear interpolation operator and later see how neural networks can approximate 1-dimensional continuous functions to arbitrary precision. +% +%\subsection{The Linear Interpolation Operator} +%\begin{definition}[Linear Interpolation Operator]\label{lio} +% Let $n \in \N$, $x_0,x_1,...,x_n, y_0,y_1,...,y_n \in \R$. Let it also be the case that $x_0 \leqslant x_1 \leqslant \cdots \leqslant x_n$. We denote by $\lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n}: \R \rightarrow \R$, the function that satisfies for $i \in \{1,2,...,n\}$, and for all $w \in \lp -\infty, x_0 \rp$, $x \in [ x_{i-1},x_i )$, $z \in [ x_n, \infty)$ that: +% \begin{enumerate}[label = (\roman*)] +% \item $\lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n}\lp w \rp = y_0$ +% \item $\lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n}\lp x \rp = y_{i-1} + \frac{y_i-y_{i-1}}{x_i-x_{i-1}}\lp x- x_{i-1} \rp $ +% \item $\lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n}\lp z \rp = y_n$ +% \end{enumerate} +%\end{definition} +%\begin{lemma} +% Let $n\in \N$, $x_0,x_1,...,x_n,y_0,y_1,...,y_n \in \R$ with $x_0 \les x_1 \les \cdots \les x_n$, it is then the case that: +% \begin{enumerate}[label = (\roman*)] +% \item for all $i \in \{0,1,...,n\}$ that: +% \begin{align}\label{7.3.1} +% \lp \lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n} \rp \lp x_i \rp = y_i +% \end{align} +% \item for all $i\in \{0,1,...,n\}$ and $x \in [x_{i-1},x_{i}]$ that: +% \begin{align}\label{7.3.2} +% \lp \lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n} \rp \lp x \rp = \lp \frac{x_i-x}{x_i - x_{i-1}} \rp y_{i-1} + \lp \frac{x-x_{i-1}}{x_i-x_{i-1}} \rp y_i +% \end{align} +% \end{enumerate} +%\end{lemma} +%\begin{proof} +% Note that (\ref{7.3.1}) is a direct consequence of Definition \ref{lio}. Item (i) then implies for all $i \in \{1,2,...,n\}$ $x \in [x_{i-1},x_i]$ that: +% \begin{align} +% \lp \lin^{y_0,y_1,...,y_n}_{x_0,x_1,...,x_n} \rp \lp x \rp &= \lb \lp \frac{x_i-x_{i-1}}{x_i-x_{i-1}} \rp - \lp \frac{x-x_{i-1}}{x_i-x_{i-1}} \rp \rb y_{i-1} + \lp \frac{x-x_{i-1}}{x_i-x_{i-1}}\rp y_i \nonumber \\ +% &= \lp \frac{x_i-x}{x_i-x_{i-1}} \rp y_{i-1} + \lp \frac{x-x_{i-1}}{x_i-x_{i-1}} \rp y_i \nonumber +% \end{align} +%\end{proof} +%\begin{lemma}\label{lem:9.4.3} +% Let $N\in \N$, $L,x_0,x_1,...,x_N \in \R$ satisfy $x_0 < x_1 < \cdots < x_N$, and set let $f:\lb x_0,x_N \rb \rightarrow \R$ satisfy for all $x,y \in \lb x_0,x_N\rb$ that $\left| f(x)-f(y)\right| \les L \left| x-y\right|$, it is then the case that: +% \begin{enumerate}[label = (\roman*)] +% \item for all $x,y \in \R$ that: +% \begin{align} +% \left| \lp \lin^{f(x_0),f(x_1),...,f(x_N)}_{x_0,x_1,...,x_N}\rp \lp x \rp - \lp \lin^{f(x_0),f(x_1),...,f(x_N)}_{x_0,x_1,...,x_N}\rp \lp y \rp \right| \les L \left| x-y \right| +% \end{align}, and +% \item that: +% \begin{align} +% \sup_{x \in \lb x_0,x_N \rb }\left| \lp \lin^{f(x_0),f(x_1),...,f(x_N)}_{x_1,x_2,...,x_N}\rp \lp x \rp -f\lp x \rp\right| \les L \lp \max_{i \in \{ 1,2,...N\}} \left| x_i-x_{i-1}\right|\rp +% \end{align} +% \end{enumerate} +%\end{lemma} +%\begin{proof} +% The assumption that for all $x,y \in \lb x_0, x_k \rb$ it is the case that $\left| f(x) - f(y) \right| \les L \left| x-y\right|$ and Lemma \ref{lem:9.3.3} prove Item (i) and Item (ii). +%\end{proof} +%\subsection{Neural Networks to Approximate the $\lin$ Operator} +%\begin{lemma}\label{7.3.3} +% Let $\alpha,\beta,h \in \R$. Denote by $\relu \in \neu$ the neural network given by $\relu = h \circledast \lp \mathsf{i}_1 \bullet \aff_{\alpha,\beta}\rp $. It is then the case that: +% \begin{enumerate}[label = (\roman*)] +% \item $\relu = \lp \lp \alpha, \beta \rp , \lp h,0 \rp \rp$ +% \item $\lay(\relu) = \lp 1,1,1 \rp \in \N^3$. +% \item $\real_{\rect}\lp \relu \rp \in C \lp \R, \R \rp$ +% \item for all $x \in \R$ that $\lp \real_{\rect} \lp \relu \rp \rp \lp x \rp = h\max \{\alpha x+\beta ,0\}$ +% \end{enumerate} +%\end{lemma} +%\begin{proof} +% Note that by Definition \ref{5.3.1} we know that $\aff_{\alpha,\beta} = \lp \lp \alpha,\beta \rp \rp$, this with Definition \ref{actnn}, and Definition \ref{5.2.1} together tell us that $\mathfrak{i}_1\bullet \aff_{\alpha,\beta} = \lp \alpha,\beta \rp$. A further application of Definition \ref{5.2.1}, and an application of Definition \ref{slm} yields that $h \circledast \lp \mathfrak{i}_1 \bullet \aff_{\alpha,\beta} \rp = \lp \lp \alpha,\beta \rp, \lp h ,0 \rp \rp$. This proves Item (i). +% +% Note that $\lay(\aff_{\alpha,\beta})= (1,1)$, $\lay(\mathfrak{i}_1) = \lp 1,1,1 \rp $, and $\lay(h)=1$. Item (i) of Lemma \ref{6.0.3} therefore tells us that $\lay (\relu) = \lay \lp h \circledast \lp \mathfrak{i}_1 \bullet \aff_{\alpha,\beta}\rp \rp$. This proves Item (ii). +% +% Note that Lemmas \ref{7.1.2} and \ref{6.0.3} tell us that: +% \begin{align} +% \forall x\in \R: \lp \real_{\rect}\lp \mathfrak{i}_1 \bullet \aff_{\alpha,\beta} \rp \rp \lp x \rp = \rect \lp \real_{\rect} \rp \lp x \rp = \max\{ \alpha x+ \beta \} +% \end{align} +% This and Lemma \ref{slm} ensures that $\real_{\rect}\lp \relu \rp \in C\lp \R, \R \rp$ and further that: +% \begin{align} +% \lp \real_{\rect} \lp \relu \rp \rp \lp x \rp = h \lp \lp \real_{\rect}\lp \mathfrak{i}_1\bullet \aff_{\alpha,\beta} \rp \rp \lp x\rp \rp = h\max\{\alpha x+\beta,0 \} +% \end{align} +% This proves Item (iii)-(iv). This completes the proof of the lemma. +%\end{proof} +%\begin{lemma}\label{9.3.4} +% Let $N\in \N$, $x_0,x_1,...,x_N,y_0,y_1,...,y_N \in \R$ and further that $x_0 \les x_2 \les \cdots \les x_N$. Let $\Phi \in \neu$ satisfy that: +% \begin{align}\label{7.3.5} +% \Phi = \aff_{1,y_0} \bullet \lp \bigoplus^N_{i=0} \lb \lp \frac{y_{\min\{i+1,N\}}-y_i}{x_{\min\{i+1,N\}}-x_{\min\{i,N-1\}}}- \frac{y_i-y_{\max\{i-1,0\}}}{x_{\max\{i,1\}}-x_{\max\{i-1,0\}}}\rp \circledast \lp \mathfrak{i}_1\bullet \aff_{1,-x_i} \rp \rb \rp +% \end{align} +% It is then the case that: +% \begin{enumerate}[label=(\roman*)] +% \item $\lay(\Phi)= \lp 1,N+1,1 \rp \in \N^3$ +% \item $\real_{\rect} \lp \Phi \rp \in C \lp \R, \R \rp$ +% \item $\lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp= \lin ^{y_0,y_1,...,y_N}_{x_0,x_1,...,x_N}\lp x \rp$ +% \item $\param(\Phi) = 3N+4$ +% \end{enumerate} +%\end{lemma} +%\begin{proof} +% For notational convenience, let it be the case that for all $i \in \{0,1,2,..., N\}$: +% \begin{align}\label{7.3.6} +% h_i = \frac{y_{\min\{i+1,N\}}-y_i}{x_{\min\{i+1,N\}}-x_{\min\{i,N-1\}}}- \frac{y_i-y_{\max\{i-1,0\}}}{x_{\max\{i,1\}}-x_{\max\{i-1,0\}}} +% \end{align} +% Note that $\lay \lp \mathfrak{i}_1 \bullet \aff_{1,-x_0} \rp= \lp1,1,1 \rp$, and further that for all $i\in \{0,1,2,...,N\}$, $h_i \in \R$. Lemma \ref{7.3.3} then tells us that for all $i \in \{0,1,2,...,N\}$, $\lay \lp h_i \circledast \lp \mathfrak{i}_1 \bullet \aff_{1,-x_i} \rp \rp = \lp 1,1,1 \rp $, $\real_{\rect}\lp h_i \circledast \lp \mathfrak{i}_1 \bullet \aff_{1,-x_i} \rp \rp \in C \lp \R,\R \rp$, and that $ \lp \real_{\act} \lp h_i \circledast \lp \mathfrak{i}_1 \bullet \aff_{1,-x_i} \rp \rp \rp \lp x \rp = h_i \max\{x-x_k,0 \}$. This, (\ref{7.3.5}), Lemma \ref{5.3.3}, and \cite[Lemma~3.28]{Grohs_2022} ensure that $\lay(\Phi) = \lp 1,N+1,1 \rp \in \N^3$ and that $\real_{\rect} \lp \Phi \rp \in C \lp \R, \R \rp$ establishing Items (i)--(ii). +% +% In addition, note that Item (i) and (\ref{widthdef}), tell us that: +% \begin{align} +%% NOTE: Ask Dr. P about this parameter +% \param(\Phi) = \overbrace{(N+1)}^{W_1}+\underbrace{(N+1)}_{b_1}+\overbrace{(N+1)}^{W_2}+\underbrace{1}_{b_2} =3N+4 +% \end{align} +% Which proves Item (iv). For all $i \in \{0,1,2,...,N\}$, let $\phi_i$ be $\phi_i = h_i \circledast \lp \mathfrak{i} \bullet \aff_{1,-x_i} \rp $. Next note that \ref{7.3.6}, Lemma \ref{5.3.3}, and \cite[Lemma~3.28]{Grohs_2022} then tell us that: +% \begin{align}\label{7.3.8} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp = y_0 + \sum^n_{i=1} \lp \real_{\act} \lp \phi_i \rp \rp\lp x \rp = y_0 + \sum^n_{i=1}h_i \max\{x-x_i,0\} +% \end{align} +% Since $x_0 \les x_i$ for all $i\in\{1,2,...,n\}$, it then is the case for all $x \in (\infty, x_0]$ that: +% \begin{align}\label{7.3.10.2} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp = y_0+0 = y_0 +% \end{align} +% \begin{claim} +% For all $i \in \{1,2,...,N\}$ it is the case that : +% \begin{align}\label{7.3.10} +% \sum_{j=0}^{i-1}h_j = \frac{y_{i}-y_{i-1}}{x_i-x_{i-1}} +% \end{align} +% \end{claim} +% We prove this claim by induction. For the base case of $i=1$, we have: +% \begin{align} +% \sum^0_{j=0} h_0 = h_0 = \frac{y_{1}-y_0}{x_{1}-x_{0}}- \frac{y_0-y_{0}}{x_{1}-x_{0}} =\frac{y_1-y_0}{x_1-x_0} +% \end{align} +% This proves the base base for (\ref{7.3.10}). Assume next that this holds for $k$, for the $(k+1)$-th induction step we have: +% \begin{align} +% \sum^{k+1}_{j=0}h_j = \sum^k_{j=0}h_j + h_{k+1} &=\frac{y_k-y_{k-1}}{x_k-x_{k-1}}+h_{k+1} \nonumber\\ +% &= \frac{y_k-y_{k-1}}{x_k-x_{k-1}} + \frac{y_{k+2}-y_{k-1}}{x_{k+2}-x_{k+1}} - \frac{y_{k+1}-y_{k}}{x_{k+1} - x_k} \nonumber\\ +% &= \frac{y_{k+1}-y_k}{x_{k+1}-x_k} +% \end{align} +%%TODO: Double-check this proof +%This proves (\ref{7.3.10}). In addition, note that (\ref{7.3.8}), (\ref{7.3.10}), and the fact that for all $i \in \{1,2,...,n\}$ it is the case that $x_{i-1} \les x_{i}$ tells us that for all $i \in \{1,2,...,n\}$ and $x \in [x_{i-1},x_i]$ it is the case that: +% \begin{align}\label{7.3.13} +% &\lp \real_{\rect}\lp \Phi \rp \rp \lp x \rp - \lp \real_{\act}\lp \Phi \rp \rp \lp x_{i-1}\rp = \sum^n_{j=0} h_j \lp \max \{ x-x_j,0 \}-\max \{x_{i-1}-x_j,0\} \rp \nonumber\\ +% &= \sum^{i-1}_{j=0}c_j \lb \lp x-x_j \rp -\lp x_{i-1}-x_j \rp \rb = \sum^{i-1}_{j=0} c_j \lp x - x_{i-1} \rp = \lp \frac{y_i-y_{i-1}}{x_i-x_{i-1}}\rp \lp x-x_{i-1} \rp +% \end{align} +% \begin{claim} +% For all $i \in \{1,2,...,N\}$, $x\in [x_{i-1},x_i]$ it is the case that: +% \begin{align}\label{7.3.14} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp = y_{i-1}+ \lp \frac{y_i-y_{i-1}}{x_i-x_{i-1}} \rp \lp x - x_{i-1} \rp +% \end{align} +% \end{claim} +% We will prove this claim by induction. For the base case of $i=1$, (\ref{7.3.13}) and (\ref{7.3.10}) tell us that: +% \begin{align} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp &=\lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp - \lp \real_{\rect}\lp \Phi \rp \rp \lp x_{i-1} \rp + \lp \real_{\rect}\lp \Phi \rp \rp \lp x_{i-1} \rp \nonumber \\ +% &= y_0 + \lp \frac{y_1-y_0}{x_i-x_{i-1}} \rp \lp x - x_{i-1} \rp +% \end{align} +% For the induction step notice that (\ref{7.3.13}) implies that for all $i \in \{2,3,...,N\}$, $x \in [x_{i-1},x_i]$, with the instantiation that $\forall x \in [x_{i-2},x_{i-1}]: \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp = y_{i-2} + \lp \frac{y_{i-1}-y_{i-2}}{x_{i-1}-x_{i-2}} \rp \lp x-x_{i-2} \rp $, it is then the case that: +% \begin{align} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp &= \lp \real_{\rect} \lp \Phi \rp \rp \lp x_{i-1}\rp + \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp - \lp \real_{\rect} \lp \Phi \rp \rp \lp x_{i-1} \rp \nonumber\\ +% &=y_{i-2} + \lp \frac{y_{i-1}-y_{i-2}}{x_{i-1}-x_{i-2}} \rp \lp x_{i-1}+x_{i-2} \rp + \lp \frac{y_i-y_{i-1}}{x_i-x_{i-1}} \rp \lp x - x_{i-1} \rp \nonumber\\ +% &= y_{i-1} + \lp \frac{y_i-y_{i-1}}{x_i-x_{i-1}} \rp \lp x-x_{i-1} \rp +% \end{align} +% Thus induction proves (\ref{7.3.14}). Furthermore note that (\ref{7.3.10}) and (\ref{7.3.6}) tell us that: +% \begin{align} +% \sum^N_{i=0} h_i = c_N +\sum^{N-1}_{i=0}h_i = -\frac{y_N-y_{N-1}}{x_N-x_{N-1}}+\frac{y_N-y_{N-1}}{x_N-x_{N-1}} = 0 +% \end{align} +% The fact that $\forall i \in \{0,1,...,N\}:x_i \les x_N$, together with (\ref{7.3.8}) imply for all $x \in [x_N,\infty)$ that: +% \begin{align} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp - \lp \real_{\rect} \lp \Phi \rp \rp \lp x_N \rp &= \lb \sum^N_{i=0} h_i \lp \max\{x-x_i,0\}-\max\{x_N-x_i,0\} \rp \rb \nonumber\\ +% &= \sum^N_{i=0} h_i \lb \lp x- x_i \rp - \lp x_N - r_i \rp \rb = \sum^N_{i=0} h_i \lp x - x_N \rp =0 \nonumber +% \end{align} +% This and (\ref{7.3.14}) tells us that for all $x \in [x_N,\infty)$ we have: +% \begin{align} +% \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp = \lp \real_{\rect} \lp \Phi \rp \rp \lp x_N \rp = y_{N-1}+ \lp \frac{y_N-y_{N-1}}{x_N - x_{N-1}} \rp \lp x_N-x_{N-1} \rp = x_N +% \end{align} +% Together with (\ref{7.3.10.2}), (\ref{7.3.14}), and Definition \ref{lio} establishes Item (iii) thus proving the lemma. +% \end{proof} +%\section{Neural Network Approximations of 1-dimensional Functions.} +% +%\begin{lemma}\label{lem:9.5.1} +% Let $N\in \N$, $L. a. x_0,x_1,...,x_N \in \R$, $b\in \lp a,\infty \rp$, satisfy for all $i \in \left\{0,1,...,N\right\}$ that $x_i = a+ \frac{i(b-a)}{N}$. Let $f:\lb a,b\rb \rightarrow \R$ satisfy for all $x,y \in \lb a,b\rb$ that $\left|f(x) - f(y) \right| \les L\left|x-y\right|$ and let $\mathsf{F} \in \neu$ satisfy: +% \begin{align} +% \F = \aff_{1,f(x_0)}\bullet \lb\bigoplus^N_{i=0} \lp \lp \frac{N \lp f \lp x_{\min \{i+1,N\}}\rp-2f\lp x_i\rp + f \lp x_{\max \{ i-1,0 \}}\rp\rp}{b-a}\rp \circledast \lp \mathsf{i}_1 \bullet \aff_{1,-x_k} \rp\rp \rb +% \end{align} +% It is then the case that: +% \begin{enumerate}[label = (\roman*)] +% \item $\lay \lp \F \rp = \lp 1, N+1,1\rp$ +% \item $\real_{\rect} \lp \F \rp\in C\lp \R, \R \rp$ +% \item $\real_{\rect} \lp \F \rp = \lin ^{f(x_0),f(x_1),...,f(x_N)}_{x_1,x_2,...,x_N}$ +% \item it holds that for all $x,y \in \R$ that $\left| \lp \real_{\rect} \lp \F \rp \rp \lp x \rp -\lp \real_{\rect} \lp \F \rp\rp\lp y \rp \right| \les L \left| x-y \right|$ +% \item it holds that $\sup_{x \in \lb a,b \rb} \left| \lp \real_{\rect} \lp \F \rp \rp\lp x\rp -f(x)\right| \les \frac{L \lp b-a\rp}{N}$, and +% \item $\param\lp \F \rp = 3N+4$. +% \end{enumerate} +%\end{lemma} +%\begin{proof} +% Note that since it is the case that for all $i \in \left\{0,1,...,N \right\}: x_{\min \{i+1,N\}} - x_{\min \{i, N-1\}} = x_{\max\{i,1\}} - x_{\max \{i-1,0\}} = \frac{b-a}{N}$, we have that: +% \begin{align} +% \frac{f\lp x_{\min\{i+1,N\}}\rp- f \lp x_i \rp}{x_{\min \{ i+1,N\}}-x_{\min \{i,N-1\}}} - \frac{f(x_i)-f\lp x_{\max\{i-1,0\}}\rp}{x_{\max \{ i,1\}}-x_{\max \{i-1,0\}}} = \frac{N \lp f \lp x_{\min \{i+1,N\}}\rp -2f\lp x_i \rp +f\lp x_{\max \{ i-1,0\}}\rp\rp}{b-a} +% \end{align} +% Thus Items (i)-(iv) of Lemma \ref{9.3.4} prove Items (i)-(iii), and (vi) of this lemma. Item (iii) combined with the assumption that for all $x,y \in \lb a,b \rb: \left| f(x) - f(y) \right| \les \left| x-y \right|$ and Item (i) in Lemma \ref{lem:9.4.3} establish Item (iv). Furthermore, note that Item (iii), the assumption that for all $x,y \in \lb a,b \rb: \left| f(x) -f(y)\right| \les L\left| x-y\right|$, Item (ii) in Lemma \ref{lem:9.4.3} and the fact that for all $i \in \{1,2,..., N\}: x_i-x_{i-1} = \frac{b-a}{N}$ demonstrate for all $x \in \lb a,b \rb$ it holds that: +% \begin{align} +% \left| \lp \real_{\rect} \lp \F \rp\rp \lp x \rp -f\lp x \rp \right| \les L \lp \max_{i \in \{1,2,...,N\}} \left| x_i - x_{i-1}\right|\rp = \frac{L(b-a)}{N} +% \end{align} +%\end{proof} +%\begin{lemma}\label{lem:9.5.2} +% Let $L,a \in \R$, $b\in \lb a, \infty \rp$, $\xi \in \lb a,b \rb$, let $f: \lb a,b \rb \rightarrow \R$ satisfy for all $x,y \in \lb a,b \rb$ that $\left| f(x) - f(y) \right| \les L\left|x-y \right|$, and let $\F \in \neu$ satisfy $\F = \aff_{1,f(\xi)} \bullet \lp 0 \circledast \lp \mathsf{i}_1 \bullet \aff_{1,-\xi} \rp \rp $, it is then the case that: +% \begin{enumerate}[label = (\roman*)] +% \item $\lay \lp \F \rp = \lp 1,1,1 \rp$ +% \item $\real_{\rect} \lp \F \rp \in C \lp \R, \R \rp$ +% \item for all $x \in \R$, we have $\lp \real_{\rect}\lp \F \rp \rp \lp x \rp = f \lp \xi \rp$ +% \item $\sup_{x \in \lb a,b\rb} \left| \lp \real_{\rect} \lp \F \rp \rp\lp x \rp -f(x)\right| \les L \max \{ \xi -a, b-\xi\}$ +% \item $\param \lp \F \rp = 4$ +% \end{enumerate} +%\end{lemma} +% +%\begin{proof} +% Note that Item (i) is a consequence of the fact that $\aff_{1,-\xi}$ is a neural network with a real number as weight and a real number as a bias and the fact that $\lay \lp \mathsf{i}_1 \rp = \lp 1,1,1 \rp$. Note also that Item (iii) of Lemma \ref{7.3.3} proves Item (iii). +% +% Note that from the construction of $\aff$ we have that: +% \begin{align}\label{(9.5.4)} +% \lp \real_{\rect} \lp \F \rp\rp \lp x \rp &= \lp \real_{\rect} \lp 0 \circledast \lp \mathsf{i}_1 \bullet \aff_{1,-\xi}\rp\rp \rp \lp x \rp + f \lp \xi \rp \nonumber \\ +% &= 0 \lp \lp \real_{\rect} \lp \mathsf{i}_1 \bullet \aff_{1,-\xi} \rp\rp \lp x \rp \rp + f \lp \xi \rp = f \lp \xi \rp +% \end{align} +% Which establishes Item (iii). Note that (\ref{(9.5.4)}), the fact that $\xi \in \lb a,b\rb$ and the fact that for all $x,y \in \lb a,b \rb$ it is the case that $\left| f(x) - f(y) \right| \les \left| x-y \right|$ give us that for all $x \in \lb a,b \rb$ it holds that: +% \begin{align} +% \left| \lp \real_{\rect} \lp \F \rp\rp \lp x \rp - f\lp x \rp\right| = \left| f\lp \xi \rp - f \lp x \rp\right| \les L \left| x- \xi \right| \les L \max\left\{ \xi -a, b-\xi \right\} +% \end{align} +% This establishes Item (iv). Note a simple parameter count yields the following: +% \begin{align} +% \param \lp \F \rp = 1(1+1)+1(1+1) = 4 +% \end{align} +% Establishing Item (v) and hence the lemma. This completes the proof. +%\end{proof} +%\begin{corollary} +% Let $\ve \in (0,\infty)$, $L,a \in \R$, $b \in \lp a,\infty \rp$, $N \in \N_0 \cap \lb \frac{L(b-a)}{\ve}, \frac{L(b-a)}{\ve}+1\rb$, $x_0, x_1,...,x_N \in \R$ satisfy for all $i \in \{ 0,1,...,N\}$ that $x_i = a + \frac{i(b-a)}{\max\{N,1\}}$, let $f: \lb a,b \rb \rightarrow \R$ satisfy for all $x,y \in \lb a,b \rb$ that $\left| f(x) - f(y) \rb \les L\left| x-y \right|$, and let $\F \in \neu$ satisfy: +% \begin{align} +% \F = \aff_{1,f(x_0)}\bullet \lb\bigoplus^N_{i=0} \lp \lp \frac{N \lp f \lp x_{\min \{i+1,N\}}\rp-2f\lp x_i\rp + f \lp x_{\max \{ i-1,0 \}}\rp\rp}{b-a}\rp \circledast \lp \mathsf{i}_1 \bullet \aff_{1,-x_k} \rp\rp \rb +% \end{align} +% It is then the case that: +% \begin{enumerate}[label = (\roman*)] +% \item $\lay\lp \F \rp = \lp 1,N+1,1 \rp$ +% \item $\real_{\rect} \lp \F \rp \in C \lp \R, \R \rp$ +% \item for all $x,y \in \R$ that $\left| \lp \real_{\rect} \lp \F \rp \rp \lp x \rp - \lp \real_{\rect} \lp \F \rp \rp \lp x \rp \right| \les L \left| x-y \right|$ +% \item $\sup_{x \in \lb a,b \rb} \left| \lp \real_{\rect} \lp \F \rp \rp \lp x \rp -f(x) \right| \les \frac{L(b-a)}{\max \{N,1\}} \les \ve$, and +% \item $\param \lp \F \rp = 3N+4 \les 3L \lb \frac{b-a}{\ve} \rb +7$. +% \end{enumerate} +%\end{corollary} +%\begin{proof} +% The fact that $N \in \N_0 \cap \lb \frac{L(b-a)}{\ve}, \frac{L(b-a)}{\ve}+1 \rb$ ensures that $\frac{L(b-a)}{\max\{ K,1\}} \les \ve$. This and Items (i),(ii),(iv), and (v) in Lemma \ref{lem:9.5.1} and Items (i)-(iii), and (iv) of Lemma $\ref{lem:9.5.2}$ establishes Items (i)-(iv). Furthermore, note that since $N\les 1 + \frac{L(b-a)}{\ve}$, Item (vi) in Lemma \ref{lem:9.5.1} and Item (v) in Lemma \ref{lem:9.5.2} tells us that: +% \begin{align} +% \param \lp \F\rp = 3N+4 \les \frac{3L\lp b-a \rp}{\ve} + 7. +% \end{align} +% Which establishes Item (v) and proves the result. +%\end{proof} +\section{$\trp^h$, $\etr^{n,h}$ and Neural Network Approximations For the Trapezoidal Rule.} +\begin{definition}[The $\trp$ neural network] + Let $h \in \R_{\ges 0}$. We define the $\trp^h \in \neu$ neural network as: + \begin{align} + \trp^h \coloneqq \aff_{\lb \frac{h}{2} \: \frac{h}{2}\rb,0} + \end{align} +\end{definition} +\begin{lemma} + Let $h\in \lp -\infty, \infty\rp$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for $x = \{x_1,x_2\} \in \R^2$ that $\lp \real_{\rect} \lp \trp^h \rp \rp \lp x \rp \in C \lp \R^2, \R \rp$ + \item for $x = \{x_1,x_2 \} \in \R^2$ that $\lp \real_{\rect} \lp \trp^h \rp \rp \lp x \rp = \frac{1}{2}h \lp x_1+x_2 \rp$ + \item $\dep \lp \trp^h \rp = 1$ + \item $\param\lp \trp^h \rp = 3$ + \item $\lay \lp \trp^h \rp = \lp 2,1 \rp$ + \end{enumerate} +\end{lemma} +\begin{proof} + This a straight-forward consequence of Lemma \ref{5.3.1} +\end{proof} +\begin{definition}[The $\etr$ neural network] + Let $n\in \N$ and $h \in \R_{\ges 0}$. We define the neural network $\etr^{n,h} \in \neu$ as: + \begin{align} + \etr^{n,h} \coloneqq \aff_{\underbrace{\lb \frac{h}{2} \ h \ h\ ... \ h \ \frac{h}{2}\rb}_{n+1-many},0} + \end{align} +\end{definition} +\begin{lemma}\label{etr_prop} + Let $n\in \N$. Let $x_0 \in \lp -\infty, \infty \rp$, and $x_n \in \lb x_0, \infty \rp$. Let $ x = \lb x_0 \: x_1 \:...\: x_n\rb \in \R^{n+1}$ and $h\in \lp -\infty, \infty\rp$ such that for all $i \in \{0,1,...,n\}$ it is the case that $x_i = x_0+i\cdot h$. Then: + \begin{enumerate}[label = (\roman*)] + \item for all $x \in \R^{n+1}$ it is the case that $\lp \real_{\rect} \lp \etr^{n,h} \rp \rp \lp x \rp \in C \lp \R^{n+1}, \R \rp$ + \item for all $n\in \N$, and $h\in \lp 0,\infty\rp$ it is the case that $\lp \real_{\rect} \lp \etr^{n,h} \rp \rp \lp x \rp = \frac{h}{2} \cdot x_0+h\cdot x_1 + \cdots + h\cdot x_{n-1} + \frac{h}{2}\cdot x_n$ + \item for all $n \in \N$, and $h \in \lp 0,\infty \rp$ it is the case that $\dep \lp \etr^{n,h} \rp = 1$ + \item for all $n \in \N$ and $h \in \lp 0,\infty\rp$ it is the case that $\param\lp \etr^{n,h} \rp = n+2$ + \item for all $n\in \N$ and $h \in \lp 0,\infty\rp$ it is the case that $\lay \lp \etr^{n,h} \rp = \lp n+1,1 \rp$ + \end{enumerate} +\end{lemma} +\begin{proof} + This a straightforward consequence of Lemma \ref{5.3.1}. +\end{proof} +\begin{remark} + Let $h \in \lp 0,\infty\rp$. Note then that $\trp^h$ is simply $\etr^{2,h}$. +\end{remark} +%\begin{lemma} +% Let $f \in C \lp \R, \R \rp$, $a\in \R, b \in \lb a,\infty\rp$, $N\in \N$, and let $h = \frac{b-a}{N}$. Assume also that $f$ has first and second derivatives almost everywhere. Let $ x = \lb x_0 \: x_1 \:...\: x_n\rb \in \R^{n+1}$ such that for all $i \in \{0,1,...,n\}$ it is the case that $x_i = x_0+i\cdot h$, as such let it also be the case that $f\lp \lb x \rb_{*,*}\rp = \lb f(x_0)\: f(x_1) \: \cdots f(x_n) \rb$. Let $a = x_0$ and $b = x_n$. It is then the case that: +% \begin{align}\label{(9.6.3)} +% \left| \int^b_a f\lp x \rp dx - \lp \real_{\rect}\lp \etr^{n,h} \rp\rp \lp f\lp \lb x \rb_{*,*}\rp\rp \right| \les \frac{\lp b-a \rp^3}{12N^2} f''\lp \xi \rp +% \end{align} +% Where $\xi \in \lb a,b \rb$. +%\end{lemma} +%\begin{proof} +% Consider the fact that we may express the left-hand side of (\ref{(9.6.3)}) as: +% \begin{align} +% \left| \int^b_af dx - \lp \real_{\rect}\lp \etr^{n,h} \rp\rp \lp x \rp \right| = \left| \sum_{i=1}^n \lb \int^{x_i}_{x_{i-1}} f\lp x \rp dx-\frac{h}{2}\lp f\lp x_{i-1} \rp + f\lp x_i\rp\rp \rb \right| +% \end{align} +% We then denote by $L_i$ the error at sub-interval $\lb x_{i-1},x_i \rb$ as given by: +% \begin{align} +% L_i = \left| \int^{x_i}_{x_{i-1}}f\lp x \rp dx - \frac{h}{2}\lp f\lp x_{i-1}\rp -f\lp x_i \rp \rp \right| +% \end{align} +% Furthermore, we denote $c_i = \frac{x_{i-1}+x_i}{2}$ as the midpoint of the interval $\lb x_{i-1}, x_i\rb$, which yields the observation that: +% \begin{align}\label{(9.6.6)} +% c_i-x_{i-1} = x_i - c_i = \frac{b-a}{2N} +% \end{align} +% Integration by parts and (\ref{(9.6.6)}) then yields that: +% \begin{align} +% \int^{x_i}_{x_{i-1}}\lp t- c_i\rp f' \lp t \rp dt &= \int^{x_i}_{x_{i-1}} \lp t-c_i\rp df\lp t \rp \nonumber \\ +% &= \lp x_i -c_i \rp f\lp x_i \rp - \lp x_{i-1} - c_i\rp f \lp x_{i-1}\rp - \int^{x_i}_{x_{i-1}}f \lp t \rp dt \nonumber \\ +% &= \frac{b-a}{2N} \lp f\lp x_{i}\rp - f\lp x_{i-1}\rp\rp - \int^{x_i}_{x_{i-1}}f\lp t \rp dt = L_i +% \end{align} +% Whence we have: +% \begin{align} +% L_i = \int^{x_i}_{x_{i-1}}\lp t-c_i\rp f'\lp t\rp dt +% \end{align} +% Integration by parts, (\ref{(9.6.6)}), and the Fundamental Theorem of Calculus then gives us: +% \begin{align} +% L_i &= \int^{x_i}_{x_{i-1}} f' \lp t \rp d \frac{\lp t-c_i \rp^2}{2} \nonumber\\ +% &= \frac{\lp x_i - c_i\rp^2}{2} f' \lp x_i \rp - \frac{\lp x_{i-1} - c_i\rp^2}{2} f' \lp x_{i-1} \rp - \frac{1}{2} \int^{x_i}_{x_{i-1}} \lp t-c_i \rp^2 f'' \lp t\rp \nonumber\\ +% &= \frac{1}{2}\lb \frac{b-a}{2N}\rb^2 \lp f'\lp x_i \rp - f' \lp x_{i-1}\rp \rp - \frac{1}{2} \int^{x_i}_{x_{i-1}} \lp t-c_i\rp^2 f'' \lp t \rp dt \nonumber\\ +% &= \frac{1}{2} \int^{x_i}_{x_{i-1}} f'' \lp t \rp dt - \frac{1}{2} \int^{x_i}_{x_{i-1}} \lp t-c_i\rp^2 f'' \lp t\rp dt \nonumber \\ +% &= \frac{1}{2} \int^{x_i}_{x_{i-1}}\lp \lb \frac{b-a}{2N} \rb^2 - \lp t-c_i\rp^2 \rp f'' \lp t\rp dt +% \end{align} +% Assuming that $f''\lp x \rp \les M$ within $\lb a,b \rb$ we then have that: +% \begin{align} +% \left| \int^b_af dx - \lp \real_{\rect}\lp \etr^{n,h} \rp\rp \lp x \rp \right| &\les \sum_{i=1}^N \left| L_i\right| \nonumber\\ +% &\les \frac{1}{2}\sum^N_{i=1} \int^{x_i}_{x_{i-1}} \left| \lp \lb \frac{b-a}{2N} \rb^2 - \lp t-c_i\rp^2 \rp \right| \left| f'' \lp t\rp dt\right| \nonumber \\ +% &\les \frac{M}{2} \sum_{i=1}^N \int^{x_i}_{x_{i-1}} \lb \frac{b-a}{2N}\rb^2 - \lp t-c_i\rp^2 dt \nonumber \\ +% &= \frac{M}{2} \lp \lb \frac{b-2}{2N}\rb^2\lp b-a \rp - \frac{2n}{3} \lb \frac{b-a}{2N}\rb^3 \rp \nonumber\\ +% &= \frac{M \lp b-a \rp^3}{12N^2} +% \end{align} +% This completes the proof of the lemma. +%\end{proof} +\section{Linear Interpolation for Multi-Dimensional Functions} +\subsection{The $\nrm^d_1$ Networks} +\begin{definition}[The $\nrm_1^d$ neural network] + We denote by $\lp \nrm_1^d \rp _{d\in \N} \subseteq \neu$ the family of neural networks that satisfy: + \begin{enumerate}[label = (\roman*)] + \item for $d=1$:\begin{align}\label{(9.7.1)} + \nrm^1_1 = \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 && 1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \in \lp \R^{2 \times 1} \times \R^2 \rp \times \lp \R^{1 \times 2} \times \R^1 \rp + \end{align} + \item for $d \in \{2,3,...\}$: \begin{align} + \nrm_1^d = \sm_{d,1} \bullet \lb \boxminus_{i=1}^d \nrm_1^1 \rb + \end{align} + \end{enumerate} +\end{definition} +\begin{lemma}\label{9.7.2}\label{lem:nrm_prop} + Let $d \in \N$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\lay \lp \nrm^d_1 \rp = \lp d,2d,1 \rp$ + \item $\lp \real_{\rect} \lp \nrm^d_1\rp \rp \lp x \rp \in C \lp \R^d,\R \rp$ + \item that for all $x \in \R^d$ that $\lp \real_{\rect}\lp \nrm^d_1 \rp \rp \lp x \rp = \left\| x \right\|_1$ + \item it holds $\hid\lp \nrm^d_1\rp=1$ + \item it holds that $\param \lp \nrm_1^d \rp \les 7d^2$ + \item it holds that $\dep\lp \nrm^d_1\rp =2 $ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that by observation, it is the case that $\lay\lp \nrm^d_1 \rp = \lp 1,2,1\rp$. This and Remark \ref{5.3.2} tells us that for all $d \in \{2,3,...\}$ it is the case that $\lay \lp \boxminus_{i=1}^d \nrm^d_1 \rp = \lp d,2d,d\rp$. This, Lemma \ref{comp_prop}, and Lemma \ref{5.3.2} ensure that for all $d \in \{2,3,4,...\}$ it is the case that $\lay\lp \nrm^d_1 \rp = \lp d,2d,1 \rp$, which in turn establishes Item (i). + + Notice now that (\ref{(9.7.1)}) ensures that: + \begin{align} + \lp \real_{\rect} \lp \nrm^d_1 \rp \rp \lp x \rp = \rect \lp x \rp + \rect \lp -x \rp = \max \{x,0 \} + \max \{ -x,0\} = \left| x \right| = \| x \|_1 + \end{align} + This along with \cite[Proposition~2.19]{grohs2019spacetime} tells us that for all $d \in \{2,3,4,...\}$ and $x = \lp x_1,x_2,...,x_d\rp \in \R^d$ it is the case that: + \begin{align} + \lp \real_{\rect} \lb \boxminus^d_{i=1} \nrm^1_1\rb\rp \lp x \rp = \lp \left| x_1 \right|, \left| x_2\right|,..., \left| x_d \right| \rp + \end{align} + This together with Lemma \ref{depthofcomposition} tells us that: + \begin{align} + \lp \real_{\rect} \lp \nrm^d_1 \rp \rp &= \lp \real_{\rect} \lp \sm_{d,1} \bullet \lb \boxminus_{i=1}^d \nrm^d_1\rb\rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\rect} \lp \sm_{d,1} \rp \rp \lp |x_1|,|x_2|,...,|x_d|\rp = \sum^d_{i=1} |x_i| =\|x\|_1 + \end{align} + Note next that by observation $\hid\lp \nrm^1_1 \rp = 1$. Remark \ref{5.3.2} then tells us that since the number of layers remains unchanged under stacking, it is then the case that $\hid \lp \nrm^1_1 \rp = \hid \lp \boxminus_{i=1}^d \nrm_1^1\rp = 1$. Note next that Lemma \ref{5.2.3} then tells us that $\hid \lp \sm_{d,1} \rp = 0$ whence Lemma \ref{comp_prop} tells us that: + \begin{align} + \hid \lp \nrm^d_1 \rp &= \hid \lp \sm_{d,1}\bullet \lb \boxminus_{i=1}^d \nrm^1_1 \rb \rp \nonumber \\ + &= \hid \lp \sm_{d,1} \rp + \hid \lp \lb \boxminus_{i=1}^d \nrm^1_1 \rb \rp = 0+1=1 + \end{align} + + + Note next that: + \begin{align} + \nrm^1_1 = \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 && 1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \in \lp \R^{2 \times 1} \times \R^2 \rp \times \lp \R^{1 \times 2} \times \R^1 \rp + \end{align} + and as such $\param\lp \nrm^1_1 \rp = 7$. This, combined with Cor \ref{cor:sameparal}, and the fact that we are stacking identical neural networks then tells us that: + \begin{align} + \param \lp \lb \boxminus_{i=1}^d \nrm_1^1 \rb \rp &\les 7d^2 + \end{align} + Then Lemma Corollary \ref{affcor}, Lemma \ref{lem:5.5.4}, and Lemma \ref{comp_prop} tells us that: + \begin{align} + \param \lp \nrm^d_1 \rp &= \param \lp \sm_{d,1} \bullet \lb \boxminus_{i=1}^d \nrm_1^1 \rb\rp \nonumber \\ + &\les \param \lp \lb \boxminus_{i=1}^d \nrm_1^1 \rb \rp \les 7d^2 + \end{align} + This establishes Item (v). + + Finally, by observation $\dep \lp \nrm^1_1\rp = 2$, we are stacking the same neural network when we have $\nrm^d_1$. Stacking has no effect on depth from Definition \ref{def:stacking}, and by Lemma \ref{comp_prop}, $\dep \lp \sm_{d,1} \bullet \lb \boxminus^d_{i=1} \nrm_1^1\rb \rp = \dep \lp \boxminus \nrm^1_1\rp$. Thus we may conclude that $\dep \lp \nrm^d_1\rp = \dep \lp \nrm_1^1\rp =2$. + + This concludes the proof of the lemma. +\end{proof} + +\subsection{The $\mxm^d$ Neural Networks} + +Given $x\in \R$, it is straightforward to find the maximum; $ x$ is the maximum. For $x \in \R^d$ we may find the maximum via network (\ref{9.7.6.1}), i.e. $\mxm^2$. The strategy is to find maxima for half our entries and half repeatedly until we have one maximum. For $x \in \R^d$ where $d$ is even we may stack $d$ copies of $\mxm^2$ to halve, and for $x \in \R^d$ where $d$ is odd and greater than $3$ we may introduce ``padding'' via the $\id_1$ network and thus require $\frac{d-1}{2}$ copies of $\mxm^2$ to halve. +\begin{definition}[Maxima ANN representations] + Let $\lp \mxm ^d\rp_{d \in \N} \subseteq \neu$ represent the neural networks that satisfy: + \begin{enumerate}[label = (\roman*)] + \item for all $d \in \N$ that $\inn \lp \mxm^d \rp = d$ + \item for all $d \in \N$ that $\out\lp \mxm^d \rp = 1$ + \item that $\mxm^1 = \aff_{1,0} \in \R^{1 \times 1} \times \R^1$ + \item that: + \begin{align}\label{9.7.6} + \mxm^2 = \lp \lp \begin{bmatrix} + 1 & -1 \\ 0 & 1 \\ 0 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1&1&-1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp\rp + \end{align} + \item it holds for all $d \in \{2,3,...\}$ that $\mxm^{2d} = \mxm^d \bullet \lb \boxminus_{i=1}^d \mxm^2\rb$, and + \item it holds for all $d \in \{ 2,3,...\}$ that $\mxm^{2d-1} = \mxm^d \bullet \lb \lp \boxminus^d_{i=1} \mxm^2 \rp \boxminus \id_1\rb$. + \end{enumerate} +\end{definition} +\begin{remark} + Diagrammatically, this can be represented as: + \begin{figure}[h] + \begin{center} + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,498); %set diagram left start at 0, and has height of 498 + +%Shape: Rectangle [id:dp977616844446347] +\draw (560,138) -- (630,138) -- (630,178) -- (560,178) -- cycle ; +%Shape: Rectangle [id:dp8371611327934396] +\draw (560,206) -- (630,206) -- (630,246) -- (560,246) -- cycle ; +%Shape: Rectangle [id:dp900733317366978] +\draw (562,274) -- (632,274) -- (632,314) -- (562,314) -- cycle ; +%Shape: Rectangle [id:dp2381571768613] +\draw (565,350) -- (635,350) -- (635,390) -- (565,390) -- cycle ; +%Shape: Rectangle [id:dp47165779567431265] +\draw (568,425) -- (638,425) -- (638,465) -- (568,465) -- cycle ; +%Shape: Rectangle [id:dp26730884303141045] +\draw (438,175) -- (508,175) -- (508,215) -- (438,215) -- cycle ; +%Shape: Rectangle [id:dp43479154744962956] +\draw (439,310) -- (509,310) -- (509,350) -- (439,350) -- cycle ; +%Shape: Rectangle [id:dp14664308815255211] +\draw (302,234) -- (372,234) -- (372,274) -- (302,274) -- cycle ; +%Straight Lines [id:da5196233580766983] +\draw (437,196.5) -- (374.51,251.18) ; +\draw [shift={(373,252.5)}, rotate = 318.81] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da6126209944613533] +\draw (559,155.5) -- (509.66,188.88) ; +\draw [shift={(508,190)}, rotate = 325.92] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da5768174542895418] +\draw (558,224.5) -- (512.81,203.35) ; +\draw [shift={(511,202.5)}, rotate = 25.08] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da12927244412488015] +\draw (560,290.5) -- (510.66,323.88) ; +\draw [shift={(509,325)}, rotate = 325.92] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da9640125892832212] +\draw (566,372.5) -- (511.72,340.03) ; +\draw [shift={(510,339)}, rotate = 30.89] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da6747547273640673] +\draw (437,331.5) -- (373.38,264.95) ; +\draw [shift={(372,263.5)}, rotate = 46.29] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp14455818861310843] +\draw (445,425) -- (515,425) -- (515,465) -- (445,465) -- cycle ; +%Shape: Rectangle [id:dp03375582603009031] +\draw (301,367) -- (371,367) -- (371,407) -- (301,407) -- cycle ; +%Shape: Rectangle [id:dp0789527597033911] +\draw (163,296) -- (233,296) -- (233,336) -- (163,336) -- cycle ; +%Straight Lines [id:da6246849218035846] +\draw (302,252.5) -- (236.47,313.14) ; +\draw [shift={(235,314.5)}, rotate = 317.22] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da5611532984284957] +\draw (299,390.5) -- (235.38,323.95) ; +\draw [shift={(234,322.5)}, rotate = 46.29] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da6903134547643467] +\draw (162,315.5) -- (108,315.5) ; +\draw [shift={(106,315.5)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da647770723003481] +\draw (567,447.5) -- (518,447.5) ; +\draw [shift={(516,447.5)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da1985911653622896] +\draw (443,448.5) -- (373.61,397.68) ; +\draw [shift={(372,396.5)}, rotate = 36.22] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da06349763555732901] +\draw (437,342) -- (373.67,383.41) ; +\draw [shift={(372,384.5)}, rotate = 326.82] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (574,150.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (574,214.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (576,283.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (579,358.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (585,428.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (453,185.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (456,322.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (316,242.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (470,434.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Id}_{1}$}; +% Text Node +\draw (317,377.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; +% Text Node +\draw (177,305.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{2}$}; + +\end{tikzpicture} + \end{center} + \caption{Neural network diagram for $\mxm^5$.} + \end{figure} +\end{remark} +\begin{lemma}\label{9.7.4}\label{lem:mxm_prop} + Let $d \in \N$, it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\hid \lp \mxm^d \rp = \lceil \log_2 \lp d \rp \rceil $ + \item for all $i \in \N$ that $\wid_i \lp \mxm^d \rp \les 3 \left\lceil \frac{d}{2^i} \right\rceil$ + \item $\real_{\rect} \lp \mxm^d\rp \in C \lp \R^d, \R \rp$ and + \item for all $x = \lp x_1,x_2,...,x_d \rp \in \R^d$ we have that $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp = \max \{x_1,x_2,...,x_d \}$. + \item $\param \lp \mxm^d \rp \les \lp \frac{4}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{\left\lceil \log_2\lp d\rp\right\rceil+1}\rp$ + \item $\dep \lp \mxm^d\rp = \left\lceil \log_2 \lp d\rp \right\rceil + 1$ + \end{enumerate} +\end{lemma} +\begin{proof} + Assume w.l.o.g. that $d > 1$. Note that (\ref{9.7.6}) ensures that $\hid \lp \mxm^d \rp = 1$. This and (\ref{5.2.5}) then tell us that for all $d \in \{2,3,4,...\}$ it is the case that: + \begin{align} + \hid \lp \boxminus_{i=1}^d \mxm^2\rp = \hid \lp \lb \boxminus_{i=1}^d \mxm^2 \rb \boxminus \id_1 \rp = \hid \lp \mxm^2 \rp = 1 \nonumber + \end{align} + This and Lemma \ref{comp_prop} tells us that for all $d \in \{3,4,5,...\}$ it holds that: + \begin{align}\label{9.7.7} + \hid \lp \mxm^d \rp = \hid \lp \mxm^{\left\lceil \frac{d}{2} \right\rceil}\rp + 1 + \end{align} + And for $d \in \{4,6,8,...\}$ with $\hid \lp \mxm^{\left\lceil \frac{d}{2} \right\rceil} \rp = \left\lceil \log_2 \lp \frac{d}{2} \rp\right\rceil$ it holds that: + \begin{align}\label{9.7.8} + \hid \lp \mxm^d \rp = \left\lceil \log_2 \lp \frac{d}{2} \rp\right\rceil + 1 = \left\lceil \log_2 \lp d \rp -1 \right\rceil +1 = \left\lceil \log_2 \lp d \rp \right\rceil + \end{align} + Moreover (\ref{9.7.7}) and the fact that for all $d \in \{3,5,7,...\}$ it holds that $\left\lceil \log_2 \lp d+1 \rp \right\rceil = \left\lceil \log_2 \lp d \rp \right\rceil$ ensures that for all $d \in \{3,5,7,...\}$ with $\hid \lp \mxm^{\left\lceil \frac{d}{2}\right\rceil}\rp = \left\lceil \log_2 \lp \left\lceil \frac{d}{2} \right\rceil\rp \right\rceil$ it holds that: + \begin{align} + \hid \lp \mxm^d\rp &= \left\lceil \log_2 \lp \left\lceil \frac{d}{2} \right\rceil\rp \right\rceil + 1 = \left\lceil \log_2 \lp \left\lceil \frac{d+1}{2} \right\rceil\rp \right\rceil + 1 \nonumber\\ + &= \left\lceil \log_2 \lp d+1\rp-1 \right\rceil + 1 = \left\lceil \log_2 \lp d+1 \rp \right\rceil = \left\lceil \log_2 \lp d \rp \right\rceil + \end{align} + This and (\ref{9.7.8}) demonstrate that for all $d \in \{3,4,5,...\}$ with $\forall k \in \{2,3,...,d-1\}: \hid \lp \mxm^d\rp = \left\lceil \log_2 \lp k \rp \right\rceil$ it holds htat $\hid \lp \mxm^d \rp = \left\lceil \log_2 \lp d \rp \right\rceil$. The fact that $\hid \lp \mxm^2 \rp =1$ and induction establish Item (i). + + We next note that $\lay \lp \mxm^2 \rp = \lp 2,3,1 \rp$. This then indicates that for all $i\in \N$ that: + \begin{align}\label{9.7.10} + \wid_i \lp \mxm^2 \rp \les 3 = 3 \left\lceil \frac{2}{2^i} \right\rceil. + \end{align} + Note then that Lemma \ref{comp_prop} and Remark \ref{5.3.2} tells us that: + \begin{align}\label{9.7.11} + \wid_i \lp \mxm^{2d} \rp = \begin{cases} + 3d &:i=1 \\ + \wid_{i-1}\lp \mxm^d \rp &:i\ges 2 + \end{cases} + \end{align} + And: + \begin{align}\label{9.7.12} + \wid_i \lp \mxm^{2d-1}\rp = \begin{cases} + 3d-1 &:i=1 \\ + \wid_{i-1}\lp \mxm^d \rp &:i \ges 2 + \end{cases} + \end{align} + This in turn assures us that for all $d \in \{ 2,4,6,...,\}$ it holds that: + \begin{align}\label{9.7.13} + \wid_1 \lp \mxm^d \rp = 3\lp \frac{d}{2} \rp \les 3 \left\lceil \frac{d}{2} \right\rceil + \end{align} + Moreover, note that (\ref{9.7.12}) tells us that for all $d \in \{3,5,7,...\}$ it holds that: + \begin{align} + \wid_1 \lp \mxm^d \rp = 3\left\lceil \frac{d}{2}\right\rceil -1 \les 3 \left\lceil \frac{d}{2} \right\rceil + \end{align} + This and (\ref{9.7.13}) shows that for all $d \in \{2,3,...\}$ it holds that: + \begin{align}\label{9.7.15} + \wid_1 \lp \mxm^d\rp \les 3 \left\lceil \frac{d}{2}\right\rceil + \end{align} + Additionally note that (\ref{9.7.11}) demonstrates that for all $d \in \{ 4,6,8,...\}$, $i \in \{2,3,...\}$ with $\wid_{i-1} \lp \mxm^{\frac{d}{2}} \rp \les 3 \left\lceil \lp \frac{d}{2}\rp \frac{1}{2^{i-1}}\right\rceil$ it holds that: + \begin{align}\label{9.7.16} + \wid_i \lp \mxm^d \rp = \wid_{i-1}\lp \mxm^{\frac{d}{2}}\rp \les 3 \left\lceil \lp \frac{d}{2}\rp \frac{1}{2^{i-1}} \right\rceil = 3 \left\lceil \frac{d}{2^i} \right\rceil + \end{align} + Furthermore note also the fact that for all $d \in \{3,5,7,...\}$, $i \in \N$ it holds that $\left\lceil \frac{d+1}{2^i} \right\rceil = \left\lceil \frac{d}{2^i}\right\rceil$ and (\ref{9.7.12}) assure that for all $d \in \{3,5,7,...\}$, $i\in \{2,3,...\}$ with $\wid_{i-1} \lp \mxm^{\left\lceil \frac{d}{2}\right\rceil}\rp \les 3 \left\lceil \left\lceil \frac{d}{2}\right\rceil \frac{1}{2^{i-1}}\right\rceil$ it holds that: + \begin{align} + \wid_i \lp \mxm^d \rp = \wid_{i-1} \lp \mxm^{\left\lceil \frac{d}{2}\right\rceil}\rp \les 3 \left\lceil \left\lceil \frac{d}{2} \right\rceil \frac{1}{2^{i-1}} \right\rceil = 3 \left\lceil \frac{d+1}{2^i}\right\rceil = 3 \left\lceil \frac{d}{2^i} \right\rceil + \end{align} + This and (\ref{9.7.16}) tells us that for all $d \in \{3,4,...\}$, $i \in \{2,3,...\}$ with $\forall k \in \{2,3,...,d-1\}$, $j \in \{1,2,...,i-1\}: \wid_j \lp \mxm^k \rp \les 3 \left\lceil \frac{k}{2^j} \right\rceil$ it holds that: + \begin{align} + \wid_i \lp \mxm^d \rp \les 3 \left\lceil \frac{d}{2^i}\right\rceil + \end{align} + This, combined with (\ref{9.7.10}), (\ref{9.7.15}), with induction establishes Item (ii). + + Next observe that (\ref{9.7.6}) tells that for $x = \begin{bmatrix} + x_1 \\ x_2 + \end{bmatrix} \in \R^2$ it becomes the case that: + \begin{align} + \lp\real_{\rect} \lp \mxm^2 \rp \rp \lp x \rp &= \max \{x_1-x_2,0\} + \max\{x_2,0 \} - \max\{ -x_2,0\} \nonumber \\ + &= \max \{x_1-x_2,0\} + x_2 = \max\{x_1,x_2\} + \end{align} + Note next that Lemma \ref{idprop}, Lemma \ref{comp_prop}, and \cite[Proposition~2.19]{grohs2019spacetime} then imply for all $d \in \{2,3,4,...\}$, $x = \{x_1,x_2,...,x_d\} \in \R^d$ it holds that $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp \in C \lp \R^d,\R \rp$. and $\lp \real_{\rect} \lp \mxm^d \rp \rp \lp x \rp = \max\{ x_1,x_2,...,x_d \}$. This establishes Items (iii)-(iv). + + Consider now the fact that Item (ii) implies that the layer architecture forms a geometric series whence we have that the number of bias parameters is bounded by: + \begin{align} + \frac{\frac{3d}{2} \lp 1 - \lp \frac{1}{2} \rp^{\left\lceil \log_2 \lp d\rp\right\rceil +1} \rp }{\frac{1}{2}} &= 3d \lp 1 - \frac{1}{2}^{\left\lceil \log_2 \lp d \rp \right\rceil +1}\rp \nonumber \\ + &\les \left\lceil 3d \lp 1 - \frac{1}{2}^{\left\lceil \log_2 \lp d \rp \right\rceil +1}\rp \right\rceil + \end{align} + For the weight parameters, consider the fact that our widths follow a geometric series with ratio $\frac{1}{2}$, and considering that we have an upper bound for the number of hidden layers, and the fact that $\wid_0 \lp \mxm^d\rp = d$, would then tell us that the number of weight parameters is bounded by: + \begin{align} + &\sum^{\left\lceil \log_2\lp d\rp\right \rceil}_{i=0} \lb \lp \frac{1}{2}\rp ^i \cdot \wid_0\lp \mxm^d\rp \cdot \lp \frac{1}{2}\rp^{i+1}\cdot \wid_0 \lp \mxm^d\rp\rb \nonumber \\ + &= \sum^{\left\lceil \log_2\lp d\rp\right\rceil}_{i=0} \lb \lp \frac{1}{2}\rp^{2i+1}\lp \wid_0 \lp \mxm^d\rp\rp^2\rb \nonumber \\ + &= \frac{1}{2} \sum^{\left\lceil \log_2 \lp d\rp \right\rceil}_{i=0} \lb \lp \lp \frac{1}{2}\rp^{i} \wid_0\lp \mxm^d\rp\rp^2\rb + = \frac{1}{2} \sum^{\left\lceil \log_2\lp d\rp\right\rceil}_{i=0} \lb \lp \frac{1}{4}\rp^id^2\rb + \end{align} + Notice that this is a geometric series with ratio $\frac{1}{4}$, which would then reveal that: + \begin{align} + \frac{1}{2} \sum^{\left\lceil \log_2\lp d\rp\right\rceil}_{i=0} \lb \lp \frac{1}{4}\rp^id^2\rb \les \frac{2}{3} d^2\lp 1- \frac{1}{2}^{2\lp \left\lceil \log_2(d)\right\rceil + 1\rp}\rp + \end{align} + + Thus, we get that: + \begin{align} + \param \lp \mxm^d\rp &\les \frac{2}{3} d^2\lp 1- \frac{1}{2}^{2\lp \left\lceil \log_2(d)\right\rceil \rp + 1}\rp + \left\lceil 3d \lp 1 - \frac{1}{2}^{\left\lceil \log_2 \lp d \rp \right\rceil +1}\rp \right\rceil \nonumber\\ + &\les \frac{2}{3} d^2\lp 1- \frac{1}{2}^{2\lp \left\lceil \log_2(d)\right\rceil \rp + 1}\rp + \left\lceil 3d \lp 1 - \frac{1}{2}^{2\lp\left\lceil \log_2 \lp d \rp \right\rceil +1\rp}\rp\right\rceil\\ + &\les \left\lceil \lp \frac{2}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{2\lp \left\lceil \log_2\lp d\rp\right\rceil+1 \rp}\rp + 1 \right\rceil + \end{align} + + This proves Item (v). + + + Item (vi) is a straightforward consequence of Item (i). This completes the proof of the lemma. +\end{proof} + +\subsection{The $\mathsf{MC}$ Neural Network and Approximations via Maximum Convolutions } + +Let $f: [a,b] \rightarrow \R$ be a continuous bounded function with Lipschitz constant $L$. Let $x_0 \les x_1 \les \cdots \les x_N$ be a set of sample points within $[a,b]$, with it being possibly the case that that for all $i \in \{0,1,\hdots, N\}$, $x_i \sim \unif([a,b])$. For all $i \in \{0,1,\hdots, N\}$, define a series of functions $f_0,f_1,\hdots f_N: [a,b] \rightarrow \R$, as such: +\begin{align} + f_i = f(x_i) - L \cdot \left| x-x_i\right| +\end{align} +We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \textit{maximum convolution approximation}. This converges to $f$, as shown in +\begin{lemma}\label{(9.7.5)}\label{lem:mc_prop} + Let $d,N\in \N$, $L\in \lb 0,\infty \rp$, $x_1,x_2,\hdots, x_N \in \R^d$, $y = \lp y_1,y_2,\hdots,y_N \rp \in \R^N$ and $\mathsf{MC} \in \neu$ satisfy that: + \begin{align}\label{9.7.20} + \mathsf{MC}^{N,d}_{x,y} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \rp \bullet \cpy_{N,d} + \end{align} + It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\inn \lp \mathsf{MC} \rp = d$ + \item $\out\lp \mathsf{MC} \rp = 1$ + \item $\hid \lp \mathsf{MC} \rp = \left\lceil \log_2 \lp N \rp \right\rceil +1$ + \item $\wid_1 \lp \mathsf{MC} \rp = 2dN$ + \item for all $i \in \{ 2,3,...\}$ we have $\wid_1 \lp \mathsf{MC} \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil$ + \item it holds for all $x \in \R^d$ that $\lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp = \max_{i \in \{1,2,...,N\}} \lp y_i - L \left\| x-x_i \right\|_1\rp$ + \item it holds that $\param \lp \mathsf{MC} \rp \les \left\lceil \lp \frac{2}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{2\lp \left\lceil \log_2\lp d\rp\right\rceil+1 \rp}\rp + 1 \right\rceil + 7N^2d^2 + 3\left\lceil \frac{N}{2}\right\rceil \cdot 2dN$ + \end{enumerate} +\end{lemma} +\begin{proof} + Throughout this proof let $\mathsf{S}_i \in \neu$ satisfy for all $i \in \{1,2,...,N\}$ that $\mathsf{S}_i = \nrm_1^d \bullet \aff_{\mathbb{I}_d,-x_i}$ and let $\mathsf{X} \in \neu$ satisfy: + \begin{align} + \mathsf{X} = \aff_{-L\mathbb{I}_N,y} \bullet \lp \lb \boxminus_{i=1}^N \mathsf{S}_i \rb \rp \bullet \cpy_{N,d} + \end{align} + Note that (\ref{9.7.20}) and Lemma \ref{comp_prop} tells us that $\out \lp \R \rp = \out \lp \mxm^N \rp = 1$ and $\inn \lp \mathsf{MC} \rp = \inn \lp \cpy_{N,d} \rp =d $. This proves Items (i)--(ii). Next observe that since it is the case that $\hid \lp \cpy_{N,d} \rp$ and $\hid \lp \nrm^d_1 \rp = 1$, Lemma \ref{comp_prop} then tells us that: + \begin{align} + \hid \lp \mathsf{X} \rp = \hid \lp\aff_{-L\mathbb{I}_N,y} \rp + \hid \lp \boxminus_{i=1}^N \mathsf{S}_i\rp + \hid \lp \cpy_{N,d} \rp = 1 + \end{align} + Thus Lemma \ref{comp_prop} and Lemma \ref{9.7.4} then tell us that: + \begin{align} + \hid \lp \mathsf{MC} \rp = \hid \lp \mxm^N \bullet \mathsf{X}\rp = \hid \lp \mxm^N \rp + \hid \lp \mathsf{X}\rp = \left\lceil \log_2 \lp N \rp \right\rceil +1 + \end{align} + Which in turn establishes Item (iii). + + Note next that Lemma \ref{comp_prop} and \cite[Proposition~2.20]{grohs2019spacetime} tells us that: + \begin{align}\label{8.3.33} + \wid_1 \lp \mathsf{MC} \rp = \wid_1 \lp \mathsf{X} \rp = \wid_1 \lp \boxminus^N_{i=1} \mathsf{S}_i\rp = \sum^N_{i=1} \wid_1 \lp \mathsf{S}_i \rp = \sum^N_{i=1} \wid_1 \lp \nrm^d_1 \rp = 2dN + \end{align} + This establishes Item (iv). + + Next observe that the fact that $\hid \lp \mathsf{X} \rp=1$, Lemma \ref{comp_prop} and Lemma \ref{9.7.4} tells us that for all $i \in \{2,3,...\}$ it is the case that: + \begin{align} + \wid_i \lp \mathsf{MC} \rp = \wid_{i-1} \lp \mxm^N \rp \les 3 \left\lceil \frac{N}{2^{i-1}} \right\rceil + \end{align} + This establishes Item (v). + + Next observe that Lemma \ref{9.7.2} and Lemma \ref{5.3.3} tells us that for all $x \in \R^d$, $i \in \{1,2,...,N\}$ it holds that: + \begin{align} + \lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp - \lp \real_{\rect}\lp \nrm^d_1 \rp \circ \real_{\rect}\lp \aff_{\mathbb{I}_d,-x_i} \rp \rp \lp x \rp = \left\| x-x_i \right\|_1 + \end{align} + This an \cite[Proposition~2.20]{grohs2019spacetime} combined establishes that for all $x \in \R^d$ it holds that: + \begin{align} + \lp \real_{\rect} \lp \lb \boxminus_{i=1}^N \mathsf{S}_i \rb \bullet \cpy_{N,d} \rp \rp \lp x \rp = \lp \| x-x_1 \|_1, \|x-x_2\|_1,...,\|x-x_N\|_1\rp \nonumber \\ + \end{align} + This and Lemma \ref{5.3.3} establishes that for all $x \in \R^d$ it holds that: + \begin{align} + \lp \real_{\rect}\lp \mathsf{X}\rp \rp \lp x \rp &= \lp \real_{\rect}\lp \aff_{-L\mathbb{I}_N,y}\rp\rp \circ \lp\real_{\rect} \lp \lb \boxminus_{i=1}^N \mathsf{S}_i\rb \bullet \cpy_{N,d}\rp \rp \lp x \rp \nonumber\\ + &= \lp y_1-L \|x-x_1 \|, y_2-L\|x-x_2\|,...,y_N-L \| x-x_N \|_1\rp + \end{align} + Then Lemma \ref{comp_prop} and Lemma \ref{9.7.4} tells us that for all $x\in \R^d$ it holds that: + \begin{align} + \lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp &= \lp \real_{\rect}\lp \mxm^N \rp \circ \lp \real_{\rect}\lp \mathsf{X} \rp \rp \rp \lp x \rp \nonumber \\ + &= \lp \real_{\rect}\lp \mxm^N \rp \rp \lp y_1-L \|x-x_1\|_1,y_2-L\|x-x_2\|_1,...,y_N-L\|x-x_N\|_1\rp \nonumber\\ + &=\max_{i\in \{1,2,...,N\} } \lp y_i - L \|x-x_i\|_1\rp + \end{align} + This establishes Item (vi). + + For Item (vii) note that Lemma \ref{lem:nrm_prop}, Remark \ref{rem:stk_remark}, Lemma \ref{lem:nrm_prop}, and Corollary \ref{affcor} tells us that for all $d\in \N$ and $x \in \R^d$ it is the case that: + \begin{align} + \param \lp \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x}\rp \les \param \lp \nrm_1^d\rp \les 7d^2 + \end{align} + This, along with Corollary \ref{cor:sameparal}, and because we are stacking identical neural networks, then tells us that for all $N \in \N$, it is the case that: + \begin{align} + \param \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \les 7N^2d^2 + \end{align} + Observe next that Corollary \ref{affcor} tells us that for all $d,N \in \N$ and $x \in \R^d$ it is the case that: + \begin{align}\label{8.3.38} + \param \lp \lp \boxminus^N_{i=1} \lb \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x}\rb\rp \bullet \cpy_{N,d}\rp \les \param \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \les 7N^2d^2 + \end{align} + Now, let $d,N \in \N$, $L \in [0,\infty)$, let $x_1,x_2,\hdots, x_N \in \R^d$ and let $y = \{y_1,y_2,\hdots, y_N \} \in \R^N$. Observe that again, Corollary \ref{affcor}, and (\ref{8.3.38}) tells us that: + \begin{align} + \param\lp \aff_{-L\mathbb{I}_N,y} \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \rp \bullet \cpy_{N,d}\rp \nonumber\\ \les \param \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \les 7N^2d^2 \nonumber + \end{align} + Finally Lemma \ref{comp_prop}, (\ref{8.3.33}), and Lemma \ref{lem:mxm_prop} yields that: + \begin{align} + \param(\mathsf{MC}) &= \param \lp \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \rp \bullet \cpy_{N,d} \rp \nonumber\\ + &\les \param \lp \mxm^N \bullet \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb \rp \rp \nonumber\\ + &\les \param \lp \mxm^N \rp + \param \lp \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \rp + \nonumber\\ &\wid_1\lp \mxm^N\rp \cdot \wid_{\hid \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp} \lp \boxminus_{i=1}^N \lb \nrm^d_1\bullet \aff_{\mathbb{I}_d, -x} \rb\rp \nonumber \\ + &\les \left\lceil \lp \frac{2}{3}d^2+3d\rp \lp 1+\frac{1}{2}^{2\lp \left\lceil \log_2\lp d\rp\right\rceil+1 \rp}\rp + 1 \right\rceil + 7N^2d^2 + 3\left\lceil \frac{N}{2}\right\rceil \cdot 2dN + \end{align} +\end{proof} +\begin{remark} + We may represent the neural network diagram for $\mxm^d$ as: +\end{remark} +\begin{figure}[h] + \begin{center} + + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,560); %set diagram left start at 0, and has height of 560 + +%Shape: Rectangle [id:dp1438938274656144] +\draw (574,235) -- (644,235) -- (644,275) -- (574,275) -- cycle ; +%Straight Lines [id:da7383135897500558] +\draw (574,241) -- (513.72,84.37) ; +\draw [shift={(513,82.5)}, rotate = 68.95] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da09141712653411305] +\draw (572,251) -- (514.14,168.14) ; +\draw [shift={(513,166.5)}, rotate = 55.08] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da19953508691566213] +\draw (573,259) -- (515.07,350.81) ; +\draw [shift={(514,352.5)}, rotate = 302.25] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da5900315817761441] +\draw (575,268) -- (515.66,436.61) ; +\draw [shift={(515,438.5)}, rotate = 289.39] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp9847868081693099] +\draw (421,59) -- (512,59) -- (512,99) -- (421,99) -- cycle ; +%Shape: Rectangle [id:dp2868551357079474] +\draw (419,330) -- (510,330) -- (510,370) -- (419,370) -- cycle ; +%Shape: Rectangle [id:dp9383613429980815] +\draw (420,150) -- (511,150) -- (511,190) -- (420,190) -- cycle ; +%Shape: Rectangle [id:dp5827241951133133] +\draw (420,420) -- (511,420) -- (511,460) -- (420,460) -- cycle ; +%Shape: Rectangle [id:dp7299058955170046] +\draw (290,60) -- (381,60) -- (381,100) -- (290,100) -- cycle ; +%Shape: Rectangle [id:dp08440877870624452] +\draw (290,150) -- (381,150) -- (381,190) -- (290,190) -- cycle ; +%Shape: Rectangle [id:dp7098854649776141] +\draw (290,330) -- (381,330) -- (381,370) -- (290,370) -- cycle ; +%Shape: Rectangle [id:dp6165394921489369] +\draw (290,420) -- (381,420) -- (381,460) -- (290,460) -- cycle ; +%Straight Lines [id:da37215648665173995] +\draw (420,80) -- (401,80) -- (382,80) ; +\draw [shift={(380,80)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da2316129338082229] +\draw (420,170) -- (401,170) -- (382,170) ; +\draw [shift={(380,170)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da40267704179559083] +\draw (419,350) -- (400,350) -- (381,350) ; +\draw [shift={(379,350)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da5321116904741454] +\draw (420,440) -- (401,440) -- (382,440) ; +\draw [shift={(380,440)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp9652335934440622] +\draw (170,60) -- (250,60) -- (250,460) -- (170,460) -- cycle ; +%Straight Lines [id:da2568661285688787] +\draw (170,240) -- (132,240) ; +\draw [shift={(130,240)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da9075024165320872] +\draw (290,80) -- (252,80) ; +\draw [shift={(250,80)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da36733568107592385] +\draw (290,170) -- (252,170) ; +\draw [shift={(250,170)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da027221622247677213] +\draw (290,350) -- (252,350) ; +\draw [shift={(250,350)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da456971589533403] +\draw (290,440) -- (252,440) ; +\draw [shift={(250,440)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp5834320101871477] +\draw (60,220) -- (130,220) -- (130,260) -- (60,260) -- cycle ; +%Straight Lines [id:da39697402951042593] +\draw (60,240) -- (22,240) ; +\draw [shift={(20,240)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da09195032177210305] +\draw (690,250) -- (652,250) ; +\draw [shift={(650,250)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (583,245.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{N}{}_{,d}$}; +% Text Node +\draw (441,66.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{I}}{}_{_{d}}{}_{-x}{}_{_{i}}$}; +% Text Node +\draw (442,158.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{I}}{}_{_{d}}{}_{-x}{}_{_{i}}$}; +% Text Node +\draw (442,338.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{I}}{}_{_{d}}{}_{-x}{}_{_{i}}$}; +% Text Node +\draw (442,428.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{I}}{}_{_{d}}{}_{-x}{}_{_{i}}$}; +% Text Node +\draw (318,72.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Nrm}_{1}^{d}$}; +% Text Node +\draw (318,159.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Nrm}_{1}^{d}$}; +% Text Node +\draw (318,339.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Nrm}_{1}^{d}$}; +% Text Node +\draw (321,427.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Nrm}_{1}^{d}$}; +% Text Node +\draw (322,237.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $}; +% Text Node +\draw (462,232.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $}; +% Text Node +\draw (181,238.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{-L}{}_{\mathbb{I}}{}_{_{N} ,y}$}; +% Text Node +\draw (71,231.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Mxm}^{N}$}; + + +\end{tikzpicture} + + + \end{center} + \caption{Neural network diagramfor the $\mxm$ network} +\end{figure} + +\subsection{Lipschitz Function Approximations}\label{(9.7.6)} +\begin{lemma}%TODO: Should we stipulate compact sets? + Let $\lp E,d \rp$ be a metric space. Let $L \in \lb 0,\infty \rp$, $D \subseteq E$, $\emptyset \neq C \subseteq D$. Let $f:D \rightarrow \R$ satisfy for all $x\in D$, $y \in C$ that $\left| f(x) -f(y)\right| \les L d \lp x,y \rp$, and let $F:E \rightarrow \R \cup \{\infty\}$ satisfy for all $x\in E$ that: + \begin{align}\label{9.7.30} + F\lp x \rp = \sup_{y\in C} \lb f\lp y \rp - Ld\lp x,y \rp \rb + \end{align} + It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for all $x \in C$ that $F(x) = f(x)$ + \item it holds for all $x \in D$, that $F(x) \les f(x)$ + \item it holds for all $x\in E$ that $F\lp x \rp < \infty$ + \item it holds for all $x,y \in E$ that $\left| F(x)-F(y)\right| \les Ld\lp x,y \rp$ and, + \item it holds for all $x \in D$ that: + \begin{align}\label{9.7.31} + \left| F\lp x \rp - f \lp x \rp \right| \les 2L \lb \inf_{y\in C} d \lp x,y \rp\rb + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + The assumption that $\forall x \in D, y \in C: \left| f(x) - f(y)\right| \les Ld\lp x,y \rp$ ensures that: + \begin{align}\label{9.7.32} + f(y) - Ld\lp x,y\rp \les f\lp x \rp \les f(y) + Ld\lp x,y \rp + \end{align} + For $x\in D$, it then renders as: + \begin{align}\label{9.7.33} + f(x) \ges \sup_{y \in C} \lb f(y) - Ld\lp x,y \rp \rb + \end{align} + This establishes Item (i). Note that (\ref{9.7.31}) then tells us that for all $x\in C$ it holds that: + \begin{align} + F\lp x \rp \ges f(x) - Ld\lp x,y \rp = f\lp x \rp + \end{align} + This with (\ref{9.7.33}) then yields Item (i). + + Note next that (\ref{9.7.32}, with $x \curvearrowleft y \text{ and } y \curvearrowleft z)$ and the triangle inequality ensure that for all $x \in E$, $y,z \in C$ it holds that: + \begin{align} + f(y) - Ld\lp x,y\rp \les f(z)+Ld\lp y,z \rp - Ld\lp x,y \rp \les f(z) + Ld\lp x,z \rp + \end{align} + We then obtain for all $x\in E, z\in C$ it holds that: + \begin{align} + F\lp x \rp = \sup_{y\in C} \lb f(y) - Ld\lp x,y \rp \rb \les f\lp x \rp + Ld\lp x,z \rp < \infty + \end{align} + This proves Item (iii). Item (iii), (\ref{9.7.30}), and the triangle inequality then shows that for all $x,y \in E$, it holds that: + \begin{align} + F(x) - F(y) &= \lb \sup_{v \in C} \lp f(v) - Ld\lp x,v \rp \rp \rb - \lb \sup_{w\in C} \lp f(w)-Ld\lp y,w \rp \rp\rb \nonumber \\ + &= \sup_{v \in C}\lb f(v) - Ld\lp x,v \rp -\sup_{w\in C} \lp f(w) - L d\lp y,w \rp \rp\rb \nonumber\\ + &\les \sup_{v \in C}\lb f(v) - Ld\lp x,v \rp - \lp f(v) - Ld\lp y,w \rp \rp\rb \nonumber\\ + &= \sup_{v\in C} \lp Ld\lp y,v \rp + Ld\lp x,v \rp -Ld\lp x,v\rp \rp = Ld \lp x,y \rp + \end{align} + This establishes Item (v). Finally, note that Items (i) and (iv), the triangle inequality, and the assumption that $\forall x \in D, y\in C: \left| f(x) - f(y) \right| \les Ld\lp x,y \rp$ ensure that for all $x\in D$ it holds that: + \begin{align} + \left| F(x) - f(x) \right| &= \inf_{y\in C} \left| F(x) - F(y) +f(y) - f(x)\right| \nonumber \\ + &\les \inf_{y\in C} \lp \left| F(x) - F(y) \right| + \left| f(y) - f(x) \right|\rp \nonumber\\ + &\les \inf_{y\in C} \lp 2Ld\lp x,y \rp\rp = 2L \lb \inf_{y\in C} d \lp x,y \rp \rb + \end{align} + This establishes Item (v) and hence establishes the Lemma. +\end{proof} +\begin{corollary}\label{9.7.6.1} + Let $\lp E,d \rp$ be a metric space, let $L \in \lb 0,\infty \rp$, $\emptyset \neq C \subseteq E$, let $f: E \rightarrow \R$ satisfy for all $x\in E$, $y \in C$ that $\left\| f(x) - f(y) \right| \les Ld \lp x,y \rp$, and let $F:E \rightarrow \R \cup \{\infty\}$ satisfy for all $x\in E$ that: + \begin{align} + F \lp x \rp = \sup_{y\in C} \lb f(y) - Ld \lp x,y \rp\rb + \end{align} + It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for all $x\in C$ that $F(x) = f(x)$ + \item for all $x\in E$ that $F(x) \les f(x)$ + \item for all $x,y \in E$ that $\left| F(x) - f(y) \right| \les L d \lp x,y \rp$ and + \item for all $x\in E$ that: \begin{align} + \left| F\lp x \rp - f\lp x \rp \right| \les 2L \lb \inf_{y\in C} d \lp x,y \rp \rb + \end{align} + \end{enumerate} +\end{corollary} +\begin{proof} + Note that Lemma \ref{(9.7.6)} establishes Items (i)\textemdash(iv). +\end{proof} +\subsection{Explicit ANN Approximations } + +\begin{lemma} + Let $d,N \in \N$, $L \in \lb 0,\infty \rp$. Let $E \subseteq \R^d$. Let $x_1,x_2,...,x_N \in E$, let $f:E \rightarrow \R$ satisfy for all $x_1,y_1 \in E$ that $\left| f(x_1) -f(y_1)\right| \les L \left\| x_1-x_2 \right\|_1$ and let $\mathsf{MC} \in \neu$ and $y = \lp f\lp x_1 \rp, f \lp x_2 \rp,...,f\lp x_N \rp\rp$ satisfy: + \begin{align} + \mathsf{MC} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lb \boxminus^N_{i=1} \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \bullet \cpy_{N,d} + \end{align} + It is then the case that: + \begin{align}\label{(9.7.42)} + \sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC} \rp \rp \lp x \rp -f\lp x \rp \right| \les 2L \lb \sup _{x\in E} \lp \min_{i\in \{1,2,...,N\}} \left\| x-x_i\right\|_1\rp\rb + \end{align} +\end{lemma} +\begin{proof} + Throughout this proof let $F: \R^d \rightarrow \R$ satisfy that: + \begin{align}\label{9.7.43} + F\lp x \rp = \max_{i \in \{1,2,...,N\}} \lp f\lp x_i \rp- L \left\| x-x_i \right\|_1 \rp + \end{align} + Note then that Corollary \ref{9.7.6.1}, (\ref{9.7.43}), and the assumption that for all $x,y \in E$ it holds that $\left| f(x) - f(y)\right| \les L \left\|x-y \right\|_1$ assures that: + \begin{align}\label{(9.7.44)} + \sup_{x\in E} \left| F(x) - f(x) \right| \les 2L \lb \sup_{x\in E} \lp \min_{i \in \{1,2,...,N\}} \left\| x-x_i\right\|_1\rp\rb + \end{align} + Then Lemma \ref{(9.7.5)} tells us that for all $x\in E$ it holds that $F(x) = \lp \real_{\rect} \lp \mathsf{MC} \rp \rp \lp x \rp$. This combined with (\ref{(9.7.44)}) establishes (\ref{(9.7.42)}). +\end{proof} +\begin{lemma} + Let $d,N \in \N$, $L \in \lb 0,\infty \rp$. Let $E \subseteq \R^d$. Let $x_1,x_2,...,x_N \in E$, let $f:E \rightarrow \R$ satisfy for all $x_1,x_2 \in E$ that $\left| f(x_1) -f(x_2)\right| \les L \left\| x_1-x_2 \right\|_1$ and let $\mathsf{MC} \in \neu$ and $y = \lp f\lp x_1 \rp, f \lp x_2 \rp,...,f\lp x_N \rp\rp$ satisfy: + \begin{align} + \mathsf{MC} = \mxm^N \bullet \aff_{-L\mathbb{I}_N,y} \bullet \lb \boxminus^N_{i=1} \nrm^d_1 \bullet \aff_{\mathbb{I}_d,-x_i} \rb \bullet \cpy_{N,d} + \end{align} +It is then the case that: + \begin{align} + \lim_{N \rightarrow \infty} \lb \mathbb{P} \lp \sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC} \rp \rp \lp x \rp -f\lp x \rp \right| >0 \rp\rb \rightarrow 0 + \end{align} +\end{lemma} +\begin{proof} + Note that $N$ is chosen uniformly from set $E \subseteq \R^d$. Note next that (\ref{(9.7.44)}) tells us that: + \begin{align} + &\lim_{N \rightarrow \infty} \lb \mathbb{P} \lp \sup_{x\in E} \left| \lp \real_{\rect}\lp \mathsf{MC} \rp \rp \lp x \rp -f\lp x \rp \right| >0 \rp\rb \nonumber \\ + &\les + \end{align} + +\end{proof} + + + + + + + + + + + + + + + + + + + + diff --git a/Dissertation/ann_product.aux b/Dissertation/ann_product.aux new file mode 100644 index 0000000..b0db3e5 --- /dev/null +++ b/Dissertation/ann_product.aux @@ -0,0 +1,143 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{\numberline {6}ANN Product Approximations}{95}{chapter.6}\protected@file@percent } +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\@writefile{toc}{\contentsline {section}{\numberline {6.1}Approximation for simple products}{95}{section.6.1}\protected@file@percent } +\newlabel{lem:6.1.1}{{6.1.1}{95}{}{theorem.6.1.1}{}} +\newlabel{lem:6.1.1@cref}{{[lemma][1][6,1]6.1.1}{[1][95][]95}} 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b/Dissertation/ann_product.pdf differ diff --git a/Dissertation/ann_product.tex b/Dissertation/ann_product.tex new file mode 100644 index 0000000..5271fe2 --- /dev/null +++ b/Dissertation/ann_product.tex @@ -0,0 +1,1640 @@ +\chapter{ANN Product Approximations} +\section{Approximation for Products of Two Real Numbers} +We will build up the tools necessary to approximate $e^x$ via neural networks in the framework described in the previous sections. While much of the foundation comes from, e.g., \cite{grohs2019spacetime} way, we will, along the way, encounter neural networks not seen in the literature, such as the $\tay$, $\pwr$, $\tun$, and finally a neural network approximant for $e^x$. For each of these neural networks, we will be concerned with at least the following: +\begin{enumerate}[label = (\roman*)] + \item whether their instantiations using the ReLU function (often just continuous functions) are continuous. + \item whether their depths are bounded, at most polynomially, on the type of accuracy we want, $\ve$. + \item whether their parameter estimates are bounded at most polynomially on the type of accuracy we want, $\ve$. + \item The accuracy of our neural networks. +\end{enumerate} +\subsection{The squares of real numbers in $\lb 0,1 \rb$} + +\begin{definition}[The $\mathfrak{i}_d$ Network]\label{def:mathfrak_i} + For all $d \in \N$ we will define the following set of neural networks as ``activation neural networks'' denoted $\mathfrak{i}_d$ as: + \begin{align} + \mathfrak{i}_d = \lp \lp \mathbb{I}_d, \mymathbb{0}_d\rp, \lp \mathbb{I}_d, \mymathbb{0}_d\rp \rp + \end{align} +\end{definition} +\begin{lemma}\label{lem:mathfrak_i} + Let $d \in \N$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \mathfrak{i}_4\rp \in C \lp \R^d, \R^d\rp$. + \item $\lay \lp \mathfrak{i}_d\rp = \lp d,d,d\rp$ + \item $\param \lp \mathfrak{i}_4\rp = 2d^2+2d$ + \end{enumerate} +\end{lemma} +\begin{proof} + Item (i) is straightforward from the fact that for all $d \in \N$ it is the case that $\real_{\rect} \lp \mathfrak{i}_d\rp = \mathbb{I}_d\lp \real_{\rect} \lp \lb \mathbb{I}_d\rb_*\rp + \mymathbb{0}_d\rp + \mymathbb{0}_d$. Item (ii) is straightforward from the fact that $\mathbb{I}_d \in \R^{d \times d}$. We realize Item (iii) by observation. +\end{proof} + + +\begin{lemma}\label{lem:6.1.1}\label{lem:phi_k} + Let $\lp c_k \rp _{k \in \N} \subseteq \R$, $\lp A_k \rp _{k \in \N} \in \R^{4 \times 4},$ $\mathbb{B}\in \R^{4 \times 1}$, $\lp C_k \rp _{k\in \N}$ satisfy for all $k \in \N$ that: + \begin{align}\label{(6.0.1)} + A_k = \begin{bmatrix} + 2 & -4 &2 & 0 \\ + 2 & -4 & 2 & 0\\ + 2 & -4 & 2 & 0\\ + -c_k & 2c_k & -c_k & 1 + \end{bmatrix} \quad B=\begin{bmatrix} + 0 \\ -\frac{1}{2} \\ -1 \\ 0 + \end{bmatrix} \quad C_k = \begin{bmatrix} + -c_k & 2c_k &-c_k & 1 + \end{bmatrix} + \end{align} + and that: + \begin{align} + c_k = 2^{1-2k} + \end{align} + Let $\Phi_k \in \neu$, $k\in \N$ satisfy for all $k \in [2,\infty) \cap \N$ that $\Phi_1 = \lp \aff_{C_1,0} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B}$, that for all $d \in \N$, $\mathfrak{i}_d = \lp \lp \mathbb{I}_d, \mymathbb{0}_d \rp, \lp \mathbb{I}_d, \mymathbb{0}_d \rp \rp$ and that: + \begin{align} + \Phi_k =\lp \aff_{C_k,0}\bullet \mathfrak{i}_4 \rp \bullet \lp \aff_{A_{k-1},B} \bullet \mathfrak{i}_4\rp \bullet \cdots \bullet \lp \aff_{A_1,B} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B} + \end{align} + It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for all $k \in \N$, $x \in \R$ we have $\real_{\rect}\lp \Phi_k\rp\lp x \rp \in C \lp \R, \R \rp $ + \item for all $k \in \N$ we have $\lay \lp \Phi_k \rp = \lp 1,4,4,...,4,1 \rp \in \N^{k+2}$ + \item for all $k \in \N$, $x \in \R \setminus \lb 0,1 \rb $ that $\lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp = \rect \lp x \rp$ + \item for all $k \in \N$, $x \in \lb 0,1 \rb$, we have $\left| x^2 - \lp \real_{\rect} \lp \xi_k \rp \rp \lp x \rp \right| \les 2^{-2k-2}$, and + \item for al $k \in \N$ , we have that $\param \lp \Phi_k \rp = 20k-7$ + \end{enumerate} +\end{lemma} +\begin{proof} + Let $g_k: \R \rightarrow \lb 0,1 \rb$, $k \in \N$ be the functions defined as such, satisfying for all $k \in \N$, $x \in \R$ that: + \begin{align}\label{(6.0.3)} + g_1 \lp x \rp &= \begin{cases} + 2x & : x \in \lb 0,\frac{1}{2} \rp \\ + 2-2x &: x\in \lb \frac{1}{2},1\rb \\ + 0 &: x \in \R \setminus \lb 0,1 \rb + \end{cases} \\ + g_{k+1} &= g_1(g_{k}) \nonumber + \end{align} + and let $f_k: \lb 0,1 \rb \rightarrow \lb 0,1 \rb$, $k \in \N_0$ be the functions satisfying for all $k \in \N_0$, $n \in \{0,1,...,2^k-1\}$, $x \in \lb \frac{n}{2^k}, \frac{n+1}{2^k} \rp$ that $f_k(1)=1$ and: + \begin{align}\label{(6.0.4.2)} + f_k(x) = \lb \frac{2n+1}{2^k} \rb x-\frac{n^2+n}{2^{2k}} + \end{align} + and let $r_k = \lp r_{k,1},r_{k,2},r_{k,3},r_{k,4} \rp: \R \rightarrow \R^4$, $k \in \N$ be the functions which which satisfy for all $x \in \R$, $k \in \N$ that: + \begin{align}\label{(6.0.5)} + r_1\lp x \rp &= \begin{bmatrix} + r_{1,1}(x) \\ r_{2,1}(x) \\ r_{3,1}(x) \\ r_{4,1}(x) + \end{bmatrix}= \rect \lp \begin{bmatrix} + x \\ x-\frac{1}{2} \\ x-1 \\ x + \end{bmatrix} \rp \\ + r_{k+1} &= A_{k+1}r_k(x) \nonumber + \end{align} + Note that since it is the case that for all $x \in \R$ that $\rect(x) = \max\{x,0\}$, (\ref{(6.0.3)}) and (\ref{(6.0.5)}) shows that it holds for all $x \in \R$ that: + \begin{align}\label{6.0.6} + 2r_{1,1}(x) -4r_{2,1}(x) + 2r_{3,1}(x) &= 2 \rect(x) -4\rect \lp x-\frac{1}{2}\rp+2\rect\lp x-1\rp \nonumber \\ + &= 2\max\{x,0\} -4\max\left\{x-\frac{1}{2} ,0\right\}+2\max\{x-1,0\} \nonumber \\ + &=g_1(x) + \end{align} + Note also that combined with (\ref{(6.0.4.2)}), the fact that for all $x\in [0,1]$ it holds that $f_0(x) = x = \max\{x,0\}$ tells us that for all $x \in \R$: + \begin{align}\label{6.0.7} + r_{4,1}(x) = \max \{x,0\} = \begin{cases} + f_0(x) & :x\in [0,1] \\ + \max\{x,0\}& :x \in \R \setminus \lb 0,1\rb + \end{cases} + \end{align} + We next claim that for all $k \in \N$, it is the case that: + \begin{align}\label{6.0.8} + \lp \forall x \in \R : 2r_{1,k}(x)-4r_{2,k}(x) + 2r_{3,k}(x) =g(x) \rp + \end{align} + and that: + \begin{align}\label{6.0.9} + \lp \forall x \in \R: r_{4,k} (x) = \begin{cases} + f_{k-1}(x) & :x \in \lb 0,1 \rb \\ + \max\{x,0\} & : x \in \R \setminus \lb 0,1\rb + \end{cases} \rp + \end{align} + We prove (\ref{6.0.8}) and (\ref{6.0.9}) by induction. The base base of $k=1$ is proved by (\ref{6.0.6}) and (\ref{6.0.7}). For the induction step $\N \ni k \rightarrow k+1$ assume there does exist a $k \in \N$ such that for all $x \in \R$ it is the case that: + \begin{align} + 2r_{1,k}(x) - 4r_{2,k}(x) + 2r_{3,k}(x) = g_k(x) + \end{align} + and: + \begin{align}\label{6.0.11} + r_{4,k}(x) = \begin{cases} + f_{k-1}(x) & : x \in [0,1] \\ + \max\{x,0\} &: x \in \R \setminus \lb 0,1 \rb + \end{cases} + \end{align} + Note that then (\ref{(6.0.3)}),(\ref{(6.0.5)}), and (\ref{6.0.6}) then tells us that for all $x \in \R$ it is the case that: + \begin{align}\label{6.0.12} + g_{k+1}\lp x \rp &= g_1(g_k(x)) = g_1(2r_{1,k}(x)+4r_{2,k}(x) + 2r_{3,k}(x)) \nonumber \\ + &= 2\rect \lp 2r_{1,k}(x)) + 4r_{2,k} +2r_{3,k}(x) \rp \nonumber \\ + &-4\rect \lp 2r_{1,k}\lp x \rp -4r_{2,k}+2r_{3,k}(x) - \frac{1}{2} \rp \nonumber \\ + &+ 2\rect \lp 2r_{1,k} (x) - 4r_{2,k}(x) + 2r_{3,k}(x)-1 \rp \nonumber \\ + &=2r_{1,k+1}(x) -4r_{2,k+1}(x) + 2r_{3,k+1}(x) + \end{align} + In addition note that (\ref{(6.0.4.2)}), (\ref{(6.0.5)}), and (\ref{6.0.7}) tells us that for all $x \in \R$: + %TODO: Ask about the extra powers of 2 and b_k + \begin{align}\label{6.0.13} + r_{4,k+1}(x) &= \rect \lp \lp -2 \rp ^{3-2 \lp k+1 \rp }r_{1,k} \lp x \rp + 2^{4-2 \lp k+1 \rp}r_{2,k} \lp x \rp + \lp -2 \rp^{3-2\lp k+1\rp }r_{3,k} \lp x \rp + r_{4,k} \lp x\rp \rp \nonumber \\ + &= \rect \lp \lp -2 \rp ^{1-2k}r_{1,k} \lp x \rp + 2^{2-2k}r_{k,2}\lp x \rp + \lp -2 \rp ^{1-2k}r_{3,k} \lp x \rp + r_{4,k}\lp x \rp \rp \nonumber \\ + &=\rect \lp 2^{-2k} \lb -2r_{1,k}\lp x \rp + 2^2r_{2,k} \lp x \rp -2r_{3,k} \lp x \rp \rb +r_{4,k}\lp x \rp \rp \nonumber \\ + &= \rect \lp - \lb 2^{-2k} \rb \lb 2r_{1,k}\lp x \rp -4r_{2,k} \lp x \rp +2r_{3,k}\lp x \rp \rb +r_{4,k}\lp x \rp \rp \nonumber \\ + &= \rect\lp -\lb 2^{-2k} \rb g_k \lp x \rp +r_{4,k}\lp x \rp \rp + \end{align} + This and the fact that for all $x\in \R$ it is the case that $\rect \lp x \rp = \max\{x,0\}$, that for all $x\in \lb 0 ,1 \rb$ it is the case that $f_k \lp x \rp \ges 0$, (\ref{6.0.11}), shows that for all $x \in \lb 0,1 \rb$ it holds that: + \begin{align}\label{6.0.14} + r_{4,k+1}\lp x \rp &= \rect \lp -2 \lb 2^{-2k} g_k \rb + f_{k-1}\lp x \rp \rp = \rect \lp -2 \lp 2^{-2k}g_k \lp x \rp \rp +x-\lb \sum^{k-1}_{j=1} \lp 2^{-2j}g_j \lp x \rp \rp \rb \rp \nonumber \\ + &= \rect \lp x - \lb \sum^k_{j=1}2^{-2j}g_j \lp x \rp \rb \rp = \rect \lp f_k \lp x \rp \rp =f_k \lp x \rp + \end{align} + Note next that (\ref{6.0.11}) and (\ref{6.0.13}) then tells us that for all $x\in \R \setminus \lb 0,1\rb$: + \begin{align} + r_{4,k+1}\lp x \rp = \max \left\{ -\lp 2^{-2k}g_x \lp x \rp \rp + r_{4,k}\lp x \rp \right\} = \max\{\max\{x,0\},0\} = \max\{x,0\} + \end{align} + Combining (\ref{6.0.12}) and (\ref{6.0.14}) proves (\ref{6.0.8}) and (\ref{6.0.9}). Note that then (\ref{(6.0.1)}) and (\ref{6.0.8}) assure that for all $k\in \N$, $x\in \R$ it holds that $\real_{\rect} \lp \Phi_k \rp \in C \lp \R,\R \rp$ and that: + \begin{align}\label{(6.0.17)} + &\lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp \nonumber \\ + &= \lp \real_{\rect} \lp \lp \aff_{C_k,0} \bullet \mathfrak{i}_4 \rp \bullet \lp \aff_{A_{k-1},B} \bullet \mathfrak{i}_4 \rp \bullet \cdots \bullet\lp \aff_{A_1,B} \bullet \mathfrak{i}_4 \rp \bullet \aff_{\mymathbb{e}_4,B} \rp \rp \lp x \rp \nonumber \\ + &= \lp -2\rp^{1-2k}r_{1,k}\lp x \rp + 2^{2-2k} r_{2,k} \lp x \rp + \lp -2 \rp ^{1-2k} r_{3,k} \lp x \rp + r_{4,k} \lp x \rp \nonumber \\ + &=\lp -2 \rp ^{2-2k} \lp \lb \frac{r_{1,k}\lp x \rp +r_{3,k} \lp x \rp }{-2} \rb + r_{2,k}\lp x \rp \rp +r_{4,k}\lp x \rp \nonumber \\ + &=2^{2-2k} \lp \lb \frac{r_{1,k}\lp x \rp+r_{3,k} \lp x \rp }{-2} \rb + r_{2,k} \lp x \rp \rp +r_{4,k} \lp x \rp \nonumber \\ + &=2^{-2k}\lp 4r_{2,k} \lp x \rp -2r_{1,k}\lp x \rp -2r_{3,k} \lp x \rp \rp +r_{4,k} \lp x \rp \nonumber \\ + &=-\lb 2^{-2k} \rb \lb 2r_{1,k} \lp x \rp -4r_{2,k} \lp x \rp +2r_{3,k} \lp x \rp \rb +r_{4,k} \lp x \rp = -\lb 2^{-2k} \rb g_k \lp x \rp + r_{4,k} \lp x \rp + \end{align} + This and (\ref{6.0.9}) tell us that: + \begin{align} + \lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp = - \lp 2^{-2k}g_k \lp x \rp \rp +f_{k-1}\lp x \rp &= -\lp 2^{-2k}g_k \lp x \rp \rp +x-\lb \sum^{k-1}_{j=1} 2^{-2j}g_j \lp x \rp \rb \nonumber \\ + &=x-\lb \sum^k_{j=1}2^{-2j}g_j \lp x \rp \rb =f_k\lp x\rp \nonumber + \end{align} + Which then implies for all $k\in \N$, $x \in \lb 0,1\rb$ that it holds that: + \begin{align} + \left\| x^2-\lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp \right\| \les 2^{-2k-2} + \end{align} + This, in turn, establishes Item (i). + + Finally observe that (\ref{(6.0.17)}) then tells us that for all $k\in \N$, $x \in \R \setminus \lb 0,1\rb$ it holds that: + \begin{align} + \lp \real_{\rect} \lp \Phi_k \rp \rp \lp x \rp = -2^{-2k}g_k \lp x \rp +r_{4,k} \lp x \rp =r_{4,k} \lp x \rp = \max\{x,0\} = \rect(x) + \end{align} + This establishes Item(iv). Note next that Item(iii) ensures for all $k\in \N$ that $\dep\lp \xi_k \rp = k+1$, and: + \begin{align} + \param \lp \Phi_k \rp = 4(1+1) + \lb \sum^k_{j=2} 4 \lp 4+1\rp \rb + \lp 4+1 \rp =8+20\lp k-1\rp+5 = 20k-7 + \end{align} + This, in turn, proves Item(vi). The proof of the lemma is thus complete. +\end{proof} +\begin{remark} + For an \texttt{R} implementation see Listing \ref{Phi_k} +\end{remark} + +\begin{figure}[h] + \includegraphics[width = \linewidth]{/Users/shakilrafi/R-simulations/Phi_k_properties/diff.png} + \caption{Plot of $\log_{10}$ of the $L^1$ difference between $\Phi_k$ and $x^2$ over $\lb 0,1\rb$ for different values of $k$} +\end{figure} + +\begin{corollary}\label{6.1.1.1}\label{cor:phi_network} + Let $\ve \in \lp 0,\infty\rp$, $M= \min \{ \frac{1}{2}\log_2 \lp \ve^{-1} \rp -1,\infty\}\cap \N$, $\lp c_k\rp_{k \in \N} \subseteq \R$, $\lp A_k\rp_{k\in\N} \subseteq \R^{4 \times 4}$, $B \in \R^{4\times 1}$, $\lp C_k\rp_{k\in \N}$ satisfy for all $k \in \N$ that: + \begin{align} + A_k = \begin{bmatrix} + 2&-4&2&0 \\ + 2&-4&2&0\\ + 2&-4&2&0\\ + -c_k&2c_k & -c_k&1 + \end{bmatrix}, \quad B = \begin{bmatrix} + 0\\ -\frac{1}{2}\quad \\ -1 \\ 0 + \end{bmatrix}\quad C_k = \begin{bmatrix} + -c_k &2c)_k&-c_k&1 + \end{bmatrix} + \end{align} + where: + \begin{align} + c_k = 2^{1-2k} + \end{align} + and let $\Phi \in \neu$ be defined as: + \begin{align} + \Phi = \begin{cases}\label{def:Phi} + \lb \aff_{C_1,0}\bullet \mathfrak{i}_4\rb \bullet \aff_{\mymathbb{e}_4,B} & M=1 \\ + \lb \aff_{C_M,0} \bullet \mathfrak{i}_4\rb\bullet \lb \aff_{A_{M-1},0} \bullet \mathfrak{i}_4 \rb \bullet \cdots \bullet \lb \aff_{A_1,B}\bullet \mathfrak{i}_4\rb \bullet \aff_{\mymathbb{e}_4,B} & M \in \lb 2,\infty \rp \cap \N + \end{cases} + \end{align} + it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \Phi\rp \in C \lp \R,\R\rp$ + \item $\lay \lp \Phi\rp = \lp 1,4,4,...,4,1\rp \in \N^{M+2} $ + \item it holds for all $x \in \R \setminus\lb 0,1 \rb$ that $\lp \real_{\rect} \lp \Phi\rp\rp \lp x \rp = \rect(x)$ + \item it holds for all $x \in \lb 0,1 \rb$ that $\left| x^2 - \lp \real_{\rect} \lp \Phi \rp \rp\lp x \rp \right| \les 2^{-2M-2} \les \ve$ + \item $\dep \lp \Phi \rp \les M+1 \les \max\{ \frac{1}{2}\log_2 \lp \ve^{-1}\rp+1,2\}$, and + \item $\param \lp \Phi\rp = 20M-7 \les \max\left\{ 10\log_2 \lp \ve^{-1}\rp-7,13\right\}$ + \end{enumerate} + \end{corollary} + \begin{proof} + Items (i)--(iii) are direct consequences of Lemma \ref{lem:6.1.1}, Items (i)--(iii). Note next the fact that $M = \min \left\{\N \cap \lb \frac{1}{2} \log_2 \lp \ve^{-1}\rp-1\rb,\infty\right\}$ ensures that: + \begin{align} + M = \min \left\{ \N \cap \lb \frac{1}{2}\log_2\lp \ve^{-1}\rp-1\rb, \infty\right\} \ges \min \left\{ \lb\max \left\{ 1,\frac{1}{2}\log_2 \lp\ve^{-1} \rp-1\right\},\infty \rb\right\} \ges \frac{1}{2}\log_2 \lp \ve^{-1}\rp-1 + \end{align} + This and Item (v) of Lemma \ref{lem:6.1.1} demonstrate that for all $x\in \lb 0,1\rb$ it then holds that: + \begin{align} + \left| x^2 - \lp \real_{\rect}\lp \Phi\rp\rp \lp x\rp \right| \les 2^{-2M-2} = 2^{-2(M+1)} \les 2^{-\log_2\lp\ve^{-1} \rp} = \ve + \end{align} + Thus establishing Item (iv). The fact that $M = \min \left\{ \N \cap \lb \frac{1}{2}\log_2 \lp \ve^{-1}\rp -1,\infty\rb\right\}$ and Item (ii) of Lemma \ref{lem:6.1.1} tell us that: + \begin{align} + \dep \lp \Phi \rp = M+1 \les \max \left\{ \frac{1}{2} \log_2 \lp \ve^{-1}\rp+1,2\right\} + \end{align} + Which establishes Item(v). This and Item (v) of Lemma \ref{lem:6.1.1} then tell us that: + \begin{align} + \param \lp \Phi_M\rp \les 20M-7 \les 20 \max\left\{ \frac{1}{2}\log_2\lp\ve^{-1}\rp,2\right\}-7 = \max\left\{ 10\log_2 \lp\ve^{-1} \rp-7,13\right\} + \end{align} + This completes the proof of the corollary. + \end{proof} +\begin{remark} + For an implementation in \texttt{R}, see Listing \ref{Phi} +\end{remark} +\begin{figure}[h] + \centering + \includegraphics[width = \linewidth]{/Users/shakilrafi/R-simulations/Phi_properties/Phi_diff_contour.png} + \caption{Contour plot of the $L^1$ difference between $\Phi$ and $x^2$ over $\lb 0,1 \rb$ for different values of $\ve$.} +\end{figure} +\begin{remark} + Note that (\ref{def:Phi}) implies that $\dep \lp \Phi \rp \ges 4$. +\end{remark} +Now that we have neural networks that perform the squaring operation inside $\lb -1,1\rb$, we may extend to all of $\R$. Note that this neural network representation differs somewhat from the ones in \cite{grohs2019spacetime}. + +\subsection{The $\sqr$ network} +\begin{lemma}\label{6.0.3}\label{lem:sqr_network} + Let $\delta,\epsilon \in (0,\infty)$, $\alpha \in (0,\infty)$, $q\in (2,\infty)$, $ \Phi \in \neu$ satisfy that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$, $\alpha = \lp \frac{\ve}{2}\rp^{\frac{1}{q-2}}$, $\real{\rect}\lp\Phi\rp \in C\lp \R,\R\rp$, $\dep(\Phi) \les \max \left\{\frac{1}{2} \log_2(\delta^{-1})+1,2\right\}$, $\param(\Phi) \les \max\left\{10\log_2\lp \delta^{-1}\rp-7,13\right\}$, $\sup_{x \in \R \setminus [0,1]} | \lp \real_{\rect} \lp \Phi \rp -\rect(x) \right| =0$, and $\sup_{x\in \lb 0,1\rb} |x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x\rp | \les \delta$, let $\Psi \in \neu$ be the neural network given by: + \begin{align} + \Psi = \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{\alpha,0} \rp \bigoplus\lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0}\rp + \end{align} + \begin{enumerate}[label = (\roman*)] + \item it holds that $\real_{\rect} \lp \Psi \rp \in C \lp \R,\R \rp$. + \item it holds that $\lp \real_{\rect} \lp \Psi \rp \rp \lp 0\rp=0$ + \item it holds for all $x\in \R$ that $0\les \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \les \ve + |x|^2$ + \item it holds for all $x \in \R$ that $|x^2-\lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp |\les \ve \max\{1,|x|^q\}$ + \item it holds that $\dep (\Psi)\les \max\left\{1+\frac{1}{q-2}+\frac{q}{2(q-2)}\log_2 \lp \ve^{-1} \rp,2\right\}$, and + \item it holds that $\param\lp \Psi \rp \les \max\left\{ \lb \frac{40q}{q-2} \rb \log_2 \lp \ve^{-1} \rp +\frac{80}{q-2}-28,52 \right\}$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that for all $x\in \R$ it is the case that: + \begin{align}\label{6.0.21} + \lp \real_{\rect}\lp \Psi \rp \rp\lp x \rp &= \lp \real_{\rect} \lp \lp \aff_{\alpha^{-2}}\bullet \Phi \bullet \aff_{\alpha,0}\rp \oplus\lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0} \rp \rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\rect}\lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{\alpha,0} \rp \rp \lp x\rp + \lp \real_{\rect}\lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0}\rp \rp \lp x\rp \nonumber \\ + &= \frac{1}{\alpha^2}\lp \real_{\rect}\lp \Phi \rp \rp \lp \alpha x\rp + \frac{1}{\alpha^2}\lp \real_{\rect} \lp \Phi \rp \rp \lp -\alpha x\rp \nonumber\\ + &= \frac{1}{\lp \frac{\ve}{2}\rp^{\frac{2}{q-2}}}\lb \lp \real_{\rect}\lp \Phi \rp \rp \lp \lp \frac{\ve}{2}\rp ^{\frac{1}{q-2}}x \rp + \lp \real_{\rect}\lp \Phi \rp \rp \lp -\lp \frac{\ve}{2}\rp^{\frac{1}{q-2}}x\rp \rb + \end{align} + This and the assumption that $\Phi \in C\lp \R, \R \rp$ along with the assumption that $\sup_{x\in \R \setminus \lb 0,1\rb } | \lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp -\rect\lp x\rp | =0$ tells us that for all $x\in \R$ it holds that: + \begin{align} + \lp \real_{\rect}\lp \Psi \rp \rp \lp 0 \rp &= \lp \frac{\ve}{2}\rp^{\frac{-2}{q-2}}\lb \lp \real_{\rect}\lp \Phi \rp \rp \lp 0 \rp +\lp \real_{\rect} \lp \Phi\rp \rp \lp 0 \rp \rb \nonumber \\ + &=\lp \frac{\ve}{2}\rp ^{\frac{-2}{q-2}} \lb \rect (0)+\rect(0) \rb \nonumber \\ + &=0 + \end{align} + This, in turn, establishes Item (i)--(ii). Observe next that from the assumption that $\real_{\rect} \lp \Phi \rp \in C\lp \R,\R \rp$ and the assumption that $\sup_{x\in \R \setminus \lb 0,1\rb} | \lp \real_{\rect}\lp \Phi \rp \rp \lp x \rp -\rect(x) |=0$ ensure that for all $x\in \R \setminus \lb -1,1 \rb$ it holds that: + \begin{align}\label{6.0.23} + \lb \real_{\rect}\lp \Phi \rp \rb \lp x\rp + \lb \real_{\rect}\lp \Phi \rp \lp -x \rp\rb = \rect\lp x\rp +\rect(-x) &= \max\{x,0\}+\max\{-x,0\} \nonumber\\ + &=|x| + \end{align} + The assumption that for all $\sup_{x\in \R \setminus \lb 0,1\rb }|\lp \real_{\rect} \lp \Phi \rp \rp \lp x\rp -\rect\lp x\rp |=0$ and the assumption that $\sup_{x\in\lb 0,1\rb} |x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x\rp |\les \delta$ show that: + \begin{align}\label{6.0.24} + &\sup_{x \in \lb -1,1\rb} \left|x^2 - \lp \lb \real_{\rect}\lp \Phi \rp \rb \lp x\rp +\lb \real_{\rect}\lp \Phi \rp \lp x \rp \rb \rp \right| \nonumber \\ + &= \max\left\{ \sup_{x\in \lb -1,0 \rb} \left| x^2-\lp \rect(x)+ \lb \real_{\rect}\lp \Phi \rp \rb \lp -x \rp \rp \right|,\sup _{x\in \lb 0,1 \rb} \left| x^2-\lp \lb \real_{\rect} \lp \Phi \rp \rb \lp x \rp + \rect \lp -x \rp \rp \right| \right\} \nonumber\\ + &= \max\left\{\sup_{x\in \lb -1,0 \rb}\left|\lp -x \rp^2 - \lp \real_{\rect}\lp \Phi \rp \rp \lp -x \rp \right|, \sup_{x\in \lb 0,1\rb} \left| x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp \right| \right\} \nonumber \\ + &=\sup_{x\in \lb 0,1 \rb}\left| x^2 - \lp \real_{\rect}\lp \Phi \rp \rp \lp x\rp \right| \les \delta + \end{align} + Next observe that (\ref{6.0.21}) and (\ref{6.0.23}) show that for all $x \in \R \setminus \lb -\lp \frac{\ve}{2} \rp^{\frac{-1}{q-2}}, \lp \frac{\ve}{2}\rp ^{\frac{-1}{q-2}} \rb$ it holds that: + \begin{align}\label{6.0.25} + 0 \les \lb \real_{\rect} \lp \Psi \rp \rb \lp x \rp &= \lp \frac{\ve}{2} \rp ^{\frac{-2}{q-2}}\lp \lb \real_{\rect} \lp \Phi \rp \rb \lp \lp \frac{\ve}{2}\rp ^{\frac{1}{q-2}}x \rp + \lb \real_{\rect} \lp \Phi \rp \rb \lp -\lp \frac{\ve}{2}\rp^{\frac{1}{q-2}} x\rp \rp \nonumber \\ + &= \lp \frac{\ve}{2} \rp ^{\frac{-2}{q-2}} \left| \lp \frac{\ve}{2} \rp^{\frac{1}{q-2}}x \right| = \lp \frac{\ve}{2} \rp^{\frac{-1}{q-2}|x|} \les |x|^2 + \end{align} + The triangle inequality then tells us that for all $x\in \R \setminus \lb - \lp \frac{\ve}{2} \rp^{\frac{-1}{q-2}}, \lp \frac{\ve}{2} \rp^{\frac{-1}{q-2}} \rb$ it holds that: + \begin{align} \label{6.0.25} + \left| x^2- \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \right| &= \left| x^2 - \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}}\left|x\right| \right| \les \lp \left|x \right|^2 + \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}} \left| x \right| \rp \nonumber\\ + &= \lp \left| x \right|^q \left|x\right|^{-(q-2)} + \lp \frac{\ve}{2} \rp^{\frac{-1}{q-2}} \left| x \right|^q\left| x \right|^{-(q-1)} \rp \nonumber \\ + &\les \lp \left| x \right|^q \lp \frac{\ve}{2} \rp^{\frac{q-2}{q-2}} + \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}} \left| x \right|^q \lp \frac{\ve}{2} \rp ^{\frac{q-1}{q-2}} \rp \nonumber \\ + &= \lp \frac{\ve}{2}+ \frac{\ve}{2} \rp \left| x \right|^q = \ve \left| x \right|^q \les \ve \max \left\{ 1, \left| x \right|^q \right\} + \end{align} + +Note that (\ref{6.0.24}), (\ref{6.0.21}) and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve^{\frac{q}{q-2}}$ then tell for all $x \in \lb -\lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}}, \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}} \rb$ it holds that: +\begin{equation} +\begin{aligned}\label{6.0.26} +% &\left| x^2-\lp \real_{\rect} \lp \Phi \rp \rp \lp x \rp \right| \\ +% &= \lp \frac{\ve}{2} \rp^{\frac{-2}{q-2}} \left| \lp \lp \frac{\ve}{2} \rp ^{\frac{1}{q-2}}x \rp^2 - \lp \lb \real_{\rect} \lp \Phi \rp \rb \lp \lp \frac{\ve}{2} \rp ^{\frac{1}{q-2}}x \rp + \lb \real_{\rect} \lp \Phi \rp \rb \lp -y \rp \rp \right| \\ +% &\les \lp \frac{\ve}{2} \rp^{\frac{-2}{q-2}} \lb \sup_{y \in \lb -1,1\rb} \left| y^2 - \left \lb \real_{\rect} \lp \Phi \rp \rb \lp y \rp + \lb \real_{\rect} \lp \Phi \rp \rb \lp -y \rp \right| \rb \\ + &\left| x^2-\left( \real_{\rect} (\Phi) \right) (x) \right| \\ + &= \left( \frac{\varepsilon}{2} \right)^{\frac{-2}{q-2}} \left| \left( \left( \frac{\varepsilon}{2} \right) ^{\frac{1}{q-2}}x \right)^2 - \left( \left[ \real_{\rect} (\Phi) \right] \left( \left( \frac{\varepsilon}{2} \right) ^{\frac{1}{q-2}}x \right) + \left[ \real_{\rect} (\Phi) \right] (-y) \right) \right| \\ + &\les \left( \frac{\varepsilon}{2} \right)^{\frac{-2}{q-2}} \left[ \sup_{y \in \left[-1,1\right]} \left| y^2 - \left[ \real_{\rect} (\Phi) \right] (y) + \left[ \real_{\rect} (\Phi) \right] (-y) \right| \right] \\ + &\les \lp \frac{\ve}{2} \rp^{\frac{-2}{q-2}} \delta = \lp \frac{\ve}{2} \rp^{\frac{-2}{q-2}} 2^{\frac{-2}{q-2}} \ve^{\frac{q}{q-2}} = \ve \les \ve \max \{ 1, \left| x \right|^q \} +\end{aligned} +\end{equation} +Now note that this and (\ref{6.0.25}) tells us that for all $x\in \R$ it is the case that: +\begin{align} + \left| x^2-\lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \right| \les \ve \max\{1,|x|^q \} +\end{align} +This establishes Item (v). Note that, (\ref{6.0.26}) tells that for all $x \in \lb - \lp \frac{\ve}{2} \rp ^{\frac{-1}{q-2}}, \lp \frac{\ve}{2} \rp ^{\frac{1}{q-2}} \rb $ it is the case that: +\begin{align} + \left| \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \right| \les \left| x^2 - \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \right| + \left| x \right|^2 \les \ve + \left| x \right| ^2 +\end{align} +This and (\ref{6.0.25}) tells us that for all $x\in \R$: +\begin{align} + \left| \lp \real_{\rect} \rp \lp x \rp \right| \les \ve + |x|^2 +\end{align} +This establishes Item (iv). + +Note next that by Corollary \ref{affcor}, Remark \ref{5.3.2}, the hypothesis, and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$ tells us that: +\begin{align} + \dep \lp \Psi \rp = \dep \lp \Phi \rp &\les \max \left\{\frac{1}{2} \log_2(\delta^{-1})+1,2\right\} \nonumber \\ + &= \max \left\{ \frac{1}{q-2} + \lb \frac{q}{q-2}\rb\log_2 \lp \ve \rp +1,2\right\} +\end{align} +This establishes Item (v). + +Notice next that the fact that $\delta = 2^{\frac{-2}{q-2}}\ve^{\frac{q}{q-2}}$ tells us that: +\begin{align} + \log_2 \lp \delta^{-1} \rp = \log_2 \lp 2^{\frac{2}{q-2}} \ve^{\frac{-q}{q-2}}\rp = \frac{2}{q-2} + \lb \lb \frac{q}{q-2}\rb \log_2 \lp \ve^{-1}\rp \rb +\end{align} +Note that by , Corollary \ref{affcor} we have that: +\begin{align} + \param \lp \Phi \bullet \aff_{-\alpha,0} \rp &\les \lb \max\left\{ 1, \frac{\inn \lp \aff_{-\alpha,0}\rp+1}{\inn\lp \Phi\rp+1}\right\}\rb \param \lp \Phi\rp = \param \lp \Phi\rp +\end{align} +and further that: +\begin{align} + \param \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0} \rp &= \lb \max\left\{ 1, \frac{\out \lp \aff_{-\alpha^2,0}\rp}{\out\lp \Phi \bullet \aff_{-\alpha,0}\rp}\right\}\rb \param \lp \Phi \bullet \aff_{-\alpha,0}\rp \nonumber\\ + &\les \param \lp \Phi\rp +\end{align} +By symmetry note also that $ \param \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{\alpha,0}\rp = \param \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0}\rp $ and also that $ \lay \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{\alpha,0}\rp = \lay \lp \aff_{\alpha^{-2},0} \bullet \Phi \bullet \aff_{-\alpha,0}\rp $. Thus Lemma \ref{paramsum}, Corollary \ref{cor:sameparal}, and the hypothesis tells us that: +\begin{align}\label{(6.1.42)} + \param \lp \Psi \rp &= \param \lp \Phi \boxminus \Phi \rp \nonumber \\ + &\les 4\param \lp \Phi\rp \nonumber \\ + &= 4\max\left\{10\log_2\lp \delta^{-1}\rp-7,13\right\} +\end{align} +This, and the fact that $\delta = 2^{\frac{-2}{q-2}}\ve ^{\frac{q}{q-2}}$ renders (\ref{(6.1.42)}) as: +\begin{align} + 4\max\left\{10\log_2\lp \delta^{-1}\rp-7,13\right\} &= 4\max\left\{10\log_2\lp \delta^{-1}\rp-7,13\right\} \nonumber\\ + &= 4\max \left\{ 10 \lp \frac{2}{q-2} +\frac{q}{q-2}\log_2 \lp \ve^{-1}\rp\rp-7,13\right\} \nonumber \\ + &=\max \left\{ \lb \frac{40q}{q-2}\rb \log_2 \lp \ve^{-1}\rp + \frac{80}{q-2}-28,52\right\} +\end{align} +\end{proof} +\begin{remark} + We will often find it helpful to refer to this network for fixed $\ve \in \lp 0, \infty \rp$ and $q \in \lp 2,\infty\rp$ as the $\sqr^{q,\ve}$ network. +\end{remark} +\begin{remark} + For an \texttt{R} implementation see Listing \ref{Sqr} +\end{remark} + + + +\begin{figure}[h] + \centering + \includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/experimental_deps.png} + \includegraphics[width = 0.45\linewidth]{/Users/shakilrafi/R-simulations/Sqr_properties/dep_theoretical_upper_limits.png} + \caption{Left: $\log_{10}$ of depths for a simulation with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points. Right: The theoretical upper limits over the same range of values} +\end{figure} + +% Please add the following required packages to your document preamble: +% \usepackage{booktabs} + +\begin{table}[h] +\begin{tabular}{@{}l|llllll@{}} +\toprule + & Min. & 1\textsuperscript{st} Qu. & Median & Mean & 3\textsuperscript{rd} Qu. & Max. \\ \midrule +Experimental $|x^2 - \real_{\rect}(\mathsf{Sqr}^{q,\ve})(x)$ & 0.000003 & 0.089438 & 0.337870 & 3.148933 & 4.674652 & 20.00 \\ \midrule +Theoretical $|x^2 - \real_{\rect}(\mathsf{Sqr})^{q,\ve}(x)$ & 0.010 & 1.715 & 10.402 & 48.063 & 45.538 & 1250.00 \\ \midrule +Difference & 0.001 & 1.6012 & 9.8655 & 44.9141 & 40.7102 & 1230 +\end{tabular} +\caption{Theoretical upper bounds for $L^1$ error, experimental $L^1$ error and their forward difference, with $q \in \lb 2.1, 4 \rb $, $\ve \in \lp 0.1, 2 \rb$, and $x \in \lb -5,5 \rb$, all with $50$ mesh-points.} +\end{table} + + +\subsection{The $\prd$ network} +We are finally ready to give neural network representations of arbitrary products of real numbers. However, this representation differs somewhat from those found in the literature, especially \cite{grohs2019spacetime}, where parallelization (stacking) is used instead of neural network sums. This will help us calculate $\wid_1$ and the width of the second to last layer. +\begin{lemma}\label{prd_network} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, $A_1,A_2,A_3 \in \R^{1\times 2}$, $\Psi \in \neu$ satisfy for all $x\in \R$ that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, $A_1 = \lb 1 \quad 1 \rb$, $A_2 = \lb 1 \quad 0 \rb$, $A_3 = \lb 0 \quad 1 \rb$, $\real_{\rect} \in C\lp \R, \R \rp$, $\lp \real_{\rect} \lp \Psi \rp \rp \lp 0\rp = 0$, $0\les \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \les \delta+|x|^2$, $|x^2-\lp \real_{\rect}\lp \Psi \rp \rp \lp x \rp |\les \delta \max \{1,|x|^q\}$, $\dep\lp \Psi \rp \les \max\{ 1+\frac{1}{q-2}+\frac{q}{2(q-2)}\log_2 \lp \delta^{-1} \rp ,2\}$, and $\param \lp \Psi \rp \les \max\left\{\lb \frac{40q}{q-2} \rb \log_2\lp \delta^{-1} \rp +\frac{80}{q-2}-28,52\right\}$, then: + \begin{enumerate}[label=(\roman*)] + \item there exists a unique $\Gamma \in \neu$ satisfying: + \begin{align} + \Gamma = \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp \bigoplus \lp \lp -\frac{1}{2}\rp \triangleright\lp \Psi \bullet \aff_{A_2,0} \rp \rp \bigoplus\lp \lp -\frac{1}{2}\rp \triangleright \lp \Psi \bullet \aff_{A_3,0} \rp \rp + \end{align} + \item it that $\real_{\rect} \lp \Gamma \rp \in C \lp \R^2,\R \rp$ + \item it holds for all $x\in \R$ that $\lp \real_{\rect}\lp \Gamma \rp \rp \lp x,0\rp = \lp \real_{\rect}\lp \Gamma \rp \rp \lp 0,y\rp =0$ + \item it holds for any $x,y \in \R$ that $\left|xy - \lp \real_{\rect} \lp \Gamma \rp \rp \lp \begin{bmatrix} + x \\ + y + \end{bmatrix} \rp \right| \les \ve \max \{1,|x|^q,|y|^q \}$ + \item it holds that $\param(\Gamma) \les \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb -252$ + \item it holds that $\dep\lp \Gamma \rp \les \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb $ + \item it holds that $\wid_1 \lp \Gamma \rp=24$ + \item it holds that $\wid_{\hid \lp\Gamma\rp} = 24$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that: + \begin{align} + &\lp \real_{\rect} \lp \Gamma \rp \rp \lp \begin{bmatrix} + x\\y + \end{bmatrix} \rp = \real_{\rect} \lp \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp \bigoplus \lp \lp -\frac{1}{2}\rp \triangleright\lp \Psi \bullet \aff_{A_2,0} \rp \rp \bigoplus \right. \\ + &\left. \lp \lp -\frac{1}{2}\rp \triangleright \lp \Psi \bullet \aff_{A_3,0} \rp \rp \rp \nonumber \lp \begin{bmatrix} + x \\ y + \end{bmatrix} \nonumber\rp\\ + &= \real_{\rect} \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp \lp \begin{bmatrix} + x\\y + \end{bmatrix} \rp + \real_{\rect}\lp \lp -\frac{1}{2}\rp \triangleright\lp \Psi \bullet \aff_{A_2,0} \rp \rp \lp \begin{bmatrix} + x \\ y + \end{bmatrix} \rp \nonumber \\ + &+\real_{\rect}\lp \lp -\frac{1}{2}\rp \triangleright \lp \Psi \bullet \aff_{A_3,0} \rp \rp \lp \begin{bmatrix} + x\\y + \end{bmatrix} \rp \nonumber \\ + &= \frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp \begin{bmatrix} + 1 && 1 + \end{bmatrix} \begin{bmatrix} + x \\ y + \end{bmatrix}\rp - \frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp \begin{bmatrix} + 1 && 0 + \end{bmatrix} \begin{bmatrix} + x \\ y + \end{bmatrix} \rp \nonumber\\ + &-\frac{1}{2} \lp \real_{\rect}\lp \Psi \rp \rp \lp \begin{bmatrix} + 0 && 1 + \end{bmatrix} \begin{bmatrix} + x \\y + \end{bmatrix} \rp \nonumber \\ + &=\frac{1}{2} \lp \real_{\rect}\lp \Psi \rp \rp \lp x+y \rp -\frac{1}{2} \lp \real_{\rect}\lp \Psi \rp \rp \lp x \rp - \frac{1}{2} \lp \real_{\rect}\lp \Psi \rp \rp \lp y \rp \label{6.0.33} + %TODO: Revisit this estimate + \end{align} + Note that this, and the assumption that $\lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \in C \lp \R, \R \rp$ and that $\lp \real_{\rect}\lp \Psi \rp \rp \lp 0 \rp = 0$ ensures: + \begin{align} + \lp \real_{\rect} \lp \Gamma \rp \rp \lp \begin{bmatrix} + x \\0 + \end{bmatrix} \rp &= \frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp x+0 \rp -\frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp - \frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp 0 \rp \nonumber \\ + &= 0 \nonumber\\ + &= \frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp 0+y \rp -\frac{1}{2} \lp \real_{\rect} \lp \Psi \rp \rp \lp 0 \rp - \frac{1}{2}\lp \real_{\rect} \lp \Psi \rp \rp \lp y \rp \nonumber \\ + &=\lp \real_{\rect} \lp \Gamma \rp \rp \lp \begin{bmatrix} + 0 \\y + \end{bmatrix} \rp + \end{align} + Next, observe that since by assumption it is the case for all $x,y\in \R$ that $|x^2 - \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp | \les \delta \max\{1,|x|^q\}$, $xy = \frac{1}{2}|x+y|^2-\frac{1}{2}|x|^2-\frac{1}{2}|y|^2$, triangle Inequality and from (\ref{6.0.33}) we have that: + \begin{align} + &\left| \lp \real_{\rect} \lp \Gamma\rp\lp x,y \rp \rp -xy\right| \nonumber\\ + &=\left|\frac{1}{2}\lb \lp \real_{\rect} \lp \Psi \rp \rp \lp x + y \rp - \left|x+y\right|^2 \rb - \frac{1}{2} \lb \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp -\left| x \right|^2\rb - \frac{1}{2} \lb \lp \real_{\rect} \lp \Psi\rp \rp \lp x \rp -\left|y\right|^2\rb \right| \nonumber \\ + &\les \left|\frac{1}{2}\lb \lp \real_{\rect} \lp \Psi \rp \rp \lp x + y \rp - \left|x+y\right|^2 \rb + \frac{1}{2} \lb \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp -\left| x \right|^2\rb + \frac{1}{2} \lb \lp \real_{\rect} \lp \Psi\rp \rp \lp x \rp -\left|y\right|^2\rb \right| \nonumber \\ + &\les \frac{\delta}{2} \lb \max \left\{ 1, |x+y|^q\right\} + \max\left\{ 1,|x|^q\right\} + \max \left\{1,|y|^q \right\}\rb\nonumber + \end{align} + Note also that since for all $\alpha,\beta \in \R$ and $p \in \lb 1, \infty \rp$ we have that $|\alpha + \beta|^p \les 2^{p-1}\lp |\alpha|^p + |\beta|^p \rp$ we have that: + \begin{align} + &\left| \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp - xy \right| \nonumber \\ + &\les \frac{\delta}{2} \lb \max \left\{1, 2^{q-1}|x|^q+ 2^{q-1}\left| y\right|^q\right\} + \max\left\{1,\left|x\right|^q \right\} + \max \left\{1,\left| y \right|^q \right\}\rb \nonumber \\ + &\les \frac{\delta}{2} \lb \max \left\{1, 2^{q-1}|x|^q \right\}+ 2^{q-1}\left| y\right|^q + \max\left\{1,\left|x\right|^q \right\} + \max \left\{1,\left| y \right|^q \right\}\rb \nonumber \\ + &\les \frac{\delta}{2} \lb 2^q + 2\rb \max \left\{1, \left|x\right|^q, \left| y \right|^q \right\} = \ve \max \left\{ 1,\left| x \right|^q, \left| x \right|^q\right\} \nonumber + \end{align} + This proves Item (iv). + + By symmetry it holds that $\param \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp = \param \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_2,0} \rp \rp = \param \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_3,0} \rp \rp$ and further that $\lay \lp \frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_1,0} \rp \rp = \lay \lp -\frac{1}{2}\triangleright \lp \Psi \bullet \aff_{A_2,0} \rp \rp = \lay \lp -\frac{1}{2}\triangleright\lp \Psi \bullet \aff_{A_3,0} \rp \rp$. + Note also that Corollary \ref{affcor} tells us that for all $i \in \{1,2,3\}$ and $a \in \{ \frac{1}{2},-\frac{1}{2}\}$ it is the case that: + \begin{align} + \param \lp a \triangleright \lp \Psi \bullet \aff_{A_i,0}\rp \rp = \param \lp \Psi \rp + \end{align} + This, together with Corollary \ref{corsum} indicates that: + \begin{align}\label{(6.1.49)} + \param \lp \Gamma \rp &\les 9\param\lp \Psi \rp \nonumber \\ + &\les 9\max\left\{\lb \frac{40q}{q-2} \rb \log_2\lp \delta^{-1} \rp +\frac{80}{q-2}-28,52\right\} + \end{align} + Combined with the fact that $\delta =\ve \lp 2^{q-1} +1\rp^{-1}$, this is then rendered as: + \begin{align}\label{(6.1.50)} + &9\max\left\{\lb \frac{40q}{q-2} \rb \log_2\lp \delta^{-1} \rp +\frac{80}{q-2}-28,52\right\} \nonumber \\ + &= 9\max \left\{ \lb \frac{40q}{q-2}\rb \lp \log_2 \lp \ve^{-1}\rp +\log_2 \lp 2^{q-1}+1\rp\rp + \frac{80}{q-2}-28,52 \right\} + \end{align} + Note that: + \begin{align} + \log_2 \lp 2^{q-1}+1\rp &= \log_2\lp 2^{q-1}+1\rp - \log_2 \lp 2^q\rp + q \nonumber\\ + &=\log_2 \lp \frac{2^{q-1}+1}{2^q}\rp + q = \log_2 \lp 2^{-1}+2^{-q}\rp +q\nonumber \\ + &\les \log_2 \lp 2^{-1} + 2^{-2}\rp + q = \log_2 \lp \frac{3}{4}\rp + q = \log_2 \lp 3\rp-2+q + \end{align} + Combine this with the fact that for all $q\in \lp 2,\infty\rp$ it is the case that $\frac{q(q-1)}{q-2} \ges 2$ then gives us that: + \begin{align} + \lb \frac{40q}{q-2}\rb \log_2 \lp 2^{q-1}+1\rp -28\ges \lb \frac{40q}{q-2}\rb \log_2 \lp 2^{q-1}\rp -28= \frac{40q(q-1)}{q-2}-28 \ges 52 + \end{align} + This then finally renders (\ref{(6.1.50)}) as: + \begin{align} + &9\max \left\{ \lb \frac{40q}{q-2}\rb \lp \log_2 \lp \ve^{-1}\rp +\log_2 \lp 2^{q-1}+1\rp\rp + \frac{80}{q-2}-28,52 \right\} \nonumber \\ + &\les 9 \lb \lb \frac{40q}{q-2}\rb \lp \log_2\lp \ve^{-1}\rp + \log_2\lp 3\rp-2+q\rp +\frac{80}{q-2}-28\rb \nonumber\\ + &= 9 \lb \lb \frac{40q}{q-2}\rb \lp \log_2\lp \ve^{-1}\rp + \log_2\lp 3\rp-2+\frac{2}{q}\rp-28\rb \nonumber\\ + &\les 9 \lb \lb \frac{40q}{q-2}\rb \lp \log_2\lp \ve^{-1}\rp + \log_2\lp 3\rp-1\rp -28\rb \nonumber\\ + &= \frac{360q}{q-2}\lb \log_2 \lp \ve^{-1} \rp +q+\log_2\lp 3\rp-1\rb -252 + \end{align} + Note that Lemma \ref{depth_prop}, Lemma \ref{5.3.3}, the hypothesis, and the fact that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$ tell us that: + \begin{align} + \dep \lp \Gamma \rp = \dep\lp \Psi \rp &\les \max\left\{ 1+\frac{1}{q-2}+\frac{q}{2(q-2)}\log_2 \lp \delta^{-1} \rp ,2\right\} \nonumber\\ + &= \max \left\{1+\frac{1}{q-2} +\frac{q}{2(q-2)}\lb \log_2\lp \ve^{-1}\rp + \log_2 \lp 2^{q-1}+1\rp\rb,2 \right\} \nonumber\\ + &= \max \left\{ 1+\frac{1}{q-2}+\frac{q}{2(q-2)} \lp \log_2\lp \ve^{-1}\rp +q-1\rp,2\right\} + \end{align} + Since it is the case that $\frac{q(q-1)}{2(q-2)} > 2$ for $q \in \lp 2, \infty \rp$ we have that: + \begin{align} + & \max \left\{ 1+\frac{1}{q-2}+\frac{q}{2(q-2)} \lp \log_2\lp \ve^{-1}\rp +q-1\rp,2\right\} \nonumber \\ + &= 1+\frac{1}{q-2}+\frac{q}{2(q-2)} \lp \log_2\lp \ve^{-1}\rp +q-1\rp \nonumber \\ + &\les \frac{q-1}{q-2} +\frac{q}{2\lp q-2\rp} \lp \log_2 \lp \ve^{-1}\rp+q\rp \nonumber \\ + & + \end{align} + +Observe next that for $q\in \lp 0,\infty\rp$, $\ve \in \lp 0,\infty \rp$, $\Gamma$ consists of, among other things, three stacked $\lp \Psi \bullet \aff_{A_i,0}\rp$ networks where $i \in \{1,2,3\}$. Corollary \ref{affcor} tells us therefore, that $\wid_1\lp \Gamma\rp = 3\cdot \wid_1 \lp \Psi \rp$. On the other hand, note that each $\Psi$ networks consist of, among other things, two stacked $\Phi$ networks, which by Corollary \ref{affcor} and Lemma \ref{lem:sqr_network}, yields that $\wid_1 \lp \Gamma\rp = 6 \cdot \wid_1 \lp \Phi\rp$. Finally from Corollary \ref{cor:phi_network}, and Corollary \ref{affcor}, we see that the only thing contributing to the $\wid_1\lp \Phi\rp$ is $\wid_1 \lp \mathfrak{i}_4\rp$, which was established from Lemma \ref{lem:mathfrak_i} as $4$. Whence we get that $\wid_1\lp \Gamma\rp = 6 \cdot 4 = 24$, and that $\wid_{\hid\lp \Gamma\rp}\lp \Gamma\rp = 24$. This proves Item (vii)\textemdash(viii). This then completes the proof of the Lemma. +\end{proof} + +\begin{corollary}\label{cor_prd} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, $A_1,A_2,A_3 \in \R^{1\times 2}$, $\Psi \in \N$ satisfy for all $x\in \R$ that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, $A_1 = \lb 1 \quad 1 \rb$, $A_2 = \lb 1 \quad 0 \rb$, $A_3 = \lb 0 \quad 1 \rb$, $\real_{\rect} \in C\lp \R, \R \rp$, $\lp \real_{\rect} \lp \Psi \rp \rp \lp 0\rp = 0$, $0\les \lp \real_{\rect} \lp \Psi \rp \rp \lp x \rp \les \delta+|x|^2$, $|x^2-\lp \real_{\rect}\lp \Psi \rp \rp \lp x \rp |\les \delta \max \{1,|x|^q\}$, $\dep\lp \Psi \rp \les \max\{ 1+\frac{1}{q-2}+\frac{q}{2(q-2)}\log_2 \lp \delta^{-1} \rp ,2\}$, and $\param \lp \Psi \rp \les \max\left\{\lb \frac{40q}{q-2} \rb \log_2\lp \delta^{-1} \rp +\frac{80}{q-2}-28,52\right\}$, and finally let $\Gamma$ be defined as in Lemma \ref{prd_network}, i.e.: + \begin{align} + \Gamma = \lp \frac{1}{2}\circledast \lp \Psi \bullet \aff_{A_1,0} \rp \rp \bigoplus \lp \lp -\frac{1}{2}\rp \circledast\lp \Psi \bullet \aff_{A_2,0} \rp \rp \bigoplus\lp \lp -\frac{1}{2}\rp \circledast \lp \Psi \bullet \aff_{A_3,0} \rp \rp + \end{align} + + + + It is then the case for all $x,y \in \R$ that: + \begin{align} + \real_{\rect} \lp \Gamma \rp \lp x,y \rp \les \frac{3}{2} \lp \frac{\ve}{3} +x^2+y^2\rp \les \ve + 2x^2+2y^2 + \end{align} +\end{corollary} +\begin{proof} + Note that the triangle inequality, the fact that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, the fact that for all $x,y\in \R$ it is the case that $|x+y|^2 \les 2\lp |x|^2+|y|^2\rp $ and (\ref{6.0.33}) tell us that: + \begin{align} + \left| \real_{\rect} \lp \Gamma \rp\lp x,y\rp \right| &\les \frac{1}{2}\left| \real_{\rect} \lp \Psi \rp\lp x+y \rp \right| + \frac{1}{2}\left| \real_{\rect} \lp \Psi \rp\lp x \rp \right| + \frac{1}{2}\left| \real_{\rect} \lp \Psi \rp\lp y \rp \right| \nonumber \\ + &\les \frac{1}{2} \lp \delta + |x+y|^2 \rp + \frac{1}{2}\lp \delta + |x|^2\rp + \frac{1}{2}\lp \delta + |y|^2\rp\nonumber \\ + &\les \frac{3\delta}{2} +\frac{3}{2}\lp |x|^2+|y|^2\rp = \lp \frac{3\ve}{2}\rp \lp 2^{q-1}+1\rp^{-1} + \frac{3}{2}\lp |x|^2+|y|^2\rp \nonumber\\ + &= \frac{3}{2}\lp \frac{\ve}{2^{q-1}+1} + |x|^2 + |y|^2 \rp \les \frac{3}{2} \lp \frac{\ve}{3}+|x|^2+|y|^2\rp \nonumber \\ + &\les \ve + 2x^2+2y^2 + \end{align} +\end{proof} +\begin{remark} + We shall refer to this neural network for a given $q \in \lp 2,\infty \rp$ and given $\ve \in \lp 0,\infty \rp$ from now on as $\prd^{q,\ve}$. +\end{remark} +\begin{remark} + For an \texttt{R} implementation see Listing \ref{Prd} +\end{remark} +\begin{remark} + Diagrammatically, this can be represented as: +\end{remark} +\begin{figure} +\begin{center} +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,475); %set diagram left start at 0, and has height of 475 + +%Shape: Rectangle [id:dp5102621452939872] +\draw (242,110.33) -- (430.67,110.33) -- (430.67,162.33) -- (242,162.33) -- cycle ; +%Shape: Rectangle [id:dp5404063577476766] +\draw (238.67,204.33) -- (427.33,204.33) -- (427.33,256.33) -- (238.67,256.33) -- cycle ; +%Shape: Rectangle [id:dp36108799479514775] +\draw (240,308.33) -- (428.67,308.33) -- (428.67,360.33) -- (240,360.33) -- cycle ; +%Shape: Rectangle [id:dp8902718451088835] +\draw (515.33,202.67) -- (600.67,202.67) -- (600.67,252.33) -- (515.33,252.33) -- cycle ; +%Shape: Rectangle [id:dp787158651575801] +\draw (74,204.67) -- (159.33,204.67) -- (159.33,254.33) -- (74,254.33) -- cycle ; +%Straight Lines [id:da7097969194866411] +\draw (515.33,202.67) -- (433.55,136.26) ; +\draw [shift={(432,135)}, rotate = 39.08] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da06987054821586158] +\draw (514.67,226) -- (432,226.98) ; +\draw [shift={(430,227)}, rotate = 359.32] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da6649718583556108] +\draw (515.33,252.33) -- (430.79,331.63) ; +\draw [shift={(429.33,333)}, rotate = 316.83] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da522975332769982] +\draw (240.67,136) -- (160.86,203.38) ; +\draw [shift={(159.33,204.67)}, rotate = 319.83] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da23420272890635796] +\draw (238.67,230.67) -- (160.67,231.64) ; +\draw [shift={(158.67,231.67)}, rotate = 359.28] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da3786949398178764] +\draw (239.33,333.33) -- (160.76,255.74) ; +\draw [shift={(159.33,254.33)}, rotate = 44.64] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da6573206574101601] +\draw (640.67,228.33) -- (602.33,228.33) ; +\draw [shift={(600.33,228.33)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da2877353538717321] +\draw (74,227.67) -- (35.67,227.67) ; +\draw [shift={(33.67,227.67)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (286,124) node [anchor=north west][inner sep=0.75pt] {$\frac{1}{2} \rhd \lp \Phi \bullet \aff_{A_1,0}\rp$}; +% Text Node +\draw (286,220) node [anchor=north west][inner sep=0.75pt] {$\frac{1}{2} \rhd \lp \Phi \bullet \aff_{A_2,0}\rp$}; +% Text Node +\draw (286,326) node [anchor=north west][inner sep=0.75pt] {$\frac{1}{2} \rhd \lp \Phi \bullet \aff_{A_2,0}\rp$}; +% Text Node +\draw (543,220) node [anchor=north west][inner sep=0.75pt] {$\cpy$}; +% Text Node +\draw (100,225) node [anchor=north west][inner sep=0.75pt] {$\sm$}; + +\end{tikzpicture} + +\end{center} +\caption{A neural network diagram of the $\sqr$. } +\end{figure} +\section{Higher Approximations}\label{sec_tun} +We take inspiration from the $\sm$ neural network to create the $\prd$ neural network. However, we first need to define a special neural network called \textit{tunneling neural network} to stack two neural networks not of the same length effectively. +\subsection{The $\tun$ Neural Networks and Their Properties} +\begin{definition}[R\textemdash,2023, The Tunneling Neural Networks]\label{def:tun} + We define the tunneling neural network, denoted as $\tun_n$ for $n\in \N$ by: + \begin{align} + \tun_n = \begin{cases} + \aff_{1,0} &:n= 1 \\ + \id_1 &: n=2 \\ + \bullet^{n-2} \id_1 & n \in \N \cap [3,\infty) + \end{cases} + \end{align} + Where $\id_1$ is as in Definition \ref{7.2.1}. +\end{definition} +\begin{remark} + For an \texttt{R} implementation see Listing \ref{Tun} +\end{remark} +\begin{lemma}\label{6.2.2}\label{tun_1} + Let $n\in \N$, $x \in \R$ and $\tun_n \in \neu$. For all $n\in \N$ and $x\in \R$, it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \tun_n \rp \in C \lp \R, \R \rp$ + \item $\dep \lp \tun_n \rp =n$ + \item $\lp \real_{\rect} \lp \tun_n \rp \rp \lp x \rp = x$ + \item $\param \lp \tun_n \rp = \begin{cases} + 2 &:n=1 \\ + 7+6(n-2) &:n \in \N \cap [2,\infty) + \end{cases}$ + \item $\lay \lp \tun_n \rp = \lp l_0, l_1,...,l_{L-1}, l_L \rp = \lp 1,2,...,2,1 \rp $ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that $\aff_{0,1} \in C \lp \R, \R\rp$ and by Lemma \ref{idprop} we have that $\id_1 \in C\lp \R, \R\rp$. Finally, the composition of continuous functions is continuous, hence $\tun_n \in C\lp \R, \R\rp$ for $n \in \N \cap \lb 2,\infty\rp$. This proves Item (i). + + Note that by Lemma \ref{5.3.2} it is the case that $\dep\lp \aff_{1,0} \rp = 1$ and by Lemma \ref{7.2.1} it is the case that $\dep \lp \id_1 \rp = 2$. + Assume now that for all $n \les N$ that $\dep\lp \tun_n \rp = n$, then for the inductive step, by Lemma \ref{comp_prop} we have that: + \begin{align} + \dep \lp \tun_{n+1} \rp &= \dep \lp \bullet^{n-1} \id_1 \rp \nonumber \\ + &= \dep \lp \lp \bullet^{n-2} \id_1 \rp \bullet \id_1 \rp \nonumber \\ + &=n+2-1 = n+1 + \end{align} + This completes the induction and proves Item (i)\textemdash(iii). + Note next that by (\ref{5.1.11}) we have that: + \begin{align} + \lp \real_{\rect} \lp \aff_{1,0} \rp \rp \lp x \rp = x + \end{align} + Lemma \ref{idprop}, Item (iii) also tells us that: + \begin{align} + \lp \real_{\rect} \lp \id_1 \rp \rp \lp x \rp = \rect(x) - \rect(-x) = x + \end{align} + Assume now that for all $n\les N$ that $\tun_n \lp x \rp = x$. For the inductive step, by Lemma \ref{idprop}, Item (iii), and we then have that: + \begin{align} + \lp \real_{\rect} \lp \tun_{n+1} \rp \rp \lp x \rp &= \lp \real_{\rect} \lp \bullet^{n-1} \id_1 \rp \rp \lp x \rp \lp x \rp \nonumber\\ + &= \lp \real_{\rect} \lp \lp \bullet^{n-2} \id_1 \rp \bullet \id_1 \rp \rp \nonumber\\ + &= \lp \lp \real_{\rect} \lp \bullet^{n-2} \id_1 \rp \rp \circ \lp \real_{\rect} \lp \id_1 \rp \rp \rp \lp x \rp \nonumber \\ + &= \lp \lp \real_{\rect} \lp \tun_n \rp \rp \circ \lp \real_{\rect} \lp \id_1 \rp \rp \rp \lp x \rp \nonumber \\ + &= x + \end{align} + This proves Item (ii). Next note that $\param\lp \tun_1\rp = \param\lp \aff_{1,0}\rp = 2$. Note also that: + \begin{align} + \param\lp \tun_2\rp = \param \lp \id_1 \rp &= \param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \rb \nonumber \\ + &= 7 \nonumber + \end{align} + And that by definition of composition: + \begin{align} + \param \lp \tun_3 \rp &= \param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \bullet \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \rb \nonumber \\ + &= \param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix} \rp, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1&-1 + \end{bmatrix},\begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \rb \nonumber \\ + &=13 \nonumber + \end{align} + Now for the inductive step assume that for all $n\les N\in \N$, it is the case that $\param\lp \tun_n \rp = 7+6(n-2)$. For the inductive step, we then have: + \begin{align} + &\param \lp \tun_{n+1} \rp = \param \lp \tun_n \bullet \id_1 \rp \nonumber\\ + &=\param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\0 + \end{bmatrix}\rp, \cdots, \lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \bullet \id_1 \rb \nonumber \\ + &= \param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\0 + \end{bmatrix}\rp, \cdots, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 + \end{bmatrix} \rp, \lp \begin{bmatrix} + 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 + \end{bmatrix}\rp \rp \rb \nonumber \\ + &=7+6(n-2)+6 = 7+6\lp \lp n+1 \rp -2 \rp + \end{align} + This proves Item (iv). + + Note finally that Item (v) is a consequence of Lemma \ref{idprop}, Item (i), and Lemma \ref{comp_prop} +\end{proof} + +\begin{definition}[R\textemdash, 2023, The Multi-dimensional Tunneling Network]\label{def:tun_mult} + We define the multi-dimensional tunneling neural network, denoted as $\tun^d_n$ for $n\in \N$ and $d \in \N$ by: + \begin{align} + \tun_n^d = \begin{cases} + \aff_{\mathbb{I}_d,\mymathbb{0}_d} &:n= 1 \\ + \id_d &: n=2 \\ + \bullet^{n-2} \id_d & :n \in \N \cap [3,\infty) + \end{cases} + \end{align} + Where $\id_d$ is as in Definition \ref{7.2.1}. +\end{definition} +\begin{remark} + We may drop the requirement for a $d$ and write $\tun_n$ where $d=1$, and it is evident from the context. +\end{remark} +\begin{lemma}\label{tun_mult} + Let $n\in \N$, $d\in \N$, $x \in \R$ and $\tun_n^d \in \neu$. For all $n\in \N$, $d\in \N$, and $x\in \R$, it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \tun_n^d \rp \in C \lp \R, \R \rp$ + \item $\dep \lp \tun_n^d \rp =n$ + \item $\lp \real_{\rect} \lp \tun_n^d \rp \rp \lp x \rp = x$ + \item $\param \lp \tun_n^d \rp = \begin{cases} + 8d^2+5d &:n=1 \\ + 4d^2+3d+ (n-1)\lp 4d^2+2d\rp &:n \in \N \cap [2,\infty) + \end{cases}$ + \item $\lay \lp \tun_n^d \rp = \lp l_0, l_1,...,l_{L-1}, l_L \rp = \lp d,2d,...,2d,d \rp$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that Items (i)\textendash(iii) are consequences of Lemma \ref{idprop} and Lemma \ref{comp_prop} respectively. Note now that by observation $\param \lp \tun^d_1\rp = d^2+d$. Next Lemma $\ref{id_param}$ tells us that $\param\lp \tun^d_2\rp = 4d^2+3d$ + Note also that by definition of neural network composition, we have the following: + \begin{align} + &\param\lp \tun_3^d\rp \\ &= \param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 \\ &\ddots \\& & 1 \\& & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1 \\ & &\ddots \\ & & & 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ \vdots \\ 0 + \end{bmatrix}\rp \rp \bullet \right.\\ &\left. \lp \lp \begin{bmatrix} + 1 \\ -1 \\ & \ddots \\ & & 1 \\ & & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 & -1\\ & &\ddots \\ & & & 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ \vdots \\ 0 + \end{bmatrix}\rp \rp \rb \nonumber \\ + &= \param \lb \lp \lp \begin{bmatrix} + 1 \\ -1 \\ & \ddots \\ & & 1 \\ & &-1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\\vdots \\ 0\\0 + \end{bmatrix} \rp, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 \\ & & \ddots \\ & & & 1 & -1 \\ & & & -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\ 0 + \end{bmatrix}\rp, \lp \begin{bmatrix} + 1 &-1 \\ & &\ddots \\ & & & 1 & -1 + \end{bmatrix},\begin{bmatrix} + 0 \\ \vdots \\ 0 + \end{bmatrix}\rp \rp \rb \nonumber \\ + &=2d \times d + 2d + 2d\times 2d +2d+2d\times d + d \nonumber \\ + &=2d^2+2d+4d^2+2d+2d^2 +d \nonumber \\ + &= 8d^2+5d + \end{align} + Suppose now that for all naturals up to and including $n$, it is the case that $\param\lp \tun_n^d\rp = 4d^2+3d + \lp n-2 \rp \lp 4d^2+2d\rp$. For the inductive step, we have the following: + + \begin{align} + & \param\lp \tun^d_{n+1}\rp = \param \lp \tun_n^d \bullet \id_d\rp \nonumber \\ + & = \param \lb \lp \begin{bmatrix} + 1 \\ -1 \\ & \ddots \\ & & 1 \\ & &-1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\ 0 + \end{bmatrix} \rp, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 \\ & \ddots \\ & & 1 & -1 \\ & & -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\ 0 + \end{bmatrix} \rp, \hdots, \lp \begin{bmatrix} + 1 &-1 \\ & \ddots \\ & & 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ \vdots \\ 0 + \end{bmatrix}\rp \right. \nonumber \\ + & \left. \bullet \id_d \rb \nonumber\\ + & = \param \lb \lp \begin{bmatrix} + 1 \\ -1 \\ & \ddots \\ & & 1 \\ & &-1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\ 0 + \end{bmatrix} \rp, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 \\ & \ddots \\ & & 1 & -1 \\ & & -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\ 0 + \end{bmatrix} \rp, \hdots, \lp \begin{bmatrix} + 1 & -1 \\ -1 & 1 \\ & \ddots \\ & & 1 & -1 \\ & & -1 & 1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ 0 \\ \vdots \\ 0 \\ 0 + \end{bmatrix} \rp, \right. \nonumber\\ &\left. \lp \begin{bmatrix} + 1 &-1 \\ & \ddots \\ & & 1 & -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ \vdots \nonumber\\ 0 + \end{bmatrix}\rp \rb \nonumber\\ + &= 4d^2+3d+ (n-2)\lp 4d^2+2d\rp + 4d^2+2d \nonumber \\ + &=4d^2+3d+\lp n-1\rp\lp 4d^2+2d\rp \nonumber +\end{align} +This proves Item (iv). Finally, Item (v) is a consequence of Lemma \ref{5.3.2} +\end{proof} + + + + + + + +\subsection{The $\pwr$ Neural Networks and Their Properties} + +\begin{definition}[R\textemdash, 2023, The Power Neural Network]\label{def:pwr} + Let $n\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. We define the power neural networks $\pwr_n^{q,\ve} \in \neu$, denoted for $n\in \N_0$ as: + \begin{align} + \pwr_n^{q,\ve} = \begin{cases} + \aff_{0,1} & :n=0\\ + \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr_{n-1}^{q,\ve})} \boxminus \pwr_{n-1}^{q,\ve} \rb \bullet \cpy_{2,1} & :n \in \N + \end{cases} \nonumber + \end{align} +Diagrammatically, this can be represented as: +\begin{figure} +\begin{center} + \begin{tikzpicture} + % Define nodes + \node[draw, rectangle] (top) at (0, 2) {$\pwr_{n-1}^{q,\ve}$}; + \node[draw, rectangle] (right) at (2, 0) {$\cpy_{2,1}$}; + \node[draw, rectangle] (bottom) at (0, -2) {$\tun_{\dep(\pwr_{n-1}^{q,\ve})}$}; + \node[draw, rectangle] (left) at (-2, 0) {$\prd^{q,\ve} $}; + + % Arrows with labels + \draw[->] (right) -- node[midway, above] {$x$} (top); + \draw[<-] (right) -- node[midway, above] {$x$} (4,0)(right); + \draw[->] (right) -- node[midway, right] {$x$} (bottom); + \draw[->] (top) -- node[midway, left] {$\lp \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp \rp \lp x \rp $} (left); + \draw[->] (bottom) -- node[midway, left] {$x$} (left); + \draw[->] (left) -- node[midway, above] {} (-5.5,0); +% \draw[->] (-3,0) -- node[midway, above] {Arrow 6} (left); +\end{tikzpicture} +\end{center} +\caption{A representation of a typical $\pwr^{q,\ve}_n$ network.} +\end{figure} +\begin{remark} + For an \texttt{R} implementation see Listing \ref{Pwr} +\end{remark} +\begin{remark} + Note that for all $i \in \N$, $q\in \lp 2,\infty\rp$, $\ve \in \lp 0, \infty \rp$, each $\pwr_i^{q,\ve}$ differs from $\pwr_{i+1}^{q,\ve}$ by atleast one $\prd^{q,\ve}$ network. +\end{remark} +\end{definition} +\begin{lemma}\label{6.2.4} + Let $x,y \in \R$, $\ve \in \lp 0,\infty \rp$ and $q \in \lp 2,\infty \rp$. It is then the case for all $x,y \in \R$ that: + \begin{align} + \ve \max \left\{ 1,|x|^q,|y|^q\right\} \les \ve + \ve |x|^q+\ve |y|^q. + \end{align} +\end{lemma} +\begin{proof} + We will do this in the following cases: + + For the case that $|x| \les 1$ and $|y| \les 1$ we then have: + \begin{align} + \ve \max \left\{ 1,|x|^q,|y|^q \right\} = \ve \les \ve + \ve |x|^q+\ve |y|^q + \end{align} + For the case that $|x| \les 1$ and $|y| \ges 1$, without loss of generality we have then: + \begin{align} + \ve \max \left\{1,|x|^q,|y|^q \right\} \les \ve | y|^q \les \ve + \ve |x|^q+\ve |y|^q: + \end{align} + For the case that $|x| \ges 1$ and $|y| \ges 1$, and without loss of generality that $|x| \ges |y|$ we have that: + \begin{align} + \ve \max\{ 1, |x|^q,|y|^q \} = \ve |x|^q \les \ve + \ve |x|^q+\ve |y|^q + \end{align} +\end{proof} +\begin{lemma} +Let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined for $\ve \in \lp 0,\infty\rp$, and $x \in \R$ as follows: + \begin{align} + \mathfrak{p}_1 &= \ve+2+2|x|^2 \nonumber\\ + \mathfrak{p}_i &= \ve +2\lp \mathfrak{p}_{i-1} \rp^2+2|x|^2 \text{ for } i \ges 2 + \end{align} + For all $n\in \N$ and $\ve \in (0,\infty)$ and $q\in (2,\infty)$ it holds for all $x\in \R$ that: + \begin{align} + \left| \real_{\rect} \lp \pwr^{q,\ve}_n \rp \lp x \rp\right| \les \mathfrak{p}_n + \end{align} + \end{lemma} + \begin{proof} + Note that by Corollary \ref{cor_prd} it is the case that: + \begin{align}\label{(6.2.31)} + \left|\real_{\rect} \lp \pwr^{q,\ve}_1 \rp \lp x \rp \right| =\left| \real_{\rect}\lp \prd^{q,\ve}\rp \lp1,x \rp \right| \les \mathfrak{p}_1 + \end{align} + and applying (\ref{(6.2.31)}) twice, it is the case that: + \begin{align} + \left| \real_{\rect} \lp \pwr_2^{q,\ve}\rp \lp x \rp \right| &= \left| \real_{\rect} \lp \prd^{q,\ve} \rp \lp \real_{\rect} \lp \prd ^{q,\ve}\lp 1,x \rp\rp,x\rp \right| \nonumber \\ + &\les \ve + 2\left| \real_{\rect} \lp \prd^{q,\ve}\rp\lp 1,x\rp \right|^2 + 2|x|^2 \nonumber \\ + &\les \ve + 2\mathfrak{p}_1^2 +2|x|^2 = \mathfrak{p}_2 + \end{align} + Let's assume this holds for all cases up to and including $n$. For the inductive step, Corollary \ref{cor_prd} tells us that: + \begin{align} + \left| \real_{\rect} \lp \pwr_{n+1}^{q,\ve}\rp \lp x\rp \right| &\les \left| \real_{\rect} \lp \prd^{q,\ve} \lp \real_{\rect} \lp \prd^{q,\ve} \lp \real_{\rect}\cdots \lp 1,x\rp,x \rp ,x\rp \cdots \rp \rp \right| \nonumber \\ + &\les \real_{\rect} \lb \prd^{q,\ve} \lp \pwr^{q,\ve}_n \lp x\rp,x \rp\rb \nonumber \\ + &\les \ve + 2\mathfrak{p}_n^2 + 2|x|^2 = \mathfrak{p}_{n+1} + \end{align} + This completes the proof of the lemma. + \end{proof} + \begin{remark} + Note that since any instance of $\mathfrak{p}_i$ contains an instance of $\mathfrak{p}_{i-1}$ for $i \in \N \cap \lb 2,\infty\rp$, we have that $\mathfrak{p}_n \in \mathcal{O}\lp \ve^{2(n-1)}\rp$ + \end{remark} + \begin{lemma}\label{param_pwr_geq_param_tun} + For all $n \in \N$, $q\in \lp 2,\infty\rp$, and $\ve \in \lp 0,\infty\rp$, it is the case that $\param \lp \tun_{\dep\lp\pwr^{q,\ve}_n\rp}\rp \les \param \lp \pwr^{q,\ve}_n\rp$. + \end{lemma} + \begin{proof} + Note that for all $n \in \N$ it is straightforwardly the case that $\param\lp \pwr_n^{q,\ve}\rp \ges \param \lp \tun_{\dep\lp \pwr^{q,\ve}_{n-1}\rp}\rp$ because for all $n\in \N$, a $\pwr^{q,\ve}_n$ network contains a $\tun_{\dep\lp \pwr^{q,\ve}_{n-1}\rp}$ network. Note now that for all $i \in \N$ we have from Lemma \ref{tun_1} that $5 \les \param\lp \tun_{i+1}\rp - \param\lp \tun_i\rp \les 6$. Recall from Corollary \ref{cor:phi_network} that every instance of the $\Phi$ network contains atleast one $\mathfrak{i}_4$ network, which by Lemma \ref{lem:mathfrak_i} has $40$ parameters, whence the $\prd^{q,\ve}$ network has atleast $40$ parameters for all $\ve \in \lp 0,\infty \rp$ and $q \in \lp 2,\infty\rp$. Note now that for all $i\in \N$, $\pwr^{q,\ve}_{i}$ and $\pwr^{q,\ve}_{i+1}$ differ by atleast as many parameters as there are in $\prd^{q,\ve}$, since, indeed, they differ by atleast one more $\prd^{q,\ve}$. Thus for every increment in $i$, $\pwr_i^{q,\ve}$ outstrips $\tun_i$ by at-least $40-6 = 34$ parameters. This is true for all $i\in \N$. Whence it is the case that for all $i \in \N$, it is the case that $\param\lp \tun_i\rp \les \param \lp \pwr^{q,\ve}_i\rp$. + \end{proof} +\begin{lemma}[R\textemdash,2023]\label{power_prop} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $n \in \N_0$, and $\pwr_n \in \neu$. It is then the case for all $n \in \N_0$, and $x \in \R$ that: + \begin{enumerate}[label = (\roman*)] + \item $\lp \real_{\rect} \lp \pwr_n^{q,\ve} \rp \rp \lp x \rp \in C \lp \R, \R \rp $ + \item $\dep(\pwr_n^{q,\ve}) \les \begin{cases} + 1 & :n=0\\ + n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 & :n \in \N + \end{cases}$ + \item $\wid_1 \lp \pwr^{q,\ve}_{n}\rp = \begin{cases} + 1 & :n=0 \\ + 24+2\lp n-1 \rp & :n \in \N + \end{cases}$ + \item $\param(\pwr_n^{q,\ve}) \les \begin{cases} + 2 & :n=0 \\ + 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp &: n\in \N + \end{cases}$\\~\\ + \item $\left|x^n -\lp \real_{\rect} \lp \pwr^{q,\ve}_n \rp \rp \lp x \rp \right| \les \begin{cases} + 0 & :n=0 \\ + \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q & :n\in \N + \end{cases}$ \\~\\ + Where we let $\mathfrak{p}_i$ for $i \in \{1,2,...\}$ be the set of functions defined as follows: + \begin{align} + \mathfrak{p}_1 &= \ve + 2 + 2|x|^2 \nonumber\\ + \mathfrak{p}_i &= \ve + 2\lp \mathfrak{p}_{i-1} \rp^2+2|x|^2 + \end{align} + And whence we get that: + \begin{align} + \left| x^{n} - \real_{\rect} \lp \pwr^{q,\ve}_n\rp \lp x\rp\right| \in \mathcal{O} \lp \ve^{2q\lp n-1\rp} \rp &\text{ for } n \ges 2 + \end{align} + \item $\wid_{\hid \lp \pwr_n^{q,\ve}\rp}\lp \pwr^{q,\ve}_n\rp = \begin{cases} + 1 & n=0 \\ + 24 & n \in \N + \end{cases}$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that Item (ii) of Lemma \ref{5.3.2} ensures that $\real_{\rect} \lp \pwr_0 \rp = \aff_{1,0} \in C \lp \R, \R \rp$. Note next that by Item (v) of Lemma \ref{comp_prop}, with $\Phi_1 \curvearrowleft \nu_1, \Phi_2 \curvearrowleft \nu_2, a \curvearrowleft \rect$, we have that: + \begin{align} + \lp \real_{\rect} \lp \nu_1 \bullet \nu_2 \rp\rp \lp x \rp = \lp\lp \real_{\rect}\lp \nu_1 \rp \rp \circ \lp \real_{\rect}\lp \nu_2 \rp \rp \rp \lp x \rp + \end{align} + This, with the fact that the composition of continuous functions is continuous, the fact the stacking of continuous instantiated neural networks is continuous tells us that $\lp \real_{\rect} \pwr_n \rp \in C \lp \R, \R \rp$ for $n \in \N \cap \lb 2,\infty \rp$. This establishes Item (i). + + Note next that by observation $\dep \lp \pwr_0^{q,\ve} \rp=1$ and by Item (iv) of Lemma \ref{idprop}, it is the case that $\dep\lp \id_1 \rp = 2$. By Lemmas $\ref{dep_cpy}$ and $\ref{depthofcomposition}$ it is also the case that: $\dep\lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr^{q,\ve}_{n-1})} \boxminus \pwr^{q,\ve}_{n-1} \rb \bullet \cpy \rp = \dep \lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr^{q,\ve}_{n-1})} \boxminus \pwr^{q,\ve}_{n-1} \rb\rp $. Note also that by Lemma we have that $\dep \lp \tun_{\dep \lp \pwr^{q,\ve}_{n-1}\rp} \boxminus \pwr^{q,\ve}_{n-1}\rp = \dep \lp \pwr^{q,\ve}_{n-1} \rp$. + This with Lemma \ref{comp_prop} then yields for $n \in \N$ that: + \begin{align} + \dep \lp \pwr^{q,\ve}_n \rp &= \dep \lp \prd \bullet \lb \tun_{\mathcal{D} \lp \pwr^{q,\ve}_{n-1} \rp } \boxminus \pwr^{q,\ve}_{n-1} \rb \bullet \cpy_{2,1} \rp \nonumber \\ + &= \dep \lp \prd \bullet \lb \tun_{\dep \lp \pwr^{q,\ve}_{n-1} \rp } \boxminus \pwr^{q,\ve}_{n-1} \rb \rp \nonumber \\ + &= \dep \lp \prd \rp + \dep \lp \tun_{\dep \lp \pwr^{q,\ve}_{n-1} \rp} \rp -1 \nonumber \\ + &\les \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb + \dep \lp \tun_{\dep\lp \pwr^{q,\ve}_{n-1} \rp} \rp - 1 \nonumber \\ + &= \frac{q}{q-2}\lb \log_2 \lp\ve^{-1} \rp + q\rb + \dep \lp \pwr^{q,\ve}_{n-1}\rp - 1 + \end{align} + And hence for all $n \in \N$ it is the case that: + \begin{align} + \dep\lp \pwr^{q,\ve}_n\rp - \dep \lp \pwr^{q,\ve}_{n-1}\rp \les \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 + \end{align} + This, in turn, indicates that: + \begin{align} + \dep \lp \pwr^{q,\ve}_n\rp &\les n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 \nonumber \\ + &\les n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 + \end{align} + This proves Item (ii). + + Note now that $\wid_1 \lp \pwr^{q,\ve}_0\rp = \wid_1 \lp \aff_{0,1}\rp = 1$. Further Lemma \ref{comp_prop}, Remark \ref{5.3.2}, tells us that for all $i,k \in \N$ it is the case that $\wid_i \lp \tun_k\rp \les 2$. Observe that since $\cpy_{2,1}, \pwr_0^{q,\ve}$, and $\tun_{\dep \lp \pwr_0^{q,\ve}\rp}$ are all affine neural networks, Lemma \ref{aff_effect_on_layer_architecture}, Corollary \ref{affcor}, and Lemma \ref{prd_network} tells us that: + \begin{align} + \wid_1 \lp \pwr_1^{q,\ve} \rp &= \wid_1 \lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr_{0}^{q,\ve})} \boxminus \pwr_{0}^{q,\ve} \rb \bullet \cpy_{2,1} \rp \nonumber \\ + &= \wid_1 \lp \prd^{q,\ve}\rp = 24 + \end{align} + And that: + \begin{align} + \wid_1 \lp \pwr_2^{q,\ve} \rp &= \wid_1 \lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr_{1}^{q,\ve})} \boxminus \pwr_{1}^{q,\ve} \rb \bullet \cpy_{2,1} \rp \nonumber \\ + &= \wid_1 \lp \lb \tun_{\dep \lp \pwr^{q,\ve}_1 \rp} \boxminus \pwr_{1}^{q,\ve} \rb \rp \nonumber\\ + &= 24+2 = 26 \nonumber + \end{align} + This completes the base case. For the inductive case, assume that for all $i$ up to and including $k\in \N$ it is the case that $\wid_1 \lp \pwr_i^{q,\ve}\rp \les \begin{cases} + 1 & :i=0 \\ + 24+2(i-1) & :i \in \N + \end{cases}$. For the case of $k+1$, we get that: + \begin{align} + \wid_1 \lp \pwr_{k+1}^{q,\ve} \rp &= \wid_1 \lp \prd^{q,\ve} \bullet \lb \tun_{\dep(\pwr_{k}^{q,\ve})} \boxminus \pwr_{k}^{q,\ve} \rb \bullet \cpy_{2,1} \rp \nonumber \\ + &=\wid_1 \lp \lb \tun_{\dep(\pwr_{k}^{q,\ve})} \boxminus \pwr_{k}^{q,\ve} \rb \rp \nonumber \\ + &=\wid_1 \lp \tun_{\dep \lp \pwr^{q,\ve}_{k}\rp}\rp + \wid_1 \lp \pwr^{q,\ve}_k\rp \nonumber \\ + &\les \begin{cases} + 2 & :k=0 \\ + 24 +2 k & :k\in \N + \end{cases} + \end{align} + This establishes Item (iii). + + For Item (iv), we will prove this in cases. + + \textbf{Case 1: $\pwr_0^{q,\ve}:$} + + Note that by Lemma \ref{5.3.2} we have that: + \begin{align} + \param\lp \pwr_0^{q,\ve} \rp = \param \lp \aff_{0,1} \rp =2 + \end{align} + This completes Case 1. + +% \textbf{Case 2: $\pwr_1^{q,\ve}:$} +% +% For this case, Lemma \ref{paramofparallel} tells us that we have: +% \begin{align} +% \param \lp \pwr_{0}^{q,\ve} \boxminus \tun_{\dep \lp \pwr_{0}^{q,\ve}\rp }\rp &= \frac{1}{2} \lp \param \lp \pwr^{q,\ve}_{0}\rp + \param \lp \tun_{ 1 } \rp\rp^2 \nonumber\\ +% &= \frac{1}{2} \lp 2+2\rp^2 \nonumber \\ +% &=8 +% \end{align} +% Notice now that by Corollary \ref{affcor}, we have that: +% \begin{align} +% \param \lp\lb \pwr_{0}^{q,\ve} \boxminus \tun_{\dep \lp \pwr_{0}^{q,\ve}\rp } \rb \bullet \cpy_{2,1}\rp &= \param \lp \pwr_{0}^{q,\ve} \boxminus \tun_{\dep \lp \pwr_{0}^{q,\ve}\rp }\rp \nonumber \\ +% &=8 +% \end{align} +% This now, coupled with Lemma \ref{comp_prop} and Lemma \ref{prd_network} tells us that: +% \begin{align}\label{(6.2.19)} +% \param \lp \prd^{q,\ve} \bullet \lb \pwr_{0}^{q,\ve} \boxminus \tun_{\dep \lp \pwr_{0}^{q,\ve}\rp } \rb \bullet \cpy_{2,1}\rp &= \param \lp \prd^{q,\ve}\bullet \lb \pwr_{0}^{q,\ve} \boxminus \tun_{\dep \lp \pwr_{0}^{q,\ve}\rp } \rb \rp\nonumber\\ +% &\les \param \lp \prd^{q,\ve}\rp + 8 + \wid_1 \lp \prd^{q,\ve} \rp \cdot \wid_0 \lp \tun_1\rp \nonumber \\ +% &=\param \lp \prd^{q,\ve}\rp + 32 \nonumber\\ +% &\les \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb -220 +% \end{align} + + \textbf{Case 2: $\pwr_n^{q,\ve}$ where $n\in \N$:} + + Note that Lemma \ref{paramofparallel}, Lemma \ref{param_pwr_geq_param_tun}, Corollary \ref{cor:sameparal}, Lemma \ref{lem:paramparal_geq_param_sum}, and Corollary \ref{cor:bigger_is_better}, tells us it is the case that: + \begin{align} + \param \lp \pwr_{n-1}^{q,\ve} \boxminus \tun_{\dep \lp \pwr_{n-1}^{q,\ve}\rp }\rp &\les \param \lp \pwr^{q,\ve}_{n-1} \boxminus \pwr^{q,\ve}_{n-1}\rp \nonumber\\ + &\les 4\param\lp \pwr^{q,\ve}_{n-1}\rp + \end{align} + Then Lemma \ref{comp_prop} and Corollary \ref{affcor} tells us that: + \begin{align}\label{(6.2.34)} + &\param \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb \bullet \cpy_{2,1}\rp \nonumber\\&= \param \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb \rp \nonumber\\ + &\les 4\param \lp \pwr^{q,\ve}_{n-1}\rp + \end{align} + Note next that by definition for all $q\in \lp 2,\infty\rp$, and $\ve \in \lp 0,\infty\rp$ it is case that $\wid_{\hid\lp \pwr_0^{q,\ve}\rp}\pwr_0^{q,\ve} = \wid_{\hid \lp \aff_{0,1}\rp} = 1$. Now, by Lemma \ref{prd_network}, and by construction of $\pwr_i^{q,\ve}$ we may say that for $i\in \N$ it is the case that: + \begin{align} + \wid_{\hid \lp \pwr^{q,\ve}_i\rp} = \wid _{\hid \lp \prd^{q,\ve}\rp} = 24 + \end{align} + + Note also that by Lemma \ref{6.2.2} it is the case that: + \begin{align} + \wid_{\hid \lp \tun_{\dep \lp \pwr_{i-1}^{q,\ve}\rp}\rp} \lp \tun_{\dep \lp \pwr^{q,\ve}_{i-1}\rp} \rp = 2 \end{align} + Furthermore, note that for $n\in \lb 2, \infty \rp \cap \N$ Lemma \ref{prd_network} tells us that: + \begin{align} + \wid_{\hid \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp} \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp = 24+2=26 + \end{align} + + Finally Lemma \ref{comp_prop}, (\ref{(6.2.34)}), and Corollary \ref{cor:sameparal}, also tells us that: + \begin{align} + &\param \lp \pwr_{n}^{q,\ve}\rp\\ &= \param \lp \prd^{q,\ve} \bullet\lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb \bullet \cpy_{2,1}\rp \nonumber \\ + &= \param \lp \prd^{q,\ve} \bullet \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp \nonumber \\ + &\les \param \lp \prd^{q,\ve} \rp + 4\param \lp \pwr_{n-1}^{q,\ve}\rp+\nonumber\\ + &+ \wid_1 \lp \prd^{q,\ve} \rp\ \cdot \wid_{\hid \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp} \lp \lb \pwr^{q,\ve}_{n-1} \boxminus\tun_{\dep \lp \pwr_{n-1}^{q,\ve} \rp}\rb\rp \nonumber \\ + &= \param\lp \prd^{q,\ve}\rp + 4\param\lp \pwr^{q,\ve}_{n-1}\rp + 624 \nonumber\\ + &= 4^{n+1}\param\lp \pwr^{q,\ve}_0\rp + \lp \frac{4^{n+1}-1}{3}\rp \lp \param\lp \prd^{q,\ve}\rp + 624\rp \nonumber\\ + &= 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp + \end{align} + + + Next note that $\lp \real_{\rect} \lp \pwr_{0,1} \rp\rp \lp x \rp$ is exactly $1$, which implies that for all $x\in \R$ we have that $|x^0-\lp \real_{\rect} \lp \pwr_{0.1}\rp\lp x \rp\rp |=0$. Note also that the instantiations of $\tun_n$ and $\cpy_{2,1}$ are exact. Note next that since $\tun_n$ and $\cpy_{2,1}$ are exact, the only sources of error for $\pwr^{q,\ve}_n$ a are $n$ compounding applications of $\prd^{q,\ve}$. + + Note also that by definition, it is the case that: + \begin{align} + \real_{\rect}\lp \pwr_n^{q,\ve} \rp = \real_{\rect} \lb \underbrace{\prd^{q,\ve} \lp \inst_{\rect} \lb \prd^{q,\ve}\lp\cdots \inst_{\rect}\lb \prd^{q,\ve} \lp 1,x\rp \rb, \cdots x\rp \rb, x \rp}_{n-copies } \rb + \end{align} + Lemma \ref{prd_network} tells us that: + \begin{align} + \left|x-\real_{\rect}\lp \prd^{q,\ve} \lp 1,x \rp \rp \right| \les \ve \max\{ 1,|x|^q\} \les \ve + \left| x\right|^q + \end{align} + The triangle inequality, Lemma \ref{6.2.4}, Lemma \ref{prd_network}, and Corollary \ref{cor_prd} then tells us that: + \begin{align} + &\left| x^2 - \real_{\rect} \lp \pwr^{q,\ve}_2 \rp \lp x \rp \right| \nonumber\\ + &=\left| x\cdot x-\real_{\rect}\lp \prd^{q,\ve}\lp \inst_{\rect}\lp \prd^{q,\ve} \lp 1,x \rp \rp,x\rp \rp\right| \nonumber\\ + &\les \left| x\cdot x - x \cdot \inst_{\rect} \lp \prd^{q,\ve}\lp 1,x\rp \rp \right| + \left| x\cdot \inst_{\rect}\lp \prd^{q,\ve} \lp 1,x \rp\rp -\inst_{\rect}\lp \prd^{q,\ve} \lp \inst_{\rect}\lp \prd^{q,\ve}\lp 1,x\rp \rp,x \rp \rp \right| \nonumber\\ + &=\left| x\lp x-\inst_{\rect}\lp \prd^{q,\ve}\lp 1,x\rp\rp\rp\right|+ \ve + \ve\left| x\right|^q+\ve \left| \inst_{\rect}\lp \prd^{q,\ve}\lp 1,x\rp\rp\right|^q \nonumber\\ + &\les \left|x\ve + x\ve\left|x\right|^q \right| + \ve + \ve\left|x\right|^q+\ve \left|\ve + 2+x^2 \right|^q \nonumber\\ + &= \left| x\ve + x\ve \left| x\right|^q\right| + \ve + \ve\left| x\right|^q + \ve \mathfrak{p}_{1}^q + \end{align} + + Note that this takes care of our base case. Assume now that for all integers up to and including $n$, it is the case that: + \begin{align}\label{(6.2.39)} + \left| x^n - \real_{\rect}\lp \pwr_n^{q,\ve}\rp \lp x \rp \right| &\les \left| x\cdot x^{n-1}-x \cdot \real_{\rect}\lp \pwr_{n-1}^{q,\ve}\rp \lp x\rp\right| + \left| x \cdot \real_{\rect}\lp \pwr_{n-1}^{q,\ve}\rp \lp x\rp -\real_{\rect} \lp \pwr_n^{q,\ve} \rp \lp x \rp \right| \nonumber \\ + &\les \left| x\lp x^{n-1}-\real_{\rect} \lp \pwr^{q,\ve}_{n-1}\rp \lp x\rp\rp\right| + \ve + \ve|x|^q + \ve\left| \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp \lp x \rp \right| ^q\nonumber \\ + &\les \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + \ve|x|^q + \ve\mathfrak{p}_{n-1}^q + \end{align} + For the inductive case, we see that: + \begin{align} + \left|x^{n+1}-\real_{\rect}\lp \pwr_{n+1}^{q,\ve}\rp\lp x\rp \right| &\les \left| x^{n+1}-x\cdot \real_{\rect}\lp \pwr_{n}^{q,\ve}\rp \lp x \rp\right| + \left| x\cdot \real_{\rect}\lp \pwr^{q,\ve}_n\rp \lp x \rp - \real_{\rect} \lp \pwr^{q,\ve}_{n+1}\rp\right| \nonumber \\ + &\les \left|x\lp x^n-\real_{\rect} \lp \pwr^{q,\ve}_n\rp \lp x\rp\rp \right| + \ve + \ve|x|^q+\ve\left| \real_{\rect} \lp \pwr^{q,\ve}_{n}\rp \lp x \rp\right|^q \nonumber \\ + &\les \left|x\lp x^n-\real_{\rect} \lp \pwr^{q,\ve}_n\rp \lp x\rp\rp \right| + \ve + \ve|x|^q + \ve\mathfrak{p}^q_n + \end{align} + Note that since $\mathfrak{p}_n \in \mathcal{O} \lp \ve^{2(n-1)}\rp$ for $n\in \N \cap \lb 2,\infty \rp$, it is the case for all $x\in \R$ then that $\left| x^{n} - \real_{\rect} \lp \pwr^{q,\ve}_n\rp \lp x\rp\right| \in \mathcal{O} \lp \ve^{2q(n-1)} \rp$ for $n \ges 2$. + + Finally note that $\wid_{\hid \lp \pwr^{q,\ve}_0\rp}\lp \pwr^{q,\ve}_0\rp = 1$ from observation. For $n\in \N$, note that the second to last layer is the second to last layer of the $\prd^{q,\ve}$ network. Thus Lemma \ref{prd_network} tells us that: + \begin{align} + \wid_{\hid\lp \pwr^{q,\ve}_m\rp} \lp \pwr^{q,\ve}_n\rp = \begin{cases} + 1 & n=0 \\ + 24 & n\in \N + \end{cases} + \end{align} + This completes the proof of the lemma. +\end{proof} +\begin{remark}\label{rem:pwr_gets_deeper} + Note each power network $\pwr_n^{q,\ve}$ is at least as big as the previous power network $\pwr_{n-1}^{q,\ve}$, one differs from the other by one $\prd^{q, ve}$ network. +\end{remark} +\subsection{$\pnm_{n,C}^{q,\ve}$ and Neural Network Polynomials.} + +\begin{definition}[Neural Network Polynomials] + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. For fixed $q,\ve$, fixed $n \in \N_0$, and for $C = \{c_0,c_1,\hdots, c_n \} \in \R^{n+1}$ (the set of coefficients), we will define the following objects as neural network polynomials: + \begin{align} + \pnm^{q,\ve}_{n,C} \coloneqq \bigoplus^n_{i=0} \lp c_i \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rp + \end{align} +\end{definition} +\begin{remark} + Diagrammatically, these can be represented as +\end{remark} +\begin{figure}[h] +\begin{center} + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,475); %set diagram left start at 0, and has height of 475 + +%Shape: Rectangle [id:dp8950407412127579] +\draw (390,52) -- (455.33,52) -- (455.33,85) -- (390,85) -- cycle ; +%Shape: Rectangle [id:dp6602004057057332] +\draw (359.33,108.67) -- (454,108.67) -- (454,141.67) -- (359.33,141.67) -- cycle ; +%Shape: Rectangle [id:dp6567335394697266] +\draw (300,168.67) -- (455.33,168.67) -- (455.33,201.67) -- (300,201.67) -- cycle ; +%Shape: Rectangle [id:dp40847692689766735] +\draw (200,255.33) -- (456,255.33) -- (456,288.33) -- (200,288.33) -- cycle ; +%Shape: Rectangle [id:dp9479406055744195] +\draw (200.67,51.33) -- (358.67,51.33) -- (358.67,84.33) -- (200.67,84.33) -- cycle ; +%Shape: Rectangle [id:dp8579663805783284] +\draw (199.33,108) -- (330,108) -- (330,141) -- (199.33,141) -- cycle ; +%Shape: Rectangle [id:dp41506308397634806] +\draw (200.67,168.67) -- (268.67,168.67) -- (268.67,201.67) -- (200.67,201.67) -- cycle ; +%Straight Lines [id:da4565055641527326] +\draw (390.67,68.33) -- (361.33,68.33) ; +\draw [shift={(359.33,68.33)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. 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(10.93,3.29) ; +%Straight Lines [id:da6814861591796668] +\draw (200,185) -- (147.29,174.07) ; +\draw [shift={(145.33,173.67)}, rotate = 11.71] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da019305885926265143] +\draw (198.67,271) -- (145.1,189.34) ; +\draw [shift={(144,187.67)}, rotate = 56.74] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da8585029210721031] +\draw (616,172.33) -- (586.67,172.33) ; +\draw [shift={(584.67,172.33)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da9805678030848519] +\draw (78.67,169.67) -- (49.33,169.67) ; +\draw [shift={(47.33,169.67)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (412,217.73) node [anchor=north west][inner sep=0.75pt] {$\vdots $}; +% Text Node +\draw (406,61.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}^{q,\ve}_{0}$}; +% Text Node +\draw (406,118.07) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}^{q,\ve}_{1}$}; +% Text Node +\draw (403.33,177.07) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}^{q,\ve}_{2}$}; +% Text Node +\draw (265.33,58.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (404,263.07) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}^{q,\ve}_{n}$}; +% Text Node +\draw (249.33,115.73) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (222,176.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (525,162.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n+1,1}$}; +% Text Node +\draw (471.33,198.4) node [anchor=north west][inner sep=0.75pt] {$\vdots $}; +% Text Node +\draw (83,163.73) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}_{n+1,1}$}; +% Text Node +\draw (230.67,214.4) node [anchor=north west][inner sep=0.75pt] {$\vdots $}; +% Text Node +\draw (172,193.73) node [anchor=north west][inner sep=0.75pt] {$\vdots $}; + + +\end{tikzpicture} +\end{center} +\caption{Neural network diagram for an elementary neural network polynomial.} +\end{figure} + +\begin{lemma}[R\textemdash,2023]\label{6.2.9}\label{nn_poly}\label{mnm_prop} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. It is then the case for all $n\in\N_0$ and $x\in \R$ that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \pnm_{n,C}^{q,\ve}\rp \in C \lp \R, \R \rp $ + \item $\dep \lp \pnm_{n,C}^{q,\ve} \rp \les \begin{cases} + 1 & :n=0\\ + n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases}$ + \item $\param \lp \pnm_{n,C}^{q,\ve} \rp \les \begin{cases} + 2 & :n =0 \\ + \lp n+1\rp\lb 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases}$ \\~\\ + \item $\left|\sum^n_{i=0} c_ix^i - \real_{\rect} \lp \pnm_{n,C}^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=1} c_i\lp \left| x \lp x^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp $\\~\\ + Where $\mathfrak{p}_i$ are the set of functions defined for $i \in \N$ as such: + \begin{align} + \mathfrak{p}_1 &= \ve+1+|x|^2 \nonumber\\ + \mathfrak{p}_i &= \ve +\lp \mathfrak{p}_{i-1} \rp^2+|x|^2 + \end{align} + Whence it is the case that: + \begin{align} + \left|\sum^n_{i=0} c_ix^i - \real_{\rect} \lp \pnm_{n,C}^{q,\ve} \rp \lp x \rp \right| \in \mathcal{O} \lp \ve^{2q(n-1)}\rp + \end{align} + \item $\wid_1 \lp \pnm_{n,C}^{q,\ve} \rp = 2+23n+n^2 $ + \item $\wid_{\hid \lp \pnm_{n,C}^{q,\ve}\rp} \lp \pnm_{n,C}^{q,\ve}\rp \les\begin{cases} + 1 &:n=0 \\ + 24 + 2n &:n\in \N \end{cases}$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that by Lemma \ref{5.6.3}, Lemma \ref{power_prop}, and Lemma \ref{comp_prop} for all $n\in \N_0$ it is the case that: + \begin{align} + \real_{\rect}\lp \pnm_{n,C}^{q,\ve} \rp &= \real_{\rect} \lp \bigoplus^n_{i=0} \lb c_i \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb \rp \nonumber\\ + &= \sum^n_{i=1}c_i \real_{\rect}\lp \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve} \rp \nonumber\\ + &= \sum^n_{i=1}c_i\real_{\rect}\lp \pwr^{q,\ve}_i \rp\nonumber + \end{align} + Since Lemma \ref{power_prop} tells us that $\lp \real_{\rect} \lp \pwr_n^{q,\ve} \rp \rp \lp x \rp \in C \lp \R, \R \rp$, for all $n\in \N_0$ and since the finite sum of continuous functions is continuous, this proves Item (i). + + Note that $\pnm_n^{q,\ve}$ is only as deep as the deepest of the $\pwr^{q,\ve}_i$ networks, which from the definition is $\pwr_n^{q,\ve}$, which in turn also has the largest bound. Therefore, by Lemma \ref{comp_prop}, Lemma $\ref{5.3.3}$, Lemma $\ref{depth_prop}$, and Lemma \ref{power_prop}, we have that: + \begin{align} + \dep \lp \pnm_{n,C}^{q,\ve} \rp &\les \dep \lp \pwr_n^{q,\ve}\rp \nonumber\\ + &\les \begin{cases} + 1 & :n=0\\ + n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases} \nonumber + \end{align} + This proves Item (ii). + + Note next that for the case of $n=0$, we have that: + \begin{align} + \pnm_n^{q,\ve} = c_i \triangleright\pwr_0^{q,\ve} + \end{align} + This then yields us $2$ parameters. + + Note that each neural network summand in $\pnm_n^{q,\ve}$ consists of a combination of $\tun_k$ and $\pwr_k$ for some $k\in \N$. Each $\pwr_k$ has at least as many parameters as a tunneling neural network of that depth, as Lemma \ref{param_pwr_geq_param_tun} tells us. This, finally, with Lemma \ref{aff_effect_on_layer_architecture}, Corollary \ref{affcor}, and Lemma \ref{power_prop} then implies that: + \begin{align} + \param\lp \pnm^{q,\ve}_{n,C} \rp &= \param \lp \bigoplus^n_{i=0} \lb c_i \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb \rp\nonumber \\ + &\les \lp n+1 \rp \cdot \param \lp c_i \triangleright \lb \tun_1 \bullet \pwr_n^{q,\ve} \rb\rp \nonumber\\ + &\les \lp n+1 \rp \cdot \param \lp \pwr_n^{q,\ve} \rp \nonumber \\ + &\les \begin{cases} + 2 & :n =0 \\ + \lp n+1\rp\lb 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases} \nonumber + \end{align} + This proves Item (iii). + + Finally, note that for all $i\in \N$, Lemma \ref{power_prop}, and the triangle inequality then tells us that it is the case for all $i \in \N$ that: + \begin{align} + \left| x^i - \real_{\rect}\lp \pwr_i^{q,\ve}\rp \lp x \rp \right| &\les \left| x^i-x \cdot \real_{\rect}\lp \pwr_{i-1}^{q,\ve}\rp \lp x\rp\right| + \left| x \cdot \real_{\rect}\lp \pwr_{i-1}^{q,\ve}\rp \lp x\rp -\real_{\rect} \lp \pwr_i^{q,\ve} \rp \lp x \rp \right| \nonumber \\ + \end{align} + This, Lemma \ref{6.2.9}, and the fact that instantiation of the tunneling neural network leads to the identity function (Lemma \ref{6.2.2} and Lemma \ref{comp_prop}), together with Lemma \ref{scalar_left_mult_distribution}, and the absolute homogeneity condition of norms, then tells us that for all $x\in \R$, and $c_0,c_1,\hdots, c_n \in \R$ it is the case that: + \begin{align} + &\left|\sum^n_{i=0} c_ix^i - \real_{\rect} \lp \pnm^{q,\ve}_{n,C} \lp x\rp \rp \right| \nonumber\\ + &= \left| \sum^n_{i=0} c_ix^i - \real_{\rect} \lb \bigoplus^n_{i=0} \lb c_i \triangleright \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve} \rb \rb\lp x \rp\right| \nonumber \\ + &=\left| \sum^n_{i=1} c_ix^i-\sum_{i=0}^n c_i \lp \inst_{\rect}\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb\lp x\rp\rp\right| \nonumber\\ + &\les \sum_{i=1}^n \left|c_i\right| \cdot\left| x^i - \inst_{\rect}\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb\lp x\rp\right| \nonumber\\ + &\les \sum^n_{i=1} \left|c_i\right|\cdot\lp \left| x \lp x^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp \nonumber + \end{align} + Note however that since for all $x\in \R$ and $i \in \N \cap \lb 2, \infty\rp$, Lemma \ref{prd_network} tells us that $\left| x^{i} - \real_{\rect} \lp \pwr^{q,\ve}_i\rp \lp x\rp\right| \in \mathcal{O} \lp \ve^{2q\lp i-1\rp} \rp$, this, and the fact that $f+g \in \mathcal{O}\lp x^a \rp$ if $f \in \mathcal{O}\lp x^a\rp$, $g \in \mathcal{O}\lp x^b\rp$, and $a \ges b$, then implies that: + \begin{align} + \sum^n_{i=1} \left| c_i\right|\cdot\lp \left| x \lp x^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp \in \mathcal{O} \lp \ve^{2q(n-1)}\rp + \end{align} + This proves Item (iv). + + Note next in our construction $\aff_{0,1}$ will require tunneling whenever $i\in \N$ in $\pwr_{i}^{q,\ve}$. Lemma \ref{aff_effect_on_layer_architecture} and Corollary \ref{affcor} then tell us that: + \begin{align} + \wid_1 \lp \pnm_n^{q,\ve} \rp &= \wid_1 \lp \bigoplus^n_{i=0} \lb c_i \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb\rp \nonumber\\ + &= \wid_1 \lp \bigoplus^n_{i=0}\pwr^{q,\ve}_i\rp \nonumber \\ + &\les \sum^n_{i=0}\wid_1 \lp \pwr^{q,\ve}_i\rp =2 + \frac{n}{2}\lp 24+24+2\lp n-1\rp\rp = 2+23n+n^2 \nonumber \\ + \end{align} + This proves Item (v). + + Finally note that from the definition of the $\pnm_{n,C}^{q,\ve}$, it is evident that $\wid_{\hid\lp \pwr_{0,C}^{q,\ve}\rp}\lp \pwr_{0,C}^{q,\ve}\rp = 1$ since $\pwr_{0,C}^{q,\ve} = \aff_{0,1}$. Other than this network, for all $i \in \N$, $\pwr_{i,C}^{q,\ve}$ end in the $\prd^{q,\ve}$ network, and the deepest of the $\pwr_i^{q,\ve}$ networks is $\pwr^{q,\ve}_n$ inside $\pnm_{n,C}^{q,\ve}$. All other $\pwr_i^{q,\ve}$ must end in tunnels. Whence in the second to last layer, Lemma \ref{prd_network} tells us that: + \begin{align} + \wid_{\hid\lp \pnm_{n,C}^{q,\ve}\rp} \les \begin{cases} + 1 &: n =0 \\ + 24+2n &:n \in \N + \end{cases} + \end{align} + This completes the proof of the Lemma. +\end{proof} +\subsection{$\xpn_n^{q,\ve}$, $\csn_n^{q,\ve}$, $\sne_n^{q,\ve}$, and Neural Network Approximations of $e^x$, $\cos(x)$, and $\sin(x)$.} +Once we have neural network polynomials, we may take the next leap to transcendental functions. Here, we will explore neural network approximations for three common transcendental functions: $e^x$, $\cos(x)$, and $\sin(x)$. + +\begin{lemma} + Let $\nu_1,\nu_2 \in \neu$, $f,g \in C \lp \R, \R \rp$, and $\ve_1,\ve_2 \in \lp 0 ,\infty \rp$ such that for all $x\in \R$ it holds that $\left| f(x) - \real_{\rect} \lp \nu_1 \rp \right| \les \ve_1 $ and $\left| g(x) - \real_{\rect} \lp \nu_2 \rp \right| \les \ve_2$. It is then the case for all $x \in \R$ that: + \begin{align}\label{6.2.14} + \left| \lb f+g \rb \lp x \rp - \real_{\rect} \lp \lb \nu_1 \oplus \nu_2 \rb \rp \lp x \rp\right| \les \ve_1 + \ve_2 + \end{align} +\end{lemma} +\begin{proof} + Note that the triangle inequality tells us: + \begin{align} + \left| \lb f+g \rb \lp x \rp - \real_{\rect} \lb \nu_1 \oplus \nu_2 \rb \lp x \rp \right| &= \left| f\lp x \rp +g\lp x \rp -\real_{\rect} \lp \nu_1\rp \lp x \rp -\real_{\rect} \lp \nu_2 \rp\lp x \rp \right|\nonumber \\ + &\les \left| f\lp x \rp -\real_{\rect}\lp \nu_1 \rp \lp x \rp \right| + \left| g\lp x \rp - \real_{\rect} \lp \nu_2 \rp \lp x \rp \right| \nonumber\\ + &\les \ve_1 + \ve_2 \nonumber + \end{align} +\end{proof} +\begin{lemma}\label{6.2.8} + Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$, $\ve_1,\ve_2,...,\ve_n \in \lp 0,\infty \rp$ and $f_1,f_2,...,f_n \in C\lp \R, \R \rp$ such that for all $i \in \{1,2,...,n\}$, and for all $x\in \R$, it is the case that, $\left| f_i\lp x \rp - \real_{\rect} \lp \nu_i \rp\lp x \rp \right| \les \ve_i$. It is then the case for all $x\in \R$, that: + \begin{align} + \left| \sum^n_{i=1} f_i \lp x \rp -\bigoplus^n_{i=1} \lp \real_{\rect}\lp \nu_i \rp \rp \lp x\rp\right| \les \sum_{i=1}^n \ve_i + \end{align} +\end{lemma} +\begin{proof} + This is a consequence of a finite number of applications of (\ref{6.2.14}). +\end{proof} +\begin{definition}[R\textemdash 2023, $\xpn_n^{q,\ve}$ and the Neural Network Taylor Approximations for $e^x$ around $x=0$] + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, and let $\pwr_n^{q,\ve}$ be as in Lemma \ref{power_prop}. We define, for all $n\in \N_0$, the family of neural networks $\xpn_n^{q,\ve} as$: + \begin{align} + \xpn_n^{q,\ve}\coloneqq \bigoplus^n_{i=0} \lb \frac{1}{i!} \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb + \end{align} +\end{definition} + +\begin{lemma}[R\textemdash,2023]\label{6.2.9}\label{tay_for_exp}\label{xpn_properties} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. It is then the case for all $n\in\N_0$ and $x\in \R$ that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \xpn_n^{q,\ve}\rp \lp x \rp\in C \lp \R, \R \rp $ + \item $\dep \lp \xpn_n^{q,\ve} \rp \les \begin{cases} + 1 & :n=0\\ + n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases}$ + \item $\param \lp \xpn_n^{q,\ve} \rp \les \begin{cases} + 2 & :n =0 \\ + \lp n+1\rp\lb 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases}$ \\~\\ + \item \begin{align*}\left|\sum^n_{i=0} \lb \frac{x^i}{i!} \rb- \real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=1} \frac{1}{i!}\lp \left| x \lp x^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp \end{align*}\\~\\ + Where $\mathfrak{p}_i$ are the set of functions defined for $i \in \N$ as such: + \begin{align} + \mathfrak{p}_1 &= \ve+1+|x|^2 \nonumber\\ + \mathfrak{p}_i &= \ve +\lp \mathfrak{p}_{i-1} \rp^2+|x|^2 + \end{align} + Whence it is the case that: + \begin{align} + \left|\sum^n_{i=0} \lb \frac{x^i}{i!} \rb- \real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right|\in \mathcal{O} \lp \ve^{2q(n-1)}\rp + \end{align} + \item $\wid_1 \lp \xpn_n^{q,\ve} \rp = 2+23n+n^2 $ + \item $\wid_{\hid \lp \xpn^n_{q,\ve} \rp}\lp \xpn_n^{q,\ve}\rp \les 24 + 2n$ + \end{enumerate} +\end{lemma} +\begin{proof} + This follows straightforwardly from Lemma \ref{nn_poly} with $c_i \curvearrowleft \frac{1}{i!}$ for all $n \in \N$ and $i \in \{0,1,\hdots, n\}$. In particular, Item (iv) benefits from the fact that for all $i \in \N_0$, it is the case that $\frac{1}{i!} \ges 0$. +\end{proof} + +\begin{lemma}[R\textemdash, 2023] + + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}.$ It is then the case for all $n\in\N_0$ and $x\in \lb a,b \rb\subsetneq \R$, where $0 \in \lb a,b\rb \subsetneq \R$ that: + \begin{align} + \left| e^x - \real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=0} \frac{1}{i!}\lp \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q \rp + \frac{e^{b}\cdot |x|^{n+1}}{(n+1)!} + \end{align} +\end{lemma} +\begin{proof} + Note that Taylor's theorem states that for $x \in \lb a,b\rb \subsetneq \R$ it is the case that: + \begin{align} + e^x = \sum^n_{i=0} \lb \frac{x^i}{i!} \rb + \frac{e^{\xi}\cdot x^{n+1}}{(n+1)!} + \end{align} + Where $\xi$ is between $0$ and $x$ in the Lagrange form of the remainder. Note then, for all $n\in \N_0$, $x\in \lb a,b \rb \subsetneq \R$, and $\xi$ between $0$ and $x$, it is the case, by monotonicity of $e^x$ that the second summand is bounded by: + \begin{align} + \frac{e^\xi \cdot x^{n+1}}{(n+1)!} \les \frac{e^b\cdot |x|^{n+1}}{(n+1)!} + \end{align} + This, and the triangle inequality, then indicates that for all $x \in \lb a,b \rb \subsetneq \R$, and $\xi$ between $0$ and $x$ that: + \begin{align} + \left| e^x -\real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right| &=\left| \sum^n_{i=0} \lb \frac{x^i}{i!} \rb + \frac{e^{\xi}\cdot x^{n+1}}{(n+1)!}-\real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp\right| \nonumber\\ + &\les \left| \sum^n_{i=0} \lb \frac{x^i}{i!} \rb - \real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right| + \frac{e^{b}\cdot |x|^{n+1}}{(n+1)!} \nonumber \\ + &\les \sum^n_{i=1} \frac{1}{i!}\lp \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q \rp + \frac{e^{b}\cdot |x|^{n+1}}{(n+1)!} \nonumber + \end{align} + Whence we have that for fixed $n\in \N_0$ and $b \in \lb 0, \infty\rp$, the last summand is constant, whence it is the case that: + \begin{align} + \left| e^x -\real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right| \in \mathcal{O} \lp \ve^{2q(n-1)}\rp + \end{align} +\end{proof} +\begin{definition}[The $\mathsf{Csn}_n^{q,\ve}$ Networks, and Neural Network Cosines] + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $\pwr^{q,\ve}_n$ be a neural networks as defined in Definition \ref{def:pwr}. We will define the neural networks $\mathsf{Csn}_{n}^{q,\ve}$ as: + \begin{align} + \mathsf{Csn}_n^{q,\ve} \coloneqq \bigoplus^n_{i=0} \lb \frac{(-1)^i}{2i!}\triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_{2i}^{q,\ve}\rb \rb + \end{align} + +\end{definition} + +\begin{lemma}[R\textemdash, 2023]\label{6.2.9}\label{csn_properties} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. It is then the case for all $n\in\N_0$ and $x\in \R$ that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \csn_n^{q,\ve}\rp \lp x\rp\in C \lp \R, \R \rp $ + \item $\dep \lp \csn_n^{q,\ve}\rp \les \begin{cases} + 1 & :n=0\\ + 2n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases}$ + \item $\param \lp \csn_n^{q,\ve} \rp \les \begin{cases} + 2 & :n =0 \\ + \lp 2n+1\rp\lb 4^{2n+\frac{3}{2}} + \lp \frac{4^{2n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases}$ \\~\\ + \item $\left|\sum^n_{i=0} \frac{(-1)^i}{2i!}x^{2i} - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| x \lp x^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{2i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{2i-1}^q \rp $\\~\\ + Where $\mathfrak{p}_i$ are the set of functions defined for $i \in \N$ as such: + \begin{align} + \mathfrak{p}_1 &= \ve+1+|x|^2 \nonumber\\ + \mathfrak{p}_i &= \ve +\lp \mathfrak{p}_{i-1} \rp^2+|x|^2 + \end{align} + Whence it is the case that: + \begin{align} + \left|\sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}x^{2i} - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| \in \mathcal{O} \lp \ve^{2q(2n-1)}\rp + \end{align} + \end{enumerate} + +\end{lemma} + +\begin{proof} + Item (i) derives straightforwardly from Lemma \ref{nn_poly}. This proves Item (i). + + Next, observe that since $\csn_n^{q,\ve}$ will contain, as the deepest network in the summand, $\pwr_{2n}^{q,\ve}$, we may then conclude that + \begin{align} + \dep \lp \csn_n^{q,\ve} \rp &\les \dep \lp \pwr_{2n}^{q,\ve}\rp \nonumber\\ + &\les \begin{cases} + 1 & :n=0\\ + 2n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases} \nonumber + \end{align} + This proves Item (ii). + + A similar argument to the above, Lemma \ref{aff_effect_on_layer_architecture}, and Corollary \ref{affcor} reveals that: + \begin{align} + \param\lp \csn_n^{q,\ve} \rp &= \param \lp \bigoplus^n_{i=0} \lb \frac{\lp -1\rp^i}{2i!} \triangleright\lb \tun_{\max_i \left\{\dep \lp \pwr_i^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_i\rp} \bullet \pwr_i^{q,\ve}\rb \rb \rp\nonumber \\ + &\les \lp n+1 \rp \cdot \param \lp c_i \triangleright \lb \tun_1 \bullet \pwr_{2n}^{q,\ve} \rb\rp \nonumber\\ + &\les \lp n+1 \rp \cdot \param \lp \pwr_{2n}^{q,\ve} \rp \nonumber \\ + &\les \begin{cases} + 2 & :n =0 \\ + \lp n+1\rp\lb 4^{2n+\frac{3}{2}} + \lp \frac{4^{2n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases} \nonumber + \end{align} + This proves Item (iii). + + In a similar vein, we may argue from Lemma \ref{nn_poly} and from the absolute homogeneity property of norms that: + \begin{align} + &\left|\sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}x^{2i} - \real_{\rect} \lp \csn_n^{q,\ve} \lp x\rp \rp \right| \nonumber\\ + &= \left| \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}x^{2i} - \real_{\rect} \lb \bigoplus^n_{i=0} \lb \frac{\lp -1\rp^i}{2i!} \triangleright \tun_{\max_{2i} \left\{\dep \lp \pwr_{2i}^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_{2i}\rp} \bullet \pwr_{2i}^{q,\ve} \rb \rb\lp x \rp\right| \nonumber \\ + &=\left| \sum^n_{i=1} \frac{\lp -1\rp^i}{2i!}x^{2i}-\sum_{i=0}^n \frac{\lp -1 \rp^i}{2i!} \lp \inst_{\rect}\lb \tun_{\max_{2i} \left\{\dep \lp \pwr_{2i}^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_{2i}\rp} \bullet \pwr_{2i}^{q,\ve}\rb\lp x\rp\rp\right| \nonumber\\ + &\les \sum_{i=1}^n \left|\frac{\lp -1\rp^i}{2i!} \right|\cdot\left| x^{2i} - \inst_{\rect}\lb \tun_{\max_{2i} \left\{\dep \lp \pwr_{2i}^{q,\ve} \rp\right\} +1 - \dep \lp \pwr^{q,\ve}_{2i}\rp} \bullet \pwr_{2i}^{q,\ve}\rb\lp x\rp\right| \nonumber\\ + &\les \sum^n_{i=1} \left|\frac{\lp -1\rp^i}{2i!}\right|\cdot \left|\lp \left| x \lp x^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{2i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{2i-1}^q \rp\right| \nonumber + \end{align} + Whence we have that: + \begin{align} + \left|\sum^n_{i=0} \lb \frac{\lp -1\rp^i x^{2i}}{2i!} \rb- \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right|\in \mathcal{O} \lp \ve^{2q(2n-1)}\rp + \end{align} + This proves Item (iv). +\end{proof} + +\begin{lemma}[R\textemdash, 2023] + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}.$ It is then the case for all $n\in\N_0$ and $x\in [a,b]\subseteq \lb 0,\infty \rp$ that: + \begin{align} + \left| \cos\lp x\rp - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| \les \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}\lp \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q \rp + + \frac{|x|^{n+1}}{(n+1)!}\nonumber + \end{align} +\end{lemma} + +\begin{proof} + Note that Taylor's theorem states that for all $x \in \lb a,b\rb \subsetneq \R$, where $0 \in \lb a,b\rb$, it is the case that: + \begin{align} + \cos\lp x \rp= \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}x^i + \frac{\cos^{\lp n+1\rp}\lp \xi \rp \cdot x^{n+1}}{(n+1)!} + \end{align} + Note further that for all $n \in \N_0$, and $x \in \R$, it is the case that $\cos^{\lp n \rp} \lp x\rp \les 1$. Whence we may conclude that for all $n\in \N_0$, $x\in \lb a,b \rb \subseteq \R$, where $0 \in \lb a,b\rb$ and $\xi$ between $0$ and $x$, we may bound the second summand by: + \begin{align} + \frac{\cos^{\lp n+1\rp}\lp \xi \rp \cdot x^{n+1}}{(n+1)!} \les \frac{|x|^{n+1}}{\lp n+1\rp!} + \end{align} + This, and the triangle inequality, then indicates that for all $x \in \lb a,b \rb \subsetneq \lb 0,\infty\rp$ and $\xi \in \lb 0,x\rb$: + \begin{align} + \left| \cos \lp x \rp -\real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| &=\left| \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}x^i + \frac{\cos^{(n+1)}\lp \xi \rp \cdot x^{n+1}}{(n+1)!}-\real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp\right| \nonumber\\ + &\les \left| \sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}x^i - \real_{\rect} \lp \csn_n^{q,\ve} \rp \lp x \rp \right| + \frac{|x|^{n+1}}{(n+1)!} \nonumber \\ + &\les \sum^n_{i=1} \left|\frac{\lp -1\rp^i}{2i!}\right|\cdot \left|\lp \left| x \lp x^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{2i-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{2i-1}^q \rp\right| \nonumber\\&+ \frac{|x|^{n+1}}{(n+1)!} \nonumber + \end{align} + This completes the proof of the Lemma. +\end{proof} + +\begin{definition}[R\textemdash, 2023, The $\mathsf{Sne}_n^{q,\ve}$ Newtorks and Neural Network Sines.]. + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $\pwr^{q,\ve}$ be a neural network defined in Definition \ref{def:pwr}. We will define the neural network $\mathsf{Csn}_{n,q,\ve}$ as: + \begin{align} + \mathsf{Sne}_n^{q,\ve} \coloneqq \csn^{q,\ve} \bullet \aff_{1, -\frac{\pi}{2}} + \end{align} +\end{definition} + +\begin{lemma}[R\textemdash, 2023]\label{6.2.9}\label{sne_properties} + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. It is then the case for all $n\in\N_0$ and $x\in \R$ that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \sne_n^{q,\ve}\rp \in C \lp \R, \R \rp $ + \item $\dep \lp \sne_n^{q,\ve}\rp \les \begin{cases} + 1 & :n=0\\ + 2n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases}$ + \item $\param \lp \sne_n^{q,\ve} \rp \les \begin{cases} + 2 & :n =0 \\ + \lp 2n+1\rp\lb 4^{2n+\frac{3}{2}} + \lp \frac{4^{2n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases}$ \\~\\ + \item \begin{align}&\left|\sum^n_{i=0} \frac{(-1)^i}{2i!}{\lp x-\frac{\pi}{2}\rp}^{2i} - \real_{\rect} \lp \sne_n^{q,\ve} \rp \lp x \rp \right| \nonumber\\ + &= \left|\sum^n_{i=0} \frac{(-1)^i}{2i!}{\lp x-\frac{\pi}{2}\rp}^{2i} - \real_{\rect} \lp \csn_n^{q,\ve} \bullet \aff_{1,-\frac{\pi}{2}}\rp \lp x \rp \right|\nonumber\\ + &\les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| \lp x -\frac{\pi}{2}\rp\lp \lp x -\frac{\pi}{2}\rp^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x-\frac{\pi}{2}\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp \nonumber \end{align}\\~\\ + Where $\mathfrak{p}_i$ are the set of functions defined for $i \in \N$ as such: + \begin{align} + \mathfrak{p}_1 &= \ve+1+|x|^2 \nonumber\\ + \mathfrak{p}_i &= \ve +\lp \mathfrak{p}_{i-1} \rp^2+|x|^2 + \end{align} + Whence it is the case that: + \begin{align} + \left|\sum^n_{i=0} \frac{\lp -1\rp^i}{2i!}\lp x-\frac{\pi}{2}\rp^{2i} - \real_{\rect} \lp \sne_n^{q,\ve} \rp \lp x \rp \right| \in \mathcal{O} \lp \ve^{2q(2n-1)}\rp + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + This follows straightforwardly from Lemma \ref{csn_properties}, and the fact that by Corollary \ref{affcor}, there is not a change to the parameter count, by Lemma \ref{comp_cont}, there is no change in depth, by Lemma \ref{aff_prop}, and Lemma \ref{csn_properties}, continuity is preserved, and the fact that $\aff_{1,-\frac{\pi}{2}}$ is exact and hence contributes nothing to the error, and finally by the fact that $\aff_{1,-\frac{\pi}{2}} \rightarrow \lp \cdot\rp -\frac{\pi}{2}$ under instantiation, assures us that the $\sne^{q,\ve}_n$ has the same error bounds as $\csn_n^{q,\ve}$. +\end{proof} + +\begin{lemma}[R\textemdash, 2023] + Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}.$ It is then the case for all $n\in\N_0$ and $x\in [a,b]\subseteq \lb 0,\infty \rp$ that: + \begin{align} + &\left| \sin\lp x\rp - \real_{\rect} \lp \sne_n^{q,\ve} \rp \lp x \rp \right|\nonumber \\ + &\les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| \lp x -\frac{\pi}{2}\rp\lp \lp x -\frac{\pi}{2}\rp^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x-\frac{\pi}{2}\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp \nonumber\\ + &+\frac{|x|^{n+1}}{(n+1)!}\label{sin_diff} + \end{align} +\end{lemma} +\begin{proof} + Note that the fact that $\sin\lp x\rp = \cos\lp x-\frac{\pi}{2}\rp$, Lemma \ref{comp_prop}, and Lemma \ref{aff_prop} then renders (\ref{sin_diff}) as: + \begin{align} + &\left| \sin\lp x\rp - \inst_{\rect}\lp \sne_n^{q,\ve}\rp\right| \nonumber\\ + &= \left| \cos \lp x - \frac{\pi}{2}\rp - \inst_{\rect}\lp \csn_n^{q,\ve}\bullet \aff_{1,-\frac{\pi}{2}}\rp\lp x\rp\right| \nonumber\\ + &=\left| \cos \lp x-\frac{x}{2}\rp - \inst_{\rect}\csn_n^{q,\ve}\lp x-\frac{\pi}{2} \rp\right| \nonumber \\ + &\les \sum^n_{i=1} \left| \frac{\lp -1\rp^i}{2i!}\right|\lp \left| \lp x -\frac{\pi}{2}\rp\lp \lp x -\frac{\pi}{2}\rp^{2i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp x-\frac{\pi}{2}\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{i-1}^q \rp+ \frac{|x|^{n+1}}{(n+1)!}\nonumber + \end{align} +\end{proof} + +\begin{remark} + Note that under these neural network architectures the famous Pythagorean identity $\sin^2\lp x\rp + \cos^2 \lp x\rp = 1$, may be rendered approximately, for fixed $n,q,\ve$ as: $\lb \sqr^{q,\ve}\bullet \csn^{q,\ve}_n \rb \oplus\lb \sqr^{q,\ve}\bullet \sne^{q,\ve}_n\rb$. 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+\setcounter{nn@lastlayerindex@counter}{4} +\setcounter{nn@thislayerindex@counter}{5} +} diff --git a/Dissertation/ann_rep_brownian_motion_monte_carlo.pdf b/Dissertation/ann_rep_brownian_motion_monte_carlo.pdf new file mode 100644 index 0000000..72f695e Binary files /dev/null and b/Dissertation/ann_rep_brownian_motion_monte_carlo.pdf differ diff --git a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex new file mode 100644 index 0000000..c82f374 --- /dev/null +++ b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex @@ -0,0 +1,1050 @@ +\chapter{ANN representations of Brownian Motion Monte Carlo} +\textbf{This is tentative without any reference to $f$.} + +\begin{lemma}[R--,2023] + Let $d,M \in \N$, $T \in (0,\infty)$ , $\act \in C(\R,\R)$, $ \Gamma \in \neu$, satisfy that $\real_{\act} \lp \mathsf{G}_d \rp \in C \lp \R^d, \R \rp$, for every $\theta \in \Theta$, let $\mathcal{U}^\theta: [0,T] \rightarrow [0,T]$ and $\mathcal{W}^\theta:[0,T] \rightarrow \R^d$ be functions , for every $\theta \in \Theta$, let $U^\theta: [0,T] \rightarrow \R^d \rightarrow \R$ satisfy satisfy for all $t \in [0,T]$, $x \in \R^d$ that: + \begin{align} + U^\theta(t,x) = \frac{1}{M} \lb \sum^M_{k=1} \lp \real_{\act} \lp \Gamma \rp \rp \lp x+ \mathcal{W}^{\lp \theta,0,-k \rp } \rp \rb + \end{align} + Let $\mathsf{U}^\theta_t \in \neu$ , $\theta \in \Theta$ satisfy for all $\theta \in \Theta$, $t \in [0,T]$ that: + \begin{align} + \mathsf{U}^\theta_t = \lb \bigoplus^M_{k=1} \lp \frac{1}{M} \triangleright \lp \mathsf{G}_d \bullet \aff_{\mathbb{I}_d, \mathcal{W}^{\lp \theta,0,-k \rp}_{T-t}} \rp \rp \rb + \end{align} + It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for all $\theta_1,\theta_2 \in \Theta$, $t_1,t_2 \in [0,T]$ that $\lay \lp \mathsf{U}^{\theta_1}_{t_1} \rp = \lay \lp \mathsf{U}^{\theta_2}_{t_2} \rp$. + \item for all $\theta \in \Theta$, $t \in [0,T]$, that $\dep \lp \mathsf{U}^\theta_t \rp \les \dep (\mathsf{G}_d)$ + \item for all $\theta \in \Theta$, $t \in [0,T]$ that: + \begin{align} + \left\| \lay\lp \mathsf{U}^\theta_t \rp \right\|_{\max} \les \|\lay \lp \mathsf{G}_d \rp \|_{\max} \lp 1+ \sqrt{2} \rp M + \end{align} + \item for all $\theta \in \Theta$, $t \in [0,T]$, $x \in \R^d$ that $U^\theta (t,x) = \lp \real_{\act} \lp \mathbf{U}^\theta_t \rp \rp \lp x \rp $ and + \item for all $\theta \in \Theta$, $t \in [0,T]$ that: + \begin{align} + \param \lp \mathsf{U}^\theta_t \rp \les 2 \dep \lp \mathsf{G}_d \rp \lb \lp 1+\sqrt{2} \rp M \left\| \lay \lp \mathsf{G}_d \rp \right\|_{\max}\rb^2 + \end{align} + \end{enumerate} +\end{lemma} + +\begin{proof} + Throughout the proof let $\mathsf{P}^\theta_t \in \neu$, $\theta \in \Theta$, $t \in [0,T]$ satisfy for all $\theta \in \Theta$, $t \in [0,T]$ that: + \begin{align} + \mathsf{P}^\theta_t = \bigoplus^M_{k=1} \lb \frac{1}{M} \triangleright \lp \mathsf{G}_d \bullet \aff_{\mathbb{I}_d, \mathcal{W}^{\theta,0,-k}_{T-t}} \rp \rb + \end{align} +Note the hypothesis that for all $\theta \in \Theta$, $t \in [0,T]$ it holds that $\mathcal{W}^\theta_t \in \R^d$ and Lemma \ref{5.6.5} applied for every $\theta \in \Theta$ $t \in [0,T]$ with $v \curvearrowleft M$, $ c_{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \frac{1}{M} \rp_{i \in \{u,u+1,...,v\}}$, $\lp B_i \rp _{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \mathcal{W}^{\lp \theta, 0 , -k \rp }_{T-t} \rp_{k \in \{1,2,...,M\}}$, $\lp \nu_i \rp_{i \in \{u,u+1,...,v\}} \curvearrowleft \lp \mathsf{G}_d \rp _{i \in \{u,u+1,...,v\}}$, $\mu \curvearrowleft \Phi^\theta_t$ and with the notation of Lemma \ref{5.6.5} tells us that for all $\theta \in \Theta$, $t \in [0,T]$, and $x \in \R^d$ it holds that: La lala +\begin{align}\label{8.0.6} + \lay \lp \mathsf{P}^\theta_t \rp = \lp d, M \wid_1 \lp \G \rp, M\wid_2 \lp \G \rp,...,M\wid_{\dep\lp \G \rp -1}\lp \G \rp ,1\rp = \lay \lp \sP^0_0 \rp \in \N^{\dep \lp \G \rp +1} +\end{align} +and that: +\begin{align}\label{8.0.7} + \lp \real_{\act} \lp \sP^\theta_t \rp \rp \lp x \rp &= \frac{1}{M} \lb \sum^M_{k=1} \lp \real_{\act} \lp \G \rp \rp \lp x + \mathcal{W}^{\lp \theta,0,-k \rp}_{T-t} \rp \rb \nonumber \\ + &= \U^\theta \lp t,x \rp +\end{align} +This proves Item (i). + +Note that (\ref{8.0.6}), and (\ref{8.0.7}) also implies that: +\begin{align} + \lay \lp \U^\theta_t \rp &= \lay \lp \sP^\theta_t \rp \nonumber \\ + &= \lp d, \wid_1 \lp \sP^\theta_t \rp, \wid_2 \lp \sP ^\theta_t \rp,..., \wid_{\dep \lp \G \rp } \lp \sP^\theta_t \rp,t \rp \nonumber\\ + &= \lay \lp \U^0_0 \rp \in \N^{\dep \lp \G \rp +1} +\end{align} +This indicates that for all $\theta \in \Theta$, $t \in [0,T]$ it is the case that: +\begin{align} + \left\| \lay \lp \U^\theta_t \rp \right\|_{\infty} &= \left\| \lay \lp \U^0_0 \rp \right\|_{\infty} \nonumber\\ + &= \max_{k \in \{1,2,...,\dep(\G)\}} \lp \wid_k \lp \sP^0_0 \rp \rp \nonumber +\end{align} +This, (\ref{8.0.6}), and Lemma \ref{comp_prop} ensure that for all $\theta \in \Theta$, $t \in [0,T]$ it is the case that: +\begin{align} + \left\| \lay \lp \U^\theta_t \rp \right\|_{\infty} &= \left\| \lay \lp \U^0_0 \rp \right\|_{\infty} \les \left\| \lay \lp \sP^0_0 \rp \right\|_{\infty} \les M \left\| \lay \lp \G \rp \right\|_{\infty} \nonumber\\ + &\les M\left\| \lay \lp \G \rp \right\|_{\infty} + M \lb \left\| \lay \lp \U^0_0 \rp \right\|_{\infty} \rb +\end{align} +Then \cite[Corollary~4.3]{hutzenthaler_strong_2021}, with $\gamma \curvearrowleft 0$, $\beta \curvearrowleft M$, $k \curvearrowleft 1$, $\alpha_0 \curvearrowleft \left\| \lay \lp \G \rp \right\|_{\infty}$, $\alpha_1 \curvearrowleft 0$, $\lp x_i \rp_{i \in \{0,1,...,k\}} \curvearrowleft \lp \left\| \lay \lp \U^0_0 \rp \right\|_{\infty} \rp _{i \in \{0,1,...,n\}} $ in the notation of \cite[Corollary~4.3]{hutzenthaler_strong_2021} yields for all $\theta \in \Theta$, $t \in [0,T]$ that: +\begin{align} + \left\| \lay \lp \U^\theta_t \rp \right\|_{\infty} &\les \frac{1}{2} \lp \left\| \lay \lp \G \rp \right\|_{\infty} \rp \lp 1+\sqrt{2} \rp M \nonumber \\ + &\les \lp \left\|\lay \lp \G \rp \right\|_{\infty} \rp \lp 1+\sqrt{2} \rp M \nonumber +\end{align} +Note that Lemma \ref{comp_prop}, Item (iii), proves that for all $\theta \in \Theta$, $t\in [0,T]$ it is the case that: +\begin{align} + \dep \lp \U^\theta_t \rp = \dep \lp \U^0_0 \rp = \dep \lp \G \rp +\end{align} + +This proves Items (ii)--(iii) and (\ref{8.0.7}) proves Item (iv). + +Items (ii)--(iii) together shows that for all $\theta \in \Theta$, $t \in [0,T]$ it is the case that: +\begin{align} + \param \lp \U^\theta_t \rp &\les \sum _{k=1}^{\dep \lp \U^\theta_t \rp } \left\|\lay \lp \U^\theta_t \rp \right\|_{\max} \nonumber\\ + &=\dep \lp \U^\theta_t \rp \left\| \lay \lp \U^\theta_t \rp \right\|_{\infty} \nonumber \\ + &\les \dep \lp \U^\theta_t \rp \lp \left\| \lay \lp \G \rp \right\|_{\infty} \rp \lp 1+\sqrt{2} \rp M \nonumber\\ + &= \dep \lp \G \rp \lp \left\| \lay \lp \G \rp \right\|_{\infty} \rp \lp 1+\sqrt{2}\rp M \nonumber +\end{align} +This proves Item (v) and hence the whole lemma. +\end{proof} +\section{The $\mathsf{E}^{N,h,q,\ve}_n$ Neural Network} +\begin{lemma}[R\textemdash, 2023]\label{mathsfE} + Let $n, N\in \N$ and $h \in \lp 0,\infty\rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{N,h,q,\ve}_{n} \in \neu$ be the neural network given by: + \begin{align} + \mathsf{E}^{N,h,q,\ve}_n = \xpn_n^{q,\ve} \bullet \etr^{N,h} + \end{align} + It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for all $x \in \R^{N+1}$ we have that $\lp \real_{\rect}\lp \mathsf{E}^{N,h,q,\ve}_n \rp\rp\lp x\rp \in C \lp \R^{N+1},\R\rp$ + \item $\dep\lp \mathsf{E}^{N,h,q,\ve}_n \rp \les \begin{cases} + 1 & :n=0\\ + n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\ges 1 + \end{cases} $ + \item \begin{align*}&\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp \\&\les \begin{cases} + N+2 & :n =0 \\ + \lp \frac{1}{2}N+1 \rp\lp n+1\rp\lb 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases} \end{align*} + \item for all $x = \{x_0,x_1,\hdots, x_N \}\in \R^{N+1}$, where $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ we have that: + \begin{align} + &\left| \exp \lb \int^b_afdx\rb - \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp\lp f \lp \lb x \rb _{*,*}\rp\rp\right| \nonumber\\ + &\les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot n^2 \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{n-1} + \nonumber \\ + &\sum^n_{i=1} \frac{1}{i!}\lp \left| \Xi \lp \Xi^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp \Xi\rp\rp\right| + \ve + |\Xi|^q + \mathfrak{p}_{i-1}^q \rp + \end{align} + \item it is the case that $\wid_{\hid \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp \les 24+2n $ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that Lemma \ref{etr_prop} tells us that $ \inst_{\rect}\lp \etr^{N,h}\rp \in C\lp \R^{N+1},\R\rp$, and Lemma \ref{xpn_properties} tells us that $\inst_{\rect}\lp \xpn^{q,\ve}_n\rp \lp x\rp \in C \lp \R, \R \rp$. Next, note that Lemma \ref{comp_prop} and the fact that the composition of continuous functions is continuous yields that: + \begin{align} + \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_n\rp &= \real_{\rect} \lp \xpn_n^{q,\ve} \bullet \aff_{\lb \frac{h}{2} \: h \:\hdots \:h \: \frac{h}{2}\rb,0}\rp \nonumber \\ + &= \real_{\rect} \lp \xpn_n^{q,\ve} \rp \circ \real_{\rect} \lp \aff_{\lb \frac{h}{2} \: h \:\hdots \:h \: \frac{h}{2}\rb,0} \rp \in C \lp \R^{N+1},\R \rp\nonumber + \end{align} + Since both component neural networks are continuous, and the composition of continuous functions is continuous, so is $\mathsf{E}^{N,h,q,\ve}_n$. This proves Item (i). + + Next note that $\dep \lp \aff_{\lb \frac{h}{2} \: h \:\hdots \:h \: \frac{h}{2}\rb} \rp = 1$, and thus Lemma \ref{comp_prop} and Lemma \ref{xpn_properties} tells us that: + \begin{align} + \dep \lp \mathsf{E}^{N,h,q,\ve}_n\rp &= \dep \lp \xpn^{q,\ve}_{n} \bullet \aff_{\lb \frac{h}{2} \: h \:\hdots \:h \: \frac{h}{2}\rb,0}\rp \nonumber \\ + &= \nonumber \dep \lp \xpn^{q,\ve}_{n} \rp + \dep \lp \aff_{\lb \frac{h}{2} \: h \:\hdots \:h \: \frac{h}{2}\rb,0} \rp -1 \nonumber \\ + &=\dep \lp \xpn^{q,\ve}_{n}\rp \nonumber \\ + &\les \begin{cases} + 1 & :n=0\\ + n\lb \frac{q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q\rb -1 \rb +1 &:n\in \N + \end{cases} \nonumber + \end{align} + This proves Item (ii). + + Next note that by Corollary \ref{affcor}, Lemma \ref{xpn_properties}, Lemma \ref{etr_prop}, and the fact that $\inn\lp \etr^{N,h}\rp = N+1$, and $\inn \lp \xpn_n^{q,\ve}\rp = 1$, tells us that, for all $N \in \N$ it is the case that: + \begin{align} + &\param \lp \mathsf{E}^{N,h,q,\ve}_n\rp \nonumber\\ + &\les \lb \max \left\{1, \frac{\inn\lp \etr^{N,h}\rp+1}{\inn\lp \xpn_n^{q,\ve}\rp+1}\right\}\rb \cdot \param\lp \xpn_n^{q,\ve}\rp \nonumber\\ + &=\lp \frac{1}{2}N+1 \rp \cdot \param \lp \xpn_n^{q,\ve}\rp \nonumber \\ + &\les \begin{cases} + N+2 & :n =0 \\ + \lp \frac{1}{2}N+1 \rp\lp n+1\rp\lb 4^{n+\frac{3}{2}} + \lp \frac{4^{n+1}-1}{3}\rp \lp \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +372\rp\rb &:n\in \N + \end{cases}\nonumber + \end{align} + This proves Item (iii). + + Note next that: + \begin{align} + \aff_{\lb \frac{h}{2} \: h \:\hdots \:h \: \frac{h}{2}\rb,0} = \etr^{N,h} + \end{align} + Thus the well-known error term of the trapezoidal rule tells us that for $\lb a,b \rb \subsetneq \R$, and for $\xi \in \lb a,b \rb$ it is the case that: + \begin{align} + \left| \int^b_a f\lp x \rp dx - \lp \real_{\rect}\lp \etr^{N,h} \rp\rp \lp f\lp \lb x \rb_{*,*}\rp\rp \right| \les \frac{ \lp b-a\rp^3}{12N^2} f''\lp \xi \rp + \end{align} + and for $n\in \N_0$, $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$ and $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$, and for $x \in \lb \mathfrak{a},\mathfrak{b}\rb \subsetneq \R$, with $0 \in \lb \mathfrak{a},\mathfrak{b}\rb$ it is the case, according to Lemma \ref{mathsfE}, that: + \begin{align} + \left| e^x - \real_{\rect} \lp \xpn_n^{q,\ve} \rp \lp x \rp \right| &\les \sum^n_{i=1} \frac{1}{i!}\lp \left| x \lp x^{n-1} - \real_{\rect}\lp \pwr^{q,\ve}_{n-1}\rp\lp x\rp\rp\right| + \ve + |x|^q + \mathfrak{p}_{n-1}^q \rp + \frac{e^{\mathfrak{b}}\cdot |x|^{n+1}}{(n+1)!} + \end{align} + Note now that for $f \in C_{ae}\lp \R,\R\rp$, $\int^b_a f dx \in \lb \mathfrak{a},\mathfrak{b}\rb \subsetneq \R$, $0 \in \lb \mathfrak{a},\mathfrak{b}\rb$, and $\xi$ between $0$ and $\int^b_a f dx$ it is the case that: + \begin{align} + \exp \lb \int_a^b f dx\rb = \sum^n_{i=1}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb + \frac{e^{\xi}\cdot \left| \int^b_a f dx\right|^{n+1}}{(n+1)!} + \end{align} + And thus the triangle inequality, Lemma \ref{comp_prop}, and Lemma \ref{xpn_properties}, tells us that for $x = \{ x_0, x_1,\hdots, x_N\}$, $a = x_0\les x_1\les \cdots \les x_N=b$ and $\lb a,b\rb \subsetneq \R$ that: + \begin{align}\label{trian_ineq_exp_real_rect} + &\left| \exp \lb \int^b_afdx\rb - \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp\lp f\lp \lb x\rb_{*,*} \rp\rp\right| \nonumber \\ + &= \left|\sum^n_{i=1}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb + \frac{e^{\xi}\cdot \lp \int^b_a f dx\rp^{n+1}}{(n+1)!} - \real_{\rect}\lp \xpn^{q,\ve}_{n} \bullet \etr^{N,h} \rp \lp f\lp \lb x \rb_{*,*}\rp\rp\right| \nonumber \\ + &\les \left|\sum^n_{i=1}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb - \real_{\rect}\lp \xpn^{q,\ve}_{n}\rp\lp x\rp \circ \real_{\rect}\lp\etr^{N,h} \rp\lp f\lp \lb x\rb_{*,*}\rp\rp\right| + \frac{e^{\xi}\cdot \left| \int^b_a f dx\right|^{n+1}}{(n+1)!} + \end{align} + Note that the instantiation of $\etr^{N,h}$ is exact as it is the instantiation of an affine neural network. For notational simplicity let $\Xi = \real_{\rect} \lp \etr^{N,h}\rp \lp f\lp \lb x\rb_{*,*}\rp\rp$. Then Lemma \ref{xpn_properties} tells us that: + \begin{align}\label{10.0.17} + \left|\sum^n_{i=0}\lb \frac{\Xi^i}{i!}\rb - \real_{\rect} \lp \xpn^{q,\ve}_n \rp \lp \Xi \rp \right| & \les \sum^n_{i=1} \frac{1}{i!}\lp \left| \Xi \lp \Xi^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp \Xi\rp\rp\right| + \ve + |\Xi|^q + \lp \mathfrak{p}_{i-1}^{\Xi}\rp^q \rp + \end{align} + Where for $i\in \N$, let $\mathfrak{p}^{\Xi}_{i-1}$ be the family of functions defined as such: + \begin{align} + \mathfrak{p}^{\Xi}_1 &= \ve+1+|\Xi|^2 \nonumber\\ + \mathfrak{p}^{\Xi}_i &= \ve +\lp \mathfrak{p}_{i-1} \rp^2+|\Xi|^2 + \end{align} + + This then leaves us with: + \begin{align}\label{(10.0.19)} + \left|\sum^n_{i=0}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb - \sum^n_{i=0}\lb \frac{\Xi^i}{i!}\rb\right| &\les \sum_{i=0}^n\left| \lb\frac{1}{i!} \lp \int^b_a fdx\rp^i -\frac{\Xi^i}{i!}\rb \right| \nonumber \\ + &\les \lp n+1\rp \max_{i \in \{0,1,...,n\}}\left|\lb \frac{1}{i!} \lp \int^b_a fdx\rp^i -\frac{\Xi^i}{i!} \rb \right|\nonumber \\ + &\les n \cdot \max_{i \in \{1,...,n\}}\frac{1}{i!} \left|\lb \lp \int^b_a fdx\rp^i -\Xi^i \rb \right| + \end{align} + + Note that for each $i \in \{1,...,n \}$ it holds that: + \begin{align}\label{(10.0.18)} + \lp \int^b_a f dx\rp^i - \Xi^i =\lp \int^b_a f dx - \Xi\rp \lb\lp \int^b_a f dx\rp^{i-1} + \lp \int^b_afdx\rp^{i-2}\cdot \Xi + \cdots +\Xi^{i-1}\rb + \end{align} + Note that the well-known error bounds for the trapezoidal rule tell us that $\Xi$ and $\int^b_afdx$ differ by at most $\frac{\lp b-a \rp^3}{12N^2} f''\lp \xi \rp$ in absolute terms, and thus: + \begin{align} + \max \left\{ \Xi, \int^b_afdx\right\} \les \Xi + \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp + \end{align} + This then renders (\ref{(10.0.18)}) as: + \begin{align} + \lp \int^b_a fdx\rp^i - \Xi^i \les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot i \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{i-1} + \end{align} + Note that this also renders (\ref{(10.0.19)}) as: + \begin{align} + \left|\sum^n_{i=0}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb - \sum^n_{i=0}\lb \frac{\Xi^i}{i!}\rb\right| &\les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot n^2 \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{n-1} + \end{align} + This, the triangle inequality and (\ref{10.0.17}), then tell us for all $x \in \lb a,b\rb \subseteq \lb 0,\infty\rp$ that: + \begin{align} + &\left| \sum^n_{i=0} \lb \frac{1}{i!} \lp \int^b_a fdx\rp^i\rb - \real_{\rect} \lp \xpn^{q,\ve}_n \rp\lp x\rp \circ \Xi \right| \nonumber\\ + &\les \left|\sum^n_{i=0}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb - \sum^n_{i=0}\lb \frac{\Xi^i}{i!}\rb\right| \nonumber+ \left|\sum^n_{i=0}\lb \frac{\Xi^i}{i!}\rb - \real_{\rect} \lp \xpn_n^{q,\ve} \rp\lp x \rp \circ \Xi \right| \nonumber \\ + &\les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot n^2 \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{n-1} + \nonumber \\ + &\sum^n_{i=1} \frac{1}{i!}\lp \left| \Xi \lp \Xi^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp \Xi\rp\rp\right| + \ve + |\Xi|^q + \lp \mathfrak{p}^{\Xi}_{i-1}\rp^{q} \rp + \end{align} + This, applied to (\ref{trian_ineq_exp_real_rect}) then gives us that: + \begin{align} + &\left| \exp \lb \int^b_afdx\rb - \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_n\rp\lp f\lp \lb x \rb_{*,*}\rp\rp\right| \nonumber\\ + &\les \left|\sum^n_{i=1}\lb \frac{1}{i!} \lp \int^b_afdx\rp^i\rb - \real_{\rect}\lp \xpn^{q,\ve}_n\rp\lp x\rp \circ \real_{\rect}\lp\etr^{N,h} \rp\lp f\lp \lb x\rb_{*,*}\rp\rp\right| + \frac{e^{\xi}\cdot \left| \int^b_a f dx\right|^{n+1}}{(n+1)!}\nonumber \\ + &\les \frac{\lp b-a\rp^3}{12N^2}f''\lp \xi \rp \cdot n^2 \cdot \lb \Xi + \frac{\lp b-a\rp^3}{12N^2} f''\lp \xi\rp\rb^{n-1} + \nonumber \\ + &\sum^n_{i=1} \frac{1}{i!}\lp \left| \Xi \lp \Xi^{i-1} - \real_{\rect}\lp \pwr^{q,\ve}_{i-1}\rp\lp \Xi\rp\rp\right| + \ve + |\Xi|^q + \lp\mathfrak{p}_{i-1}^{\Xi}\rp^{q} \rp + \frac{e^{\xi}\cdot \left| \int^b_a f dx\right|^{n+1}}{(n+1)!} + \end{align} + This proves Item (iv). + + Finally note that Lemma \ref{xpn_properties} tells us that: + \begin{align} + \wid_{\hid\lp \mathsf{E}^{N,h,q,\ve}_n\rp} \lp \mathsf{E}^{N,h,q,\ve}_n\rp &= \wid_{\hid \lp \xpn^{q,\ve}_n\rp} \lp \xpn^{q,\ve}_n\rp \nonumber\\ + &\les 24+2n + \end{align} +\end{proof} +\begin{remark} + We may represent the $\mathsf{E}^{N,h,q,\ve}_n$ diagrammatically as follows: +\end{remark} +\begin{figure}[h] + \begin{center} + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,477); %set diagram left start at 0, and has a height of 477 + +%Shape: Rectangle [id:dp8133807694586985] +\draw (570,80) -- (640,80) -- (640,380) -- (570,380) -- cycle ; +%Straight Lines [id:da5644202293112723] +\draw (670,90) -- (642,90) ; +\draw [shift={(640,90)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da3958568883282644] +\draw (670,130) -- (642,130) ; +\draw [shift={(640,130)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da36159599804808873] +\draw (670,360) -- (642,360) ; +\draw [shift={(640,360)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da8399916298816871] +\draw (570,220) -- (532,220) ; +\draw [shift={(530,220)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp36371899004668184] +\draw (460,200) -- (530,200) -- (530,240) -- (460,240) -- cycle ; +%Straight Lines [id:da6087697346260741] +\draw (460,210) -- (430.7,131.87) ; +\draw [shift={(430,130)}, rotate = 69.44] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da09350943957233837] +\draw (460,230) -- (430.39,378.04) ; +\draw [shift={(430,380)}, rotate = 281.31] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp5858773975599592] +\draw (360,110) -- (430,110) -- (430,150) -- (360,150) -- cycle ; +%Shape: Rectangle [id:dp6593787630372464] +\draw (300,170) -- (430,170) -- (430,210) -- (300,210) -- cycle ; +%Shape: Rectangle [id:dp8981897806517504] +\draw (200,360) -- (430,360) -- (430,400) -- (200,400) -- cycle ; +%Shape: Rectangle [id:dp8722515968192108] +\draw (200,170) -- (270,170) -- (270,210) -- (200,210) -- cycle ; +%Shape: Rectangle [id:dp6009233720093313] +\draw (200,110) -- (330,110) -- (330,150) -- (200,150) -- cycle ; +%Straight Lines [id:da25919556276229416] +\draw (360,130) -- (332,130) ; +\draw [shift={(330,130)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da5118244891967815] +\draw (300,190) -- (272,190) ; +\draw [shift={(270,190)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp8302038959734496] +\draw (130,110) -- (170,110) -- (170,150) -- (130,150) -- cycle ; +%Straight Lines [id:da3444887884321911] +\draw (200,130) -- (172,130) ; +\draw [shift={(170,130)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp830705563112423] +\draw (130,170) -- (170,170) -- (170,210) -- (130,210) -- cycle ; +%Shape: Rectangle [id:dp32656481150478633] +\draw (130,360) -- (170,360) -- (170,400) -- (130,400) -- cycle ; +%Straight Lines [id:da3058371632279333] +\draw (190,190) -- (200,190) -- (172,190) ; +\draw [shift={(170,190)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da7847718449576501] +\draw (200,380) -- (172,380) ; +\draw [shift={(170,380)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp4662565884782496] +\draw (30,240) -- (100,240) -- (100,280) -- (30,280) -- cycle ; +%Straight Lines [id:da10776626554102664] +\draw (30,260) -- (2,260) ; +\draw [shift={(0,260)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da09899836992792843] +\draw (130,130) -- (100.53,238.07) ; +\draw [shift={(100,240)}, rotate = 285.26] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da35627707263158237] +\draw (130,190) -- (100.79,258.16) ; +\draw [shift={(100,260)}, rotate = 293.2] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da27409517502644487] +\draw (460,220) -- (431.41,191.41) ; +\draw [shift={(430,190)}, rotate = 45] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da1945630670120475] +\draw (130,380) -- (100.53,271.93) ; +\draw [shift={(100,270)}, rotate = 74.74] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (581,206.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Etr}{^{N}}^{h}$}; +% Text Node +\draw (671,82.4) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}$}; +% Text Node +\draw (674,122.4) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}$}; +% Text Node +\draw (674,352.4) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}$}; +% Text Node +\draw (661,212.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $}; +% Text Node +\draw (469,212.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n}{}_{,}{}_{1}$}; +% Text Node +\draw (371,115.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}_{0}^{q}$}; +% Text Node +\draw (331,175.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}_{1}^{q}$}; +% Text Node +\draw (311,365.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Pwr}_{n}^{q}$}; +% Text Node +\draw (322,262.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $}; +% Text Node +\draw (182,262.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $}; +% Text Node +\draw (238,122.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (208,182.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (136,115.4) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] {$\frac{1}{0!} \rhd $}; +% Text Node +\draw (141,173.4) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] {$\frac{1}{1!} \rhd $}; +% Text Node +\draw (141,363.4) node [anchor=north west][inner sep=0.75pt] [font=\scriptsize] {$\frac{1}{n!} \rhd $}; +% Text Node +\draw (122,262.4) node [anchor=north west][inner sep=0.75pt] [font=\LARGE] {$\vdots $}; +% Text Node +\draw (41,250.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Cpy}_{n}{}_{,}{}_{1}$}; + + +\end{tikzpicture} + + \end{center} + \caption{Diagram of $\mathsf{E}^{N,h,q,\ve}_n$.} +\end{figure} +\section{The $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$ Neural Network} +\begin{lemma}[R\textemdash,2023]\label{UE-prop} + +Let $n, N,h\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{\exp}_{n,h,q,\ve} \in \neu$ be the neural network given by: + \begin{align} + \mathsf{E}^{N,h,q,\ve}_n = \xpn_n^{q,\ve} \bullet \etr^{N,h} + \end{align} + + Let $\mathsf{G}_d \in \neu$ be the neural network which instantiates as $\mathfrak{u}_d =\real_{\rect}\lp \mathsf{G}_d\rp\lp x\rp \in C \lp \R^d, \R \rp$ for all $x \in \R^d$. + + Let $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$ be the neural network given as: + \begin{align} + \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} = \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_n \DDiamond \mathsf{G}_d \rb + \end{align} + It is then the case that for all $\fx = \{x_0,x_1,\hdots, x_N\} \in \R^{N+1}$ and $x \in \R^d$ that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp\lp f\lp \lb \fx\rb_{*}\rp \frown x\rp \in C \lp \R^{N+1} \times \R^{d}, \R \rp$ + \item $\dep \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp \les \begin{cases} + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\dep \lp \mathsf{G}_d\rp-1 &:n = 0\\ + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\max\left\{\dep \lp \mathsf{E}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp,\dep \lp \mathsf{G}_d\rp\right\}-1 &:n \ges 1\\ + \end{cases}$ + \item It is also the case that:\begin{align} + \param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp &\les \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\nonumber\\ &+24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} + \end{align} + \item It is also the case that: + \begin{align} + &\left| \exp \lp \int^b_a fdx\rp \fu_d\lp x\rp - \real_{\rect}\lp\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp \lp f(\lb \fx\rb_* \frown x \rp \right|\nonumber\\ &\les 3\ve +2\ve \left| \fu\lp x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\fu\lp x \rp\nonumber + \end{align} + Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as: + \begin{align} + \left| \mathsf{E}^{N,h,q,\ve}_{n}\lp f\lp \lb \fx\rb_*\rp\rp - \exp \lp \int^b_afdx\rp\right| \les \mathfrak{e} + \end{align} + \end{enumerate} +\end{lemma} +\begin{remark} + Diagrammatically $\mathsf{UE}^{N,h,q,\ve}_{n}$ can be represented as: +\end{remark} + + +\begin{center} + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,475); %set diagram left start at 0, and has a height of 475 + +%Shape: Rectangle [id:dp5014556157804896] +\draw (477.6,119) -- (581.4,119) -- (581.4,159) -- (477.6,159) -- cycle ; +%Shape: Rectangle [id:dp6888177801681734] +\draw (479.2,220.6) -- (583,220.6) -- (583,260.6) -- (479.2,260.6) -- cycle ; +%Shape: Rectangle [id:dp5733062989346852] +\draw (274.2,120.2) -- (422.8,120.2) -- (422.8,160.2) -- (274.2,160.2) -- cycle ; +%Shape: Rectangle [id:dp3644206511797573] +\draw (278.2,221.8) -- (426.8,221.8) -- (426.8,261.8) -- (278.2,261.8) -- cycle ; +%Shape: Rectangle [id:dp872163975205418] +\draw (128,168.6) -- (198,168.6) -- (198,208.6) -- (128,208.6) -- cycle ; +%Straight Lines [id:da0837610283606276] +\draw (479.2,139.6) -- (425.8,139.79) ; +\draw [shift={(423.8,139.8)}, rotate = 359.79] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da9682427637559707] +\draw (477.6,242) -- (430.6,242.19) ; +\draw [shift={(428.6,242.2)}, rotate = 359.77] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da03790523286924763] +\draw (274.2,139) -- (200,174.93) ; +\draw [shift={(198.2,175.8)}, rotate = 334.16] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da20598256940944215] +\draw (278.2,242.2) -- (203.16,201.55) ; +\draw [shift={(201.4,200.6)}, rotate = 28.44] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da23636707453012662] +\draw (128.6,190.2) -- (67.4,189.43) ; +\draw [shift={(65.4,189.4)}, rotate = 0.73] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da988244371641313] +\draw (621.6,140.6) -- (583.6,139.84) ; +\draw [shift={(581.6,139.8)}, rotate = 1.15] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da5522974642204443] +\draw (623.2,243) -- (585.2,242.24) ; +\draw [shift={(583.2,242.2)}, rotate = 1.15] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (508.4,126.2) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}^{N,h,q,\ve}_n$}; +% Text Node +\draw (526,230.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{G}_d$}; +% Text Node +\draw (333.2,131.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (342,233) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (143.6,179.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Prd}^{q,\varepsilon }$}; +% Text Node +\draw (626,234.2) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}^{d}$}; +% Text Node +\draw (626.4,131) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}^{N+1}$}; + + +\end{tikzpicture} +\end{center} + +\begin{proof} + Note that from Lemma \ref{comp_prop}, and Lemma \ref{inst_of_stk}, we have that for $\fx \in \R^{N+1}$, and $x \in \R^d$ it is the case that $\real_{\rect} \lp \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb \rp\lp f\lp \lb \fx\rb_* \frown x\rp\rp = \real_{\rect} \lp \prd^{q,\ve}\rp \circ \real_{\rect}\lp \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb \rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp $. Then Lemma \ref{prd_network} tells us that $\real_{\rect} \lp \prd^{q,\ve}\rp \in C \lp \R^2,\R\rp$. Lemma \ref{mathsfE} tells us that $\real_{\rect }\lp \mathsf{E}^{N,h,q,\ve}_{n} \rp \in C \lp \R^{N+1},\R\rp$ and by hypothesis it is the case that $\real_{\rect} \lp \mathsf{G}_d\rp \in C \lp \R^d,\R\rp $. Thus, by the stacking properties of continuous instantiated networks and the fact that the composition of continuous functions is continuous, we have that $\real_{\rect} \lp \mathsf{UE}^{N, h,q,\ve}_{n,\mathsf{G}_d}\rp \in C \lp \R^{N+1} \times \R^d,\R \rp$. + + Note that by Lemma \ref{comp_prop} it is the case that: + \begin{align} + \dep \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp = \dep \lp \prd^{q,\ve}\rp + \dep \lp \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rp -1 + \end{align} + Lemma \ref{mathsfE} and Lemma \ref{prd_network} then tell us that: + \begin{align} + &\dep \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp \nonumber \\ + &\les \begin{cases} + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\dep \lp \mathsf{G}_d\rp-1 &:n = 0\\ + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\max\left\{\dep \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp,\dep \lp \mathsf{G}_d\rp\right\}-1 &:n \ges 1\\ + \end{cases} + \end{align} + Note that then Lemma \ref{comp_prop}, Lemma \ref{xpn_properties}, and Lemma \ref{mathsfE} tell us that: + \begin{align} + \param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp &\les \param \lp \prd^{q,\ve}\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\}\nonumber\\&+ \wid_1\lp \prd^{q,\ve}\rp\cdot \wid_{\hid\lp \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d\rp } \lp \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d\rp\nonumber \\ + &\les \param \lp \prd^{q,\ve}\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\}\nonumber \\ + &+ 24\lb \lp 24+2n\rp\nonumber + \wid_{\hid \lp \mathsf{G}_d\rp} \lp \mathsf{G}_d\rp \rb \\ + &= \param \lp \prd^{q,\ve}\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\}\nonumber \\ + &+ 576+48n + 24\cdot\wid_{\hid \lp \mathsf{G}_d\rp} \lp \mathsf{G}_d\rp \nonumber\\ + &\les \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\nonumber\\ &+24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} + \end{align} + Now note that Lemma \ref{comp_prop}, and Lemma \ref{inst_of_stk} tells us that for all $\fx = \{x_1,x_2,\hdots, x_n\} \in \R^n$ and $x \in \R^d$: + \begin{align} + \real_{\rect} \lp \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb\rp \lp f\lp \lb \fx\rb_*\rp \frown x \rp = \real_{\rect}\lp \prd^{q,\ve}\rp \lp \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n} \rp , \real_{\rect} \lp \mathsf{G}_d \rp \rp \lp f \lp \lb \fx\rb_*\rp \frown x \rp. + \end{align} + Note then that the triangle inequality tells us that: + \begin{align} + &\left| \exp \lp \int^b_a f dx\rp \mathfrak{u}_d \lp x\rp - \real_{\rect}\lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp \lp f\lp\lb \fx\rb_*\rp \frown x\rp\right|\nonumber\\ + &= \left| \exp \lp \int^b_a fdx\rp \cdot \mathfrak{u}_d\lp x\rp- \real_{\rect}\lp \prd^{q,\ve}\rp \lp \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n} \rp , \real_{\rect} \lp \mathsf{G}_d \rp \rp \lp f\lp \lb \fx\rb_* \rp \frown x\rp\right| \nonumber \\ + &\les \left| \exp \lp \int^b_afdx \rp \cdot \mathfrak{u}_d \lp x\rp -\real_{\rect}\lp \prd^{q,\ve} \rp \lp \exp \lp \int^b_afdx\rp ,\mathfrak{u}_d\lp x\rp \rp\right| \nonumber \\ + &+ \left| \real_{\rect}\lp \prd^{q,\ve}\rp \lp \exp \lp \int^b_afdx \rp, \mathfrak{u}_d\lp x\rp\rp - \real_{\rect}\lp \prd^{q,\ve}\rp \lp \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n} \rp , \real_{\rect} \lp \mathsf{G}_d \rp \rp \lp f\lp \lb \fx\rb\rp\frown x\rp\right| \label{(10.0.35)} + \end{align} + Note that Lemma \ref{prd_network} bounds the first summand. Note that by hypothesis $\real_{\rect} \lp \mathsf{G}_d\rp$ is exactly $\mathfrak{u}_d \lp x\rp$. Note also that by Lemma \ref{mathsfE}, Lemma \ref{prd_network}, we realize that the second summand can be bounded as such: + \begin{align} + &\left| \real_{\rect}\lp \prd^{q,\ve}\rp \lp \exp \lp \int^b_afdx \rp, \mathfrak{u}_d\lp x\rp\rp - \real_{\rect}\lp \prd^{q,\ve}\rp \lp \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n} \rp , \real_{\rect} \lp \mathsf{G}_d \rp \rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp\right| \nonumber \\ + &\les \exp \lp \int^b_afdx\rp \mathfrak{u}_d\lp x\rp + \ve+\ve \left| \exp \lp \int ^b_a fdx\rp\right|^q + \ve \left| \mathfrak{u}_d\lp x \rp\right|^q\nonumber \\ + &- \lb\real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp\lp f \lp \lb \fx \rb_* \rp\rp\real_{\rect} \lp \mathsf{G}_d\rp\lp x\rp - \ve -\ve \left| \real_{\rect}\lp \mathsf{E}^{N,h,q,\ve}_{n}\rp \lp f\lp \lb\fx\rb_*\rp\rp\right|^q-\ve \left| \real_{\rect}\lp \mathsf{G}_d\rp\lp x\rp\right|^q\rb \label{(10.0.36)} + \end{align} + Per Lemma \ref{mathsfE}, let $\mathfrak{e}$ represent the error in approximation of $\mathsf{E}^{N,h,q,\ve}_{n}$, that is to say for all $\fx \in \R^{N+1}$ and corresponding $f\lp \lb \fx\rb_*\rp$, let it be the case that: + \begin{align} + \left| \mathsf{E}^{N,h,q,\ve}_{n}\lp f\lp \lb \fx\rb_*\rp\rp - \exp \lp\int^b_a fdx\rp\right| \les \mathfrak{e} + \end{align} + Thus $\mathsf{E}^{N,h,q,\ve}_{n}\lp f\lp \lb \fx\rb_*\rp\rp$ is maximally $\mathfrak{e} + \exp \lp \int^b_a f dx\rp$ and minimally $\exp \lp \int^b_afdx\rp - \mathfrak{e}$. Thus (\ref{(10.0.36)}) is rendered as: + \begin{align} + &\exp \lp \int^b_afdx\rp \mathfrak{u}_d\lp x\rp + \ve+\ve \left| \exp \lp \int ^b_a fdx\rp\right|^q + \ve \left| \mathfrak{u}_d\lp x \rp\right|^q\nonumber \\ + &- \lb\real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp\real_{\rect} \lp \mathsf{G}_d\rp\lp f\lp \lb \fx\rb_*\rp \frown x\rp - \ve -\ve \left| \real_{\rect}\lp \mathsf{E}^{N,h,q,\ve}_{n}\rp \lp f\lp \lb \fx \rb_*\rp\rp\right|^q-\ve \left| \real_{\rect}\lp \mathsf{G}_d\rp\lp x\rp\right|^q\rb \nonumber \\ + &\les \exp \lp \int^b_afdx\rp \mathfrak{u}_d\lp x\rp + \ve+\ve \left| \exp \lp \int ^b_a fdx\rp\right|^q + \ve \left| \mathfrak{u}_d\lp x \rp\right|^q \nonumber \\ + &-\lb \lp \mathfrak{e}+\exp \lp \int^b_afdx\rp \rp \mathfrak{u}_d\lp x\rp - \ve -\ve \left| \exp \lp \int^b_afdx \rp - \mathfrak{e}\right|^q -\ve \left|\mathfrak{u}_d\lp x\rp \right|^q \rb \nonumber \\ + &= \cancel{\exp \lp \int^b_afdx\rp \mathfrak{u}_d\lp x\rp} + \ve+\ve \left| \exp \lp \int ^b_a fdx\rp\right|^q + \ve \left| \mathfrak{u}_d\lp x \rp\right|^q \nonumber \\ + &-\mathfrak{e}u\lp t,x\rp-\cancel{\exp \lp \int^b_a fdx\rp \mathfrak{u}_d\lp x\rp} + \ve +\ve \left| \exp \lp \int^b_af dx\rp - \mathfrak{e}\right|^q+\ve \left| \mathfrak{u}_d\lp x\rp\right|^q \nonumber \\ + &=2\ve +2\ve \left| \mathfrak{u}_d\lp x\rp\right|^q + \ve \left| \exp \lp \int^b_a fdx\rp -\mathfrak{e}\right|^q +\ve \left| \exp \lp \int^b_afdx\rp\right|^q -\mathfrak{e}\mathfrak{u}_d\lp x \rp + \end{align} + This, together with (\ref{(10.0.35)}), then tells us that: + \begin{align} + & \left| \exp \lp \int^b_a f dx\rp \mathfrak{u}_d \lp x\rp - \real_{\rect}\lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp\right| \nonumber \\ + &\les \left| \exp \lp \int^b_afdx \rp \mathfrak{u}_d \lp x\rp -\real_{\rect}\lp \prd^{q,\ve} \rp \lp \exp \lp \int^b_afdx\rp ,\mathfrak{u}_d\lp x\rp \rp\right| \nonumber \\ + &+ \left| \real_{\rect}\lp \prd^{q,\ve}\rp \lp \exp \lp \int^b_afdx \rp, \mathfrak{u}_d\lp x\rp\rp - \real_{\rect}\lp \prd^{q,\ve}\rp \lp \real_{\rect} \lp \mathsf{E}^{N,h,q,\ve}_{n} \rp\lp f\lp \lb \fx\rb_*\rp\rp , \real_{\rect} \lp \mathsf{G}_d \rp \lp x\rp\rp \right| \nonumber \\ + &\les \ve +\ve \left| \exp \lp \int^b_a fdx\rp\right|^q + \ve \left| \mathfrak{u}_d \lp x\rp\right|^q \nonumber \\ + &+2\ve +2\ve \left| \mathfrak{u}_d\lp x\rp\right|^q + \ve \left| \exp \lp \int^b_a fdx\rp - \mathfrak{e} \right|^q +\ve \left| \exp \lp \int^b_afdx\rp\right|^q -\mathfrak{e}\mathfrak{u}_d\lp x \rp \nonumber\\ + &= 3\ve +2\ve \left| \mathfrak{u}_d\lp x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d\lp x \rp \nonumber + \end{align} +\end{proof} +\section{The $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ network}\label{UEX} +\begin{lemma}[R\textemdash,2023]\label{UE-prop} + +Let $n, N,h\in \N$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $a\in \lp -\infty,\infty \rp$, $b \in \lb a, \infty \rp$. Let $f:[a,b] \rightarrow \R$ be continuous and have second derivatives almost everywhere in $\lb a,b \rb$. Let $a=x_0 \les x_1\les \cdots \les x_{N-1} \les x_N=b$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{b-a}{N}$, and $x_i = x_0+i\cdot h$ . Let $x = \lb x_0 \: x_1\: \cdots x_N \rb$ and as such let $f\lp\lb x \rb_{*,*} \rp = \lb f(x_0) \: f(x_1)\: \cdots \: f(x_N) \rb$. Let $\mathsf{E}^{\exp}_{n,h,q,\ve} \in \neu$ be the neural network given by: + \begin{align} + \mathsf{E}^{N,h,q,\ve}_n = \xpn_n^{q,\ve} \bullet \etr^{N,h} + \end{align} + + Let $\mathsf{G}_d \subsetneq \neu$ be the neural networks which, for $d \in \N$, instantiate as $\mathfrak{u}_d =\real_{\rect}\lp \mathsf{G}_d\rp\lp x\rp \in C \lp \R^d, \R \rp$ for all $x \in \R^d$. + + Let $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \subsetneq \neu$ be the neural networks given as: + \begin{align} + \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} = \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_n \DDiamond \mathsf{G}_d \rb + \end{align} + + Finally let $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \subsetneq \neu$ be given the neural networks given by: + \begin{align} + \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} = \mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb + \end{align} + It is then the case that for all $\fx = \{x_0,x_1,\hdots, x_N\} \in \R^{N+1}$ and $x \in \R^d$ that: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \in C \lp \R^{N+1}\times \R^{d}, \R \rp$ + \item \begin{align}\dep \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp &= \dep \lp \mathsf{UE}^{N,n,h,q,\ve }_{n,\mathsf{G}_d}\rp\nonumber\\ &\les \begin{cases} + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\dep \lp \mathsf{G}_d\rp-1 &:n = 0\nonumber\\ + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\max\left\{\dep \lp \mathsf{E}^{\exp,f}_{N,n,h,q,\ve}\rp,\dep \lp \mathsf{G}_d\rp\right\}-1 &:n \in \N\\ + \end{cases}\end{align} + \item It is also the case that:\begin{align} + \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp = \param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp &\les \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\nonumber\\ &+24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} + \end{align} + \item It is also the case that: + \begin{align} + &\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\ &\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber + \end{align} + Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as: + \begin{align} + \left| \mathsf{E}^{N,h,q,\ve}_{n}\lp f\lp \lb \fx\rb_* \rp\rp - \exp \lp \int^b_afdx\rp\right| \les \mathfrak{e} + \end{align} + \end{enumerate} +\end{lemma} + +\begin{proof} + Note that (\ref{fc-assertion}) is an assertion of Feynman-Kac. LetNow notice that for $x \in \R^{N+1} \times \R^d$ it is the case that: + \begin{align} + \real_{\rect} \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp x\rp &= \real_{\rect} \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb \rp \lp x\rp\nonumber \\ + &=\real_{\rect} \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp \circ \real_{\rect}\lp \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb\rp \lp x \rp \nonumber + \end{align} + Note that by Lemma \ref{UE-prop} it holds that $\real_{\rect} \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \rp \in C\lp \R^{N+1}\times \R^d,\R \rp$. Note also that by Lemma \ref{tun_mult}, $\tun^{N+1}_1$ is continuous and by Lemma \ref{5.3.2}, $\aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}}$ is continuous, and whence by Lemma \ref{tun_mult} and Lemma \ref{aff_effect_on_layer_architecture} it is the case that $\real_{\rect}\lp \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb\rp \lp x \rp \in C\lp \R^{N+1}\times \R^d, \R^{N+1}\times \R^d\rp$. Finally, since the composition of continuous functions is continuous, and since we have functions $\lp \R^{N+1}\times \R^d\rp \mapsto \lp \R^{N+1}\times \R^d\rp \mapsto \R$ we have that $\real_{\rect} \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \in C \lp \R^{N+1}\times \R^{d}, \R \rp$. This proves Item (i). + + Note next that by Lemma \ref{tun_mult}, it is the case that $\dep \lp \tun^{N+1}_1\rp = \dep \lp \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}}\rp$ = 1. Thus by Lemma \ref{comp_prop} it is the case that $\dep \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp = \dep \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp$. This proves Item (ii) + + Next note that: + \begin{align} + \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp = \param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb \rp + \end{align} + + Note carefully that Definition \ref{def:tun_mult} tells us that $\tun^{N+1}_1 = \aff_{\mathbb{I}_{N+1,N+1},\mymathbb{0}_{N+1}}$, and so by Lemma \ref{aff_stack_is_aff}, it must be the case that $\tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}}$ is also an affine neural network. Whence, Corollary \ref{affcor} tells us that: + \begin{align} + \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp &= \param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb \rp \nonumber \\ + &\les \lb \max \left\{ 1,\frac{\inn\lp \tun_1^{N+1}\boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}}\rp+1}{\inn\lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp+1}\right\}\rb\cdot \param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp \nonumber\\ + &=\param \lp \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}\rp \nonumber\\ + &\les \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\nonumber\\ &+24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} + \end{align} + + Finally, note that both $\aff_{W,b}$ and $\tun^d_n$ are exact and contribute nothing to the uncertainty. Thus $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ has the same error bounds as $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$. That is to say that: + \begin{align} + &\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\ &\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber + \end{align} +\end{proof} + +\begin{corollary}[R\textemdash, 2024, Approximants for Brownian Motion] +Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $n,N\in \N$, and $h \in \lp 0, \infty \rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $f:[t, T] \rightarrow \R$ be continuous almost everywhere in $\lb t, T \rb$. Let it also be the case that $f = g \circ \fh$, where $\fh: \lb t,T\rb \rightarrow \R^d$, and $g: \R^d \rightarrow \R$. Let $t=t_0 \les t_1\les \cdots \les t_{N-1} \les t_N=T$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{T-t}{N}$, and $t_i = t_0+i\cdot h$ . Let $\mathbf{t} = \lb t_0 \: t_1\: \cdots t_N \rb$ and as such let $f\lp\lb \mathbf{t} \rb_{*,*} \rp = \lb f(t_0) \: f(t_1)\: \cdots \: f(t_N) \rb$. Let $\mathsf{E}^{N,h,q,\ve}_{n} \in \neu$ be the neural network given by: + \begin{align} + \mathsf{E}^{N,h,q,\ve}_{n} = \xpn^{q,\ve}_{n} \bullet \etr^{N,h} + \end{align} + Let $u_d \in C^{1,2} \lp [0,T] \times \R^d,\R\rp$ satisfy for all $d \in \N$, $t \in \lb 0,T\rb$, $x \in \R^d$ that: + \begin{align} + \lp \frac{\partial}{\partial t} u_d\rp \lp t,x\rp + \lp \nabla_x^2 u_d\rp \lp t,x \rp + \alpha_d\lp x\rp u_d \lp t,x\rp = 0 + \end{align} + Furthermore, let $\mathfrak{u}_d^T(x) = u_d(T,x)$. Let $\mathsf{G}_d \subsetneq \neu$ be the neural networks which instantiate as $\mathfrak{u}_d^T =\real_{\rect}\lp \mathsf{G}_d\rp \in C \lp \R^d, \R \rp$. + + + Let $\mathcal{W}^d: \lb 0, T \rb \times \Omega \rightarrow \R^d$, $d \in \N$ be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: \lb t, T \rb \times \Omega \rightarrow \R^d$, $d \in \N$, $t \in \lb 0, T\rb$, $x\in \R^d$ be stochastic processes with continuous sample paths satisfying that for all $d \in \N$, $t\in \lb 0, T\rb$, $s \in \lb t, T \rb$, $x \in \R^d$ we have $\mathbb{P}$-a.s. that: + \begin{align} + \mathcal{X}^{d,t,x}_s = x + \int^t_s \sqrt{2} d\mathcal{W}^d_r + \end{align} + It is then the case that for all $d \in \N$, $t\in \lb 0,T \rb$, $x \in \R^d$ it holds that: + \begin{align}\label{fc-assertion} + u_d \lp t,x\rp = \mathbb{E} \lb \exp \lp \int^T_t \lp \alpha_d \circ \mathcal{X}^{d,t,x}_r\rp dr\rp u_d \lp T,\mathcal{X}_T^{d,t,x}\rp\rb + \end{align} + Let $\mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d}$ be the neural network given as: + \begin{align} + \mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d} = \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n} \DDiamond \mathsf{G}_d \rb + \end{align} + Finally let $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ be given by: + \begin{align} + \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} = \mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb + \end{align} + It is then the case that for all $\fx = \{x_0,x_1,\hdots, x_N\} \in \R^{N+1}$ and $x \in \R^d$ that: + \item It is also the case that: + \begin{align} + &\left| \exp \lp \int^T_t fds\rp \mathfrak{u}_d^T\lp x\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \lp f\lp \lb \fx\rb_*\rp \frown x\rp \right|\nonumber\\ &\les 3\ve +2\ve \left| \mathfrak{u}_d^T\lp t,x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\mathfrak{u}_d^T\lp x \rp\nonumber + \end{align} + Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as: + \begin{align} + \left| \mathsf{E}^{N,h,q,\ve}_{n}\lp f\lp \lb \fx\rb_* \rp\rp - \exp \lp \int^b_afdx\rp\right| \les \mathfrak{e} + \end{align} +\end{corollary} + +\begin{proof} +Note that for a fixed $T \in \lp 0,\infty \rp$ it is the case that $u_d\lp t,x \rp \in C^{1,2}\lp \lb 0,T\rb \times \R^d, \R \rp$ projects down to a function $\mathfrak{u}_d^T\lp x\rp \in C^2\lp \R^d, \R\rp$. Furthermore given a probability space $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ and a stochastic process $\mathcal{X}^{d,t,x}: \lb t,T\rb \times \Omega \rightarrow \R^d$, for a fixed outcome space $\omega_i \in \Omega$ it is the case that $\mathcal{X}^{d,t,x}$ projects down to $\mathcal{X}^{d,t,x}_{\omega_i}: \lb t,T\rb \rightarrow \R^d$. Thus given $\alpha_d: \R^d \rightarrow \R$ that is infinitely often differentiable, we get that $\alpha_d\circ \mathcal{X}_{\omega_i}^{d,t,x}: \lb t,T\rb \rightarrow\R$. + + Taken together with Lemma \ref{UE-prop} with $x \curvearrowleft \mathcal{X}^{d,t,x}_{r,\omega}, f \curvearrowleft \alpha_d\circ \mathcal{X}_{\omega_i}^{d,t,x}$, $b \curvearrowleft T$, $a \curvearrowleft t$, and $\mathfrak{u}_d^T\lp x\rp \curvearrowleft u_d \lp T,\mathcal{X}^{d,t,x}_{\omega_i}\rp$, our error term is rendered as is rendered as: + \begin{align} + &\left| \exp \lp \int^T_t \lp \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds\rp \rp u_d\lp T,\mathcal{X}^{d,t,x}_{\omega_i}\rp - \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \right|\nonumber\\ + &\les 3\ve +2\ve \left| u_d\lp T,\mathcal{X}_{r,\omega_i}^{d,t,x}\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}u_d\lp T,\mathcal{X}^{d,t,x}_{r,\omega_i} \rp\nonumber + \end{align} + This completes the proof of the lemma. +\end{proof} +\begin{remark} + Diagrammatically, this can be represented as: + +\begin{center} +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,475); %set diagram left start at 0, and has a height of 475 + +%Shape: Rectangle [id:dp5014556157804896] +\draw (328.6,130) -- (432.4,130) -- (432.4,170) -- (328.6,170) -- cycle ; +%Shape: Rectangle [id:dp6888177801681734] +\draw (324.2,209.6) -- (428,209.6) -- (428,249.6) -- (324.2,249.6) -- cycle ; +%Shape: Rectangle [id:dp5733062989346852] +\draw (182.2,129.2) -- (291.22,129.2) -- (291.22,169.2) -- (182.2,169.2) -- cycle ; +%Shape: Rectangle [id:dp3644206511797573] +\draw (177.2,206.8) -- (285.22,206.8) -- (285.22,246.8) -- (177.2,246.8) -- cycle ; +%Shape: Rectangle [id:dp872163975205418] +\draw (63,170.6) -- (133,170.6) -- (133,210.6) -- (63,210.6) -- cycle ; +%Straight Lines [id:da0837610283606276] +\draw (326.22,149.31) -- (293.8,148.83) ; +\draw [shift={(291.8,148.8)}, rotate = 0.85] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da03790523286924763] +\draw (180.22,149.31) -- (138.83,178.64) ; +\draw [shift={(137.2,179.8)}, rotate = 324.67] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da20598256940944215] +\draw (175.22,222.31) -- (137.12,199.62) ; +\draw [shift={(135.4,198.6)}, rotate = 30.77] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da23636707453012662] +\draw (61.22,190.31) -- (20.4,190.4) ; +\draw [shift={(18.4,190.4)}, rotate = 359.88] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da988244371641313] +\draw (613.22,147.31) -- (583.6,146.83) ; +\draw [shift={(581.6,146.8)}, rotate = 0.93] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da9656780010361206] +\draw (323.22,231.31) -- (290.8,230.83) ; +\draw [shift={(288.8,230.8)}, rotate = 0.85] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Shape: Rectangle [id:dp5990472515240087] +\draw (476.6,131) -- (580.4,131) -- (580.4,171) -- (476.6,171) -- cycle ; +%Shape: Rectangle [id:dp7593448401492562] +\draw (475.6,211) -- (579.4,211) -- (579.4,251) -- (475.6,251) -- cycle ; +%Straight Lines [id:da9567070863735319] +\draw (614.22,233.31) -- (584.6,232.83) ; +\draw [shift={(582.6,232.8)}, rotate = 0.93] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da34491136618106866] +\draw (477.22,149.31) -- (436.6,148.82) ; +\draw [shift={(434.6,148.8)}, rotate = 0.69] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da2710750822166835] +\draw (473.22,229.31) -- (432.6,228.82) ; +\draw [shift={(430.6,228.8)}, rotate = 0.69] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (359.4,134.2) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}_{N,n,h,q,\varepsilon }^{\exp ,f}$}; +% Text Node +\draw (373,220.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{G}_d$}; +% Text Node +\draw (222.2,139.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (222,217) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (77.6,180.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Prd}^{q,\varepsilon }$}; +% Text Node +\draw (620,223.2) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}^{d}$}; +% Text Node +\draw (616.4,136) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}^{N+1}$}; +% Text Node +\draw (506.2,140.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}_{1}^{N+1}$}; +% Text Node +\draw (490,225.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mymathbb{0}_{d,d} ,\mathcal{X}^{d,t,x}_{r,\omega_i}}$}; + + +\end{tikzpicture} +\end{center} +\end{remark} +\section{The $\mathsf{UES}$ network} +\begin{lemma}\label{lem:sm_sum} + Let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$ such that for all $i \in \{1,2,\hdots, n\}$ it is the cast that $\out\lp \nu_i\rp = 1$, and it is also the case that $\dep \lp \nu_1 \rp = \dep \lp \nu_2 \rp = \cdots =\dep \lp \nu_n\rp$. Let $x_1 \in \R^{\inn\lp \nu_1\rp},x_2 \in \R^{\inn\lp \nu_2\rp},\hdots x_n \in \R^{\inn\lp \nu_n\rp}$ and $\fx \in \R^{\sum_{i=1}^n \inn \lp \nu_i\rp}$. It is then the case that we have that: + \begin{align} + \real_{\rect}\lp \sm_{n,1} \bullet \lb \boxminus_{i=1}^n \nu_i \rb \rp \lp \fx\rp = \sum^n_{i=1} \real_{\rect} \lp \nu_i\rp \lp x_i\rp + \end{align} +\end{lemma} +\begin{proof} + Throughout the proof let $x_1 \in \R^{\inn\lp \nu_1\rp}, x_2 \in \R^{\inn\lp \nu_2\rp},\hdots, x_n \in \R^{\inn \lp \nu_n\rp}$ and $\fx \in \R^{\sum_{i=1}^n \inn \lp \nu_i\rp}$ such that $\fx = x_1 \frown x_2 \frown \cdots \frown x_n$. + Observe that by Lemma \ref{comp_prop} we have that: + \begin{align} + \real_{\rect}\lp \sm_{n,1} \bullet \lb \boxminus_{i=1}^n \nu_i \rb \rp\lp \fx \rp = \real_{\rect}\lp \sm_{n,1}\rp \circ \real_{\rect}\lp \boxminus_{i=1}^n \nu_i\rp \lp \fx \rp + \end{align} + Note however that Defiition \ref{def:rlz} yields that $\real_{\rect} \lp \sm_{n,1}\rp\lp x\rp = \lb 1 \: 1\: \cdots \: 1\rb\cdot x + 0$ for $x \in \R^n$. On the other hand $\out \lp \boxminus_{i=1}^n\nu_i\rp =n$ and furthermore by Lemma \ref{inst_of_stk} it is the case for $\fx \in \R^{\sum^n_{i=1}\inn \lp \nu_i\rp}$ that $\real_{\rect}\lp \boxminus_{i=1}^n \nu_i\rp\lp \fx \rp = \real_{\rect} \lp \nu_1\rp \lp x_1\rp\frown \real_{\rect}\lp \nu_2\rp\lp x_2\rp \frown \cdots \frown \real_{\rect}\lp \nu_n\rp\lp x_n\rp$. Thus $\real_{\rect}\lp \sm_{n,1} \bullet \lb \boxminus_{i=1}^n \nu_i \rb \rp\lp \fx \rp$ is rendered as: + \begin{align} + \begin{bmatrix} + 1 & 1 & \cdots & 1 + \end{bmatrix} \begin{bmatrix} + \real_{\rect}\lp \nu_1\rp\lp x_1\rp \\ + \real_{\rect}\lp \nu_2\rp\lp x_2\rp \\ + \vdots \\ + \real_{\rect}\lp \nu_n\rp\lp x_n\rp + \end{bmatrix} + 0 = \sum^n_{i=1} \real_{\rect} \lp \nu_i\rp \lp x_i\rp + \end{align} + This completes the proof of the lemma. +\end{proof} + +\begin{lemma}\label{sum_of_errors_of_stacking} + Let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$ with $\inn \lp \nu_1\rp = \inn \lp \nu_2\rp = \hdots =\inn \lp \nu_n\rp$ and $\out\lp \nu_1\rp = \out\lp \nu_2\rp= \hdots = \out\lp \nu_n\rp = 1$ such that for all $i \in \{1,2,\hdots,n\}$ it is the case that there exists $f_i \in C\lp \R^{\inn\lp \nu_1\rp},\R\rp$, and $\ve_i \in \lp 0, \infty \rp$, where for all $x_i \in \R^{\inn\lp \nu_1\rp}$, it is the case that $|\inst_{\rect}\lp \nu_i\rp\lp x_i\rp - f\lp x_i\rp|\les \ve_i$. It is then the case that for all $\fx \in \R^{n\cdot \inn\lp \nu_1\rp}$ and $x_i \in \R^{\inn\lp \nu_i\rp}$ with $\fx = x_1\frown x_2 \frown \cdots \frown x_n$ that: + \begin{align} + \left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 \les \sum_{i=1}^n\ve_i + \end{align} +\end{lemma} + +\begin{proof} + We will prove this with induction. This is straight-forward for the case where we have just one neural network where for all $x \in \R^{\inn\lp \nu_1\rp}$ it is the case that $\left\|\inst_{\rect}\lp \nu_1\rp \lp x\rp - f\lp x\rp\right\|_1 \les \ve_1 = \sum_{i=1}^1\ve_i$. Suppose now, that, $\left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 \les \sum_{i=1}^n\ve_i$ holds true for all cases upto and including $n$. Consider what happens when we have a triple, a function $f_{n+1}$, a neural network $\nu_{n+1}$, and $\ve_{n+1}\in \lp 0,\infty \rp$ with a maximum error over all $x \in \R^{\inn\lp \nu_1\rp}$ of $|f_{n+1}\lp x\rp - \inst_{\rect}\lp \nu_{n+1}\rp\lp x\rp | \les \ve_{n+1}$. Then Lemma \ref{inst_of_stk}, Corollary \ref{sum_of_frown_frown_of_sum}, and the triangle inequality tells us that: + \begin{align} + &\left\|\inst_{\rect}\lp \boxminus_i^{n+1}\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^{n+1} f_i\rb\lp \fx \rp\right\|_1 \nonumber \\ + &\les \left\|\inst_{\rect}\lp \boxminus_i^n\nu_i\rp \lp \fx \rp - \lb \frown_{i=1}^n f_i\rb\lp \fx \rp\right\|_1 + |f_{n+1}\lp x\rp - \inst_{\rect}\lp \nu_{n+1}\rp\lp x\rp | \nonumber \\ + &\les \sum_{i=1}^{n+1}\ve_i + \end{align} + This proves the inductive case and hence the Lemma. +\end{proof} + +\begin{lemma}[R\textemdash, 2024, Approximants for Brownian Motion] + +Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $n,N\in \N$, and $h \in \lp 0, \infty \rp$. Let $\delta,\ve \in \lp 0,\infty \rp $, $q\in \lp 2,\infty \rp$, satisfy that $\delta = \ve \lp 2^{q-1} +1\rp^{-1}$. Let $f:[t, T] \rightarrow \R$ be continuous almost everywhere in $\lb t, T \rb$. Let it also be the case that $f = g \circ \fh$, where $\fh: \lb t,T\rb \rightarrow \R^d$, and $g: \R^d \rightarrow \R$. Let $t=t_0 \les t_1\les \cdots \les t_{N-1} \les t_N=T$ such that for all $i \in \{0,1,...,N\}$ it is the case that $h = \frac{T-t}{N}$, and $t_i = t_0+i\cdot h$ . Let $\mathbf{t} = \lb t_0 \: t_1\: \cdots t_N \rb$ and as such let $f\lp\lb \mathbf{t} \rb_{*,*} \rp = \lb f(t_0) \: f(t_1)\: \cdots \: f(t_N) \rb$. Let $u_d \in C \lp \R^d,\R\rp$ satisfy for all $d \in \N$, $t \in \lb 0,T\rb$, $x \in \R^d$ that: + \begin{align} + \lp \frac{\partial}{\partial t} u_d\rp \lp t,x\rp + \lp \nabla_x^2 u_d\rp \lp t,x \rp + \alpha_d\lp x\rp u_d \lp t,x\rp = 0 + \end{align} + Furthermore, let $\mathfrak{u}_d^T(x) = u_d(T,x)$. Let $\mathsf{G}_d \subsetneq \neu$ be the neural network which instantiates as $\mathfrak{u}_d^T =\real_{\rect}\lp \mathsf{G}_d\rp \in C \lp \R^d, \R \rp$. + + Let $\mathcal{W}^d: \lb 0, T \rb \times \Omega \rightarrow \R^d$, $d \in \N$ be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: \lb t, T \rb \times \Omega \rightarrow \R^d$, $d \in \N$, $t \in \lb 0, T\rb$, $x\in \R^d$ be stochastic processes with continuous sample paths satisfying that for all $d \in \N$, $t\in \lb 0, T\rb$, $s \in \lb t, T \rb$, $x \in \R^d$ we have $\mathbb{P}$-a.s, that: + \begin{align} + \mathcal{X}^{d,t,x}_s = x + \int^t_s \sqrt{2} d\mathcal{W}^d_r + \end{align} + It is then the case that for all $d \in \N$, $t\in \lb 0,T \rb$, $x \in \R^d$ it holds that: + \begin{align}\label{fc-assertion} + u_d \lp t,x\rp = \mathbb{E} \lb \exp \lp \int^T_t \lp \alpha_d \circ \mathcal{X}^{d,t,x}_r\rp dr\rp u_d \lp T,\mathcal{X}_T^{d,t,x}\rp\rb + \end{align} + Let $\mathsf{E}^{N,h,q,\ve}_{n} \subsetneq \neu$ be neural networks given by: + \begin{align} + \mathsf{E}^{N,h,q,\ve}_{n} = \xpn^{q,\ve}_{n} \bullet \etr^{N,h} + \end{align} + Furthermore, let $\mathsf{G}_d \in \neu \subsetneq \neu$ be neural networks which instantiate as $u_d =\real_{\rect}\lp \mathsf{G}_d\rp \in C \lp \R^d, \R \rp$. + + Furthermore, let $\mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d}\subsetneq \neu$ be neural networks given by: + \begin{align} + \mathsf{UE}^{N,h,q,\ve}_{n,\mathsf{G}_d} = \prd^{q,\ve} \bullet \lb \mathsf{E}^{N,h,q,\ve}_{n,h,q,\ve} \DDiamond \mathsf{G}_d \rb + \end{align} + Futhermore, let $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \subsetneq \neu$ be neural networks given by: + \begin{align} + \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} = \mathsf{UE}^{N,h,q,\ve}_{n, \mathsf{G}_d} \bullet \lb \tun^{N+1}_1 \boxminus \aff_{\mymathbb{0}_{d,d},\mathcal{X}_{\omega_i}} \rb + \end{align} + Finally let $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn} \subsetneq \neu$ be neural networks which, for $\omega_i \in \Omega$, is given as: + \begin{align} + \mathsf{UES}^{N,h,q,\ve}_{n, \mathsf{G}_d, \Omega, \fn} = \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d, \omega_i}\rb\rp + \end{align} + + It is then the case that for all $\fX \in \R^{\fn \lp N+1\rp} \times \R^{\fn d}$: + \begin{enumerate}[label = (\roman*)] + \item $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \rp \in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$ + \item $\dep \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn}\rp \les \begin{cases} + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\dep \lp \mathsf{G}_d\rp-1 &:n = 0\\ + \frac{q}{q-2} \lb \log_2 \lp \ve^{-1}\rp +q \rb +\max\left\{\dep \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp,\dep \lp \mathsf{G}_d\rp\right\}-1 &:n \in \N\\ + \end{cases}$ + \item It is also the case that:\begin{align} + \param \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp &\les \param \lp \prd^{q,\ve}\rp + 2\lp\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\}\rp^2 \nonumber\\ + &+ 8 \max\left\{\lp 1+4n\rp, \wid_{\hid \lp \mathsf{G}_d\rp} \lp \mathsf{G}_d\rp \right\}\nonumber + \end{align} + \item It is also the case that: + \begin{align} + &\left| \mathbb{E} \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r}\rp ds\rp u_d\lp T,\mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn}\rp\right| \\ + &\les 3\ve +2\ve \left| \fu^T_d\lp x\rp\right|^q+2\ve \left| \exp \lp \int^b_afdx\rp\right|^q + \ve \left| \exp \lp \int^b_afdx\rp - \mathfrak{e}\right|^q -\mathfrak{e}\fu^T_d\lp x \rp\nonumber + \end{align} + Where, as per Lemma \ref{mathsfE}, $\mathfrak{e}$ is defined as: + \begin{align} + \left| \mathsf{E}^{N,h,q,\ve}_{n} - \exp \lp \int^b_afdx\rp\right| \les \fe + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + Note that for all $i \in \{ 1,2,\hdots, \mathfrak{n}\}$, Lemma \ref{UEX} tells us that $\real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \in C\lp \R^{N+1} \times \R^d, \R\rp$. Lemma \ref{nn_sum_cont} and Lemma \ref{nn_sum_is_sum_nn}, thus tells us that $\real_{\rect}\lp \lp \bigoplus_{i=1}^{\mathfrak{n}}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rp = \sum_{i=1}^\mathfrak{n}\lb \real_{\rect}\lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp\rb $. The sum of continuous functions is continuous. Note next that $\frac{1}{\mathfrak{n}}\triangleright$ is an affine neural network, and hence, by Lemma \ref{aff_prop}, must be continuous. + + Then Lemmas \ref{comp_prop}, \ref{5.3.4}, and the fact that by Lemma \ref{UEX} each of the individual stacked $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ neural networks is continuous then ensures us that it must therefore be the case that: $\real_{\rect} \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega,\fn}\rp \in C \lp \R^{\mathfrak{n}\lp N+1 \rp}\times \R^{\mathfrak{n} d}, \R \rp$. This proves Item (i). + + Next note that by construction each $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ has the same depth, indeed for each $i$ the only thing different for each of the $\mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}$ is the parameters themselves and not the count or depth or layer architecture. Note that $\dep \lp \sm_{\mathfrak{n},1}\rp = \dep \lp \frac{1}{\fn} \triangleright\rp = \dep \lp \aff_{\frac{1}{\mathfrak{n}},0}\rp = 1$. + + Whence by Lemma \ref{comp_prop} it is the case that $\dep \lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega,\fn} \rp = \dep \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp$. This then proves Item (ii). + + Next, observe that each of the $\mathsf{UEX}$ networks has the same architecture by construction. Corollary \ref{cor:sameparal} then yields that: + \begin{align} + \param \lp \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp + \end{align} + Note for instance also that by Remark \ref{param_of_aff}, it is the case that $\param\lp \sm_{\mathfrak{n},1} \rp = \mathfrak{n}+1$. Furthermore, since the output of the $\sm$ neural network has length one, by Definition \ref{slm} it is the case that $\param\lp \frac{1}{\mathfrak{n}}\triangleright\rp = 2$. Then Corollary \ref{affcor} leads us to conclude that: + \begin{align} + &\param \lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \les \param \lp \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \nonumber\\ + &\les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \nonumber\\ + &\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. +24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb + \end{align} + and therefore that: + \begin{align} + &\param \lp \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d, \omega_i}\rb\rp \rp \nonumber\\ + &\les \param \lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \nonumber\\ + &\les \param \lp \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i} \rp \nonumber\\ + &\les \mathfrak{n}^2\cdot \param \lp \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rp \nonumber \\ + &\les \fn^2 \cdot \lb \frac{360q}{q-2} \lb \log_2 \lp \ve^{-1} \rp +q+1 \rb +324+ 48n\right. \nonumber\\ &\left. +24 \wid_{\hid\lp \mathsf{G}_d\rp}\lp \mathsf{G}_d\rp + 4\max \left\{\param \lp \mathsf{E}^{N,h,q,\ve}_{n}\rp, \param \lp \mathsf{G}_d\rp \right\} \rb + \end{align} + Now observe that by the triangle inequality, we have that: + \begin{align} + &\left| \E \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp ds\rp u_d^T\lp \mathcal{X}^{d,t,x}_{r,\Omega}\rp\rb - \real_{\rect}\lp \mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d,\Omega, \fn}\rp \right| \nonumber \\ + &=\left| \E \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds\rp u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \inst_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb\right| \nonumber\\ + &\les \left| \E \lb \exp \lp \int^T_t f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds\rp u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb \right|\nonumber \\ + &+\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb\right| \nonumber \\ + \end{align} + Observe that by the triangle inequality, the absolute homogeneity condition for norms, the fact that the Brownian motions are independent of each other, Lemma \ref{lem:sm_sum}, the fact that $\mathfrak{n}\in \N$, the fact that the upper limit of error remains bounded by the same bound for all $\omega_i \in \Omega$, and Lemma \ref{sum_of_errors_of_stacking}, then renders the second summand as: + \begin{align} + &\left| \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp\rb \rb - \real_{\rect}\lb \frac{1}{\mathfrak{n}} \triangleright\lp \sm_{\mathfrak{n},1}\bullet\lb \boxminus_{i=1}^{\mathfrak{n}} \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp\rb\right| \nonumber \\ + &\les \left|\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1} \exp \lp \int_t^T f\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp \rb - \frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lp \real_{\rect}\lb \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\rb\rp \rb \right| \nonumber \\ + &\les \cancel{\frac{1}{\mathfrak{n}} \sum^{\mathfrak{n}}_{i=1}}\left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\right| \nonumber\\ + &\les \left| \exp \lp \int^T_tf\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot u^T_d\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rp - \mathsf{UEX}^{N,h,q,\ve}_{n,\mathsf{G}_d,\omega_i}\right| \nonumber\\ + \end{align} +% Note that by Lemma \ref{iobm} each of the $\mathcal{X}^{d,t,x}_{r,\omega_i}$ are pairwise independent of each other for all $i \in \{1,2,\hdots,\mathfrak{n}\}$. Note also that by Definition \ref{def:brown_motion} it is the case, for all $\omega_i \in \Omega$ that $\mathcal{X}^{d,t,x}_{T,\omega_i} \sim \norm \lp \mymathbb{0}_d, \diag_d(T) \rp$ + +Note for the first summand that it is in $\mathcal{O}\lp \frac{1}{\sqrt{\mathfrak{n}}}\rp$. Notice that both $f$ and $\fu^T_d$ are continuous functions for $d\in \N$. Note also that $F:[t,T] \rightarrow \R$ defined as: +\begin{align} + F(\fx) \coloneqq \int_t^\ft f\lp\fx\rp dx +\end{align} +is continuous on $\lb t,T\rb$. Thus , notice that \cite[Theorem~2.1]{rio_moment_2009} with $k$ + +\end{proof} +\begin{remark} + Note that diagrammatically, this can be represented as in figure below. +\begin{figure}[h] +\begin{center} + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,475); %set diagram left start at 0, and has a height of 475 + +%Shape: Rectangle [id:dp5014556157804896] +\draw (414.07,18) -- (489.31,18) -- (489.31,58) -- (414.07,58) -- cycle ; +%Shape: Rectangle [id:dp6888177801681734] +\draw (410.88,97.6) -- (486.12,97.6) -- (486.12,137.6) -- (410.88,137.6) -- cycle ; +%Shape: Rectangle [id:dp5733062989346852] +\draw (307.95,17.2) -- (386.97,17.2) -- (386.97,57.2) -- (307.95,57.2) -- cycle ; +%Shape: Rectangle [id:dp3644206511797573] +\draw (304.33,94.8) -- (382.62,94.8) -- (382.62,134.8) -- (304.33,134.8) -- cycle ; +%Shape: Rectangle [id:dp872163975205418] +\draw (221.55,58.6) -- (272.29,58.6) -- (272.29,98.6) -- (221.55,98.6) -- cycle ; +%Straight Lines [id:da0837610283606276] +\draw (412.34,37.31) -- (389.39,36.84) ; +\draw [shift={(387.39,36.8)}, rotate = 1.18] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. 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(10.93,3.29) ; + +% Text Node +\draw (428.28,22.2) node [anchor=north west][inner sep=0.75pt] {$\mathsf{E}_{N,n,h,q,\varepsilon }^{\exp ,f}$}; +% Text Node +\draw (444.46,108.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{G}_d$}; +% Text Node +\draw (332.68,27.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (332.53,105) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; +% Text Node +\draw (226.08,68.6) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Prd}^{q,\varepsilon }$}; +% Text Node +\draw (622.13,111.2) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}^{d}$}; +% Text Node +\draw (617.18,24) node [anchor=north west][inner sep=0.75pt] {$\mathbb{R}^{N+1}$}; +% Text Node +\draw (535.1,28.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}_{1}^{N+1}$}; +% Text Node +\draw (534.54,108.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Aff}_{\mathbb{0}}{}_{_{d}{}_{,}{}_{d} ,\mathcal{X}}$}; +% Text Node 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[anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $}; +% Text Node +\draw (553,216.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $}; +% Text Node +\draw (336,215.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $}; +% Text Node +\draw (619,214.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $}; +% Text Node +\draw (234,217.4) node [anchor=north west][inner sep=0.75pt] [font=\Large] {$\vdots $}; +% Text Node +\draw (132,221.4) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Sum}$}; +% Text Node +\draw (202,127.4) node [anchor=north west][inner sep=0.75pt] {$\vdots $}; +% Text Node +\draw (202,312.4) node [anchor=north west][inner sep=0.75pt] {$\vdots $}; +% Text Node +\draw (56,225.4) node [anchor=north west][inner sep=0.75pt] {$\frac{1}{\mathfrak{n}} \rhd $}; + + +\end{tikzpicture} +\end{center} +\caption{Neural network diagram for the $\mathsf{UES}$ network.} +\end{figure} +\end{remark} +\begin{remark} + It may be helpful to think of this as a very crude form of ensembling. +\end{remark} + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/Dissertation/appendices.pdf b/Dissertation/appendices.pdf new file mode 100644 index 0000000..77d6f28 Binary files /dev/null and b/Dissertation/appendices.pdf differ diff --git a/Dissertation/appendices.tex b/Dissertation/appendices.tex new file mode 100644 index 0000000..b7f56de --- /dev/null +++ b/Dissertation/appendices.tex @@ -0,0 +1,69 @@ + +\section{Code Listings} + +Parts of this code have been released on \texttt{CRAN} under the package name \texttt{nnR}, and can be found in \cite{nnR-package}, with the corresponding repository being found at \cite{Rafi_nnR_2024}: + +\lstinputlisting[language = R, style = rstyle, label = nn_creator, caption = {R code for neural network generation}]{"/Users/shakilrafi/R-simulations/nn_creator.R"} + +\lstinputlisting[language = R, style = rstyle, label = aux_fun, caption = {R code for auxilliary functions}]{"/Users/shakilrafi/R-simulations/aux_fun.R"} + +\lstinputlisting[language = R, style = rstyle, label = activations, caption = {R code for activation functions ReLU and Sigmoid}]{"/Users/shakilrafi/R-simulations/activations.R"} + +\lstinputlisting[language = R, style = rstyle, label = instantiation, caption = {R code for realizations}]{"/Users/shakilrafi/R-simulations/instantiation.R"} + +\lstinputlisting[language = R, style = rstyle, label = stk, caption = {R code for parallelizing two neural networks}]{"/Users/shakilrafi/R-simulations/stacking.R"} + + +\lstinputlisting[language = R, style = rstyle, label = Aff, caption = {R code for affine neural networks}]{"/Users/shakilrafi/R-simulations/Aff.R"} + + +\lstinputlisting[language = R, style = rstyle, label = comp, caption = {R code for composition of two neural networks}]{"/Users/shakilrafi/R-simulations/comp.R"} + + +\lstinputlisting[language = R, style = rstyle, label = scalar_mult, caption = {R code for scalar multiplication}]{"/Users/shakilrafi/R-simulations/scalar_mult.R"} + + +\lstinputlisting[language = R, style = rstyle, label = nn_sum, caption = {R code for sum of two neural networks}]{"/Users/shakilrafi/R-simulations/nn_sum.R"} + + +\lstinputlisting[language = R, style = rstyle, label = i, caption = {R code for $\mathfrak{i}$}]{"/Users/shakilrafi/R-simulations/i.R"} +\lstinputlisting[language = R, style = rstyle, label = Id, caption = {R code for $\mathsf{Id}$ neural networks}]{"/Users/shakilrafi/R-simulations/Id.R"} +\lstinputlisting[language = R, style = rstyle, label = Tun, caption = {R code for $\tun$}]{"/Users/shakilrafi/R-simulations/Tun.R"} + + +\lstinputlisting[language = R, style = rstyle, label = Phi_k, caption = {R code for $\Phi_k$}]{"/Users/shakilrafi/R-simulations/Phi_k.R"} +\includegraphics{"/Users/shakilrafi/R-simulations/Phi_k_properties/Phi_k_diff.png"} + + +\lstinputlisting[language = R, style = rstyle, label = Phi_k_properties, caption = {R code for simulations involving $\Phi_k$}]{"/Users/shakilrafi/R-simulations/Phi_k_properties.R"} + + +\lstinputlisting[language = R, style = rstyle, label = Phi, caption = {R code for $\Phi$}]{"/Users/shakilrafi/R-simulations/Phi.R"} + + +\includegraphics{"/Users/shakilrafi/R-simulations/Phi_properties/Phi_diff_contour.png"} + + +\lstinputlisting[language = R, style = rstyle, label = Phi_properties, caption = {R code for simulations involving $\Phi$}]{"/Users/shakilrafi/R-simulations/Phi_properties.R"} + + +\lstinputlisting[language = R, style = rstyle, label = Sqr, caption = {R code for $\sqr$}]{"/Users/shakilrafi/R-simulations/Sqr.R"} + +\lstinputlisting[language = R, style = rstyle, label = Sqr_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Sqr_properties.R"} + +\lstinputlisting[language = R, style = rstyle, label = Pwr, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Pwr.R"} + +\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Pwr_3_properties.R"} + +\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Nrm.R"} + +\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Mxm.R"} + +\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\tay$}]{"/Users/shakilrafi/R-simulations/Tay.R"} + +\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\etr$}]{"/Users/shakilrafi/R-simulations/Etr.R"} + + + + + diff --git a/Dissertation/brownian_motion_monte_carlo_non_linear_case.aux b/Dissertation/brownian_motion_monte_carlo_non_linear_case.aux new file mode 100644 index 0000000..ac528f2 --- /dev/null +++ b/Dissertation/brownian_motion_monte_carlo_non_linear_case.aux @@ -0,0 +1,60 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{Beck_2021} +\@writefile{toc}{\contentsline {chapter}{\numberline {4}Brownian motion Monte Carlo of the non-linear case}{59}{chapter.4}\protected@file@percent } +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\newlabel{4.0.1}{{4.0.1}{59}{Subsequent Setting}{equation.4.0.1}{}} +\newlabel{4.0.1@cref}{{[equation][1][4,0]4.0.1}{[1][59][]59}} +\newlabel{4.0.2}{{4.0.2}{59}{Subsequent Setting}{equation.4.0.2}{}} +\newlabel{4.0.2@cref}{{[equation][2][4,0]4.0.2}{[1][59][]59}} +\newlabel{def:1.18}{{4.0.1}{60}{Subsequent Setting}{equation.4.0.4}{}} +\newlabel{def:1.18@cref}{{[definition][1][4,0]4.0.1}{[1][60][]60}} +\newlabel{Setting 1.1}{{4.0.1}{60}{Subsequent Setting}{equation.4.0.4}{}} +\newlabel{Setting 1.1@cref}{{[definition][1][4,0]4.0.1}{[1][60][]60}} +\newlabel{(1.12)}{{4.0.5}{60}{Subsequent 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+1,43 @@ +\chapter{Brownian motion Monte Carlo of the non-linear case} + +We now seek to apply the techniques introduced in Chapter 2 on \ref{3.3.21}. To do so we need a variation of Setting \ref{Setting 1.1}. To that end we define such a setting. Assume $v,f,\alpha$ from Lemma \ref{3.3.2}. +\begin{definition}[Subsequent Setting] +Let $g \in C\lp \R^d, \R \rp$ be the function defined by: +\begin{align}\label{4.0.1} + g(x) = v(T,x) +\end{align} +Let $F: C\lp \lb 0,T\rb \times \R^d, \R \rp \rightarrow C \lp \lb ,T \rb \times \R^d, \R \rp$ be the functional defined as: +\begin{align}\label{4.0.2} + \lp F \lp v \rp \rp \lp t,x \rp = f\lp t,x,v\lp t,x\rp \rp +\end{align} +Note also that by Claim \ref{3.3.5} it is the case that: +\begin{align} + \lv f \lp t,x,w\rp -f\lp t,x,\mathfrak{w}\rp \rv \leqslant L \lv w-\mathfrak{w} \rv +\end{align} +Note also that since $f\lp t,x,0 \rp =0$, and since by \cite[Corollary~3.9]{Beck_2021}, $v$ is growing at most polynomially, it is then the case that: +\begin{align} + \max \left\{ \lv f \lp t,x,0 \rp \rv, \lv g \lp x \rp \rv \right\} \leqslant \mathfrak{L}\lp 1 + \|x\|^p \rp +\end{align} + +Substituting (\ref{4.0.1}) and (\ref{4.0.2}) into (\ref{3.3.20}) renders (\ref{3.3.20}) as: +\begin{align} + v(t,x) &= \E \lb v\lp T, \mathcal{X}_T^{t,x} \rp + \int ^T_t f \lp s, \mathcal{X}^{t,x}_s, v \lp s, \mathcal{X}^{t,x}_s \rp \rp ds\rb \nonumber\\ + v(t,x) &= \E \lb v\lp T, \mathcal{X}_T^{t,x} \rp \rb + \E \lb \int ^T_t f \lp s, \mathcal{X}^{t,x}_s, v \lp s, \mathcal{X}^{t,x}_s \rp \rp ds\rb \nonumber\\ + v(t,x) &= \E \lb v\lp T, \mathcal{X}_T^{t,x} \rp \rb + \int ^T_t \E \lb f \lp s, \mathcal{X}^{t,x}_s, v \lp s, \mathcal{X}^{t,x}_s \rp \rp ds\rb \nonumber\\ + v\lp t,x \rp &= \E \lb g\lp \mathcal{X}^{t,x}_T \rp \rb+ \int^T_t \E \lb \lp F \lp v \rp \rp \lp s,\mathcal{X}^{t,x}_s\rp \rb ds\nonumber +\end{align} +\label{def:1.18}\label{Setting 1.1} Let $d,m \in \mathbb{N}$, $T, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \bigcup_{n\in \mathbb{N}}\mathbb{Z}^n$, $f \in C\lp \lb 0,T \rb \times \R^d \times \R \rp $, $g \in C(\mathbb{R}^d,\mathbb{R})$, let $F: C \lp \lb 0,T \rb \times \R^d, \R \rp \rightarrow C \lp \lb 0,T \rb \times \R^d, \R \rp$ assume for all $t \in [0,T],x\in \mathbb{R}^d$ that: +\begin{align}\label{(1.12)} + \lv f\lp t,x,w \rp - f\lp t,x,\mathfrak{w} \rp \rv \leqslant L \lv w - \mathfrak{w} \rv &&\max\left\{\lv f \lp t,x,0 \rp \rv, \lv g(x) \rv \right\} \leqslant \mathfrak{L} \lp 1+\|x\|_E^p \rp +\end{align} + +and let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, let $\mathfrak{u}^\theta: \Omega \rightarrow \lb 0,1 \rb$, $\theta \in \Theta$ be i.i.d. random variables, and suume for all $\theta \in \Theta$, $r \in \lp 0,1 \rp$ that $\mathbb{P}\lp \mathfrak{u}^\theta \leqslant r \rp = r$, let $\mathcal{U}^\theta: \lb 0,T \rb \times \Omega \rightarrow \lb 0,T\rb$, $\theta \in \Theta$ satisty for all $t \in \lb 0,T \rb$, $\theta \in \Theta$ that $\mathcal{U}^\theta_t = t + \lp T-t \rp \mathfrak{u}^\theta$, let $\mathcal{W}^{\theta}: [0,T] \times \Omega \rightarrow \mathbb{R}^d$, $\theta \in \Theta$ be independent standard Brownian motions, let $u \in C([0,T] \times \mathbb{R}^d,\mathbb{R})$ satisfy for all $t \in [0,T]$, $x\in \mathbb{R}^d$, that $\mathbb{E} \lb \lv g \lp x+\mathcal{W}^0_{T-t} \rp\rv \rb + \int^T_t \E \lb \lp F \lp u \rp \rp \lp s,x+\mathcal{W}^0_{s-t} \rp \rb < \infty$ and: +\begin{align}\label{(1.12)} + u(t,x) &= \mathbb{E} \lb g \lp x+\mathcal{W}^0_{T-t} \rp \rb + \int^T_t \E \lb \lp F \lp u \rp \rp \lp s,x+ \mathcal{W}^0_{s-t} \rp \rb ds +\end{align} +and let let $U^\theta:[0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$, $\theta \in \Theta$, $n \in \Z$ satisfy for all $\theta \in \Theta$, $t \in [0,T]$, $x\in \mathbb{R}^d$, $n \in \N_0$ that: +\begin{align}\label{(1.14)} + U^\theta_n(t,x) &= \frac{\mathbbm{1}_\N \lp n \rp}{m^n}\left[\sum^{m^n}_{k=1}g\left(x+\mathcal{W}^{(\theta,0,-k)}_{T-t}\right)\right] \nonumber\\ + &+ \sum^{n-1}_{i=1} \frac{T-t}{m^{n-i}} \lb \sum^{m^{n-i}}_{k=1} \lp F \lp U^{\lp \theta,i,k \rp }_i \rp \rp \lp \mathcal{U}^{\lp \theta,i,k \rp }, x+ \mathcal{W}^{\lp \theta,i,k \rp }_{\mathcal{U}_t^{\lp \theta,i,k \rp}} \rp \rb +\end{align} +\end{definition} diff --git a/Dissertation/categorical_neural_network.aux b/Dissertation/categorical_neural_network.aux new file mode 100644 index 0000000..537ddb1 --- /dev/null +++ b/Dissertation/categorical_neural_network.aux @@ -0,0 +1,52 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {chapter}{\numberline {8}Some categorical ideas about neural networks}{132}{chapter.8}\protected@file@percent } +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\@setckpt{categorical_neural_network}{ +\setcounter{page}{133} +\setcounter{equation}{0} +\setcounter{enumi}{7} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{0} +\setcounter{mpfootnote}{0} +\setcounter{part}{2} +\setcounter{chapter}{8} +\setcounter{section}{0} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} 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+\newcommand{\scrK}{\mathscr{K}} +\newcommand{\scrL}{\mathscr{L} \cfadd{def:lin_interp}} +\newcommand{\scrM}{\mathscr{M}} +\newcommand{\scrN}{\mathscr{N}} +\newcommand{\scrO}{\mathscr{O}} +\newcommand{\scrP}{\mathscr{P}} +\newcommand{\scrQ}{\mathscr{Q}} +\newcommand{\scrR}{\mathscr{R}} +\newcommand{\scrS}{\mathscr{S}} +\newcommand{\scrT}{\mathscr{T}} +\newcommand{\scrU}{\mathscr{U}} +\newcommand{\scrV}{\mathscr{V}} +\newcommand{\scrW}{\mathscr{W}} +\newcommand{\scrX}{\mathscr{X}} +\newcommand{\scrY}{\mathscr{Y}} +\newcommand{\scrZ}{\mathscr{Z}} diff --git a/Dissertation/conclusions-further-research.tex b/Dissertation/conclusions-further-research.tex new file mode 100644 index 0000000..cf3e285 --- /dev/null +++ b/Dissertation/conclusions-further-research.tex @@ -0,0 +1,46 @@ +\chapter{Conclusions and Further Research} + +We will present three avenues of further research and related work on parameter estimates here. + +\section{Further operations and further kinds of neural networks} + +Note, for instance, that several classical operations are done on neural networks that have yet to be accounted for in this framework and talked about in the literature. We will discuss two of them \textit{dropout} and \textit{dilation} and provide lemmas that may be useful to future research. + +\subsection{Mergers and Dropout} + +\begin{definition}[Hadamard Product] + Let $m,n \in \N$. Let $A,B \in \R^{m \times n}$. For all $i \in \{ 1,2,\hdots,m\}$ and $j \in \{ 1,2,\hdots,n\}$ define the Hadamard product $\odot: \R^{m\times n} \times \R^{m \times n} \rightarrow \R^{m \times n}$ as: + \begin{align} + A \odot B \coloneqq \lb A \odot B \rb _{i,j} = \lb A \rb_{i,j} \times \lb B \rb_{i,j} \quad \forall i,j + \end{align} +\end{definition} + +\begin{definition}[Scalar product of weights] + Let $\nu \in \neu$, $L\in \N$, $i,j,k \in \N$, and $c\in \R$. Assume also that $\lay \lp \nu\rp = \lp l_0,l_1,l_2,\hdots, l_L\rp$. Assume then that the neural network is given by $\nu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$. We will denote by $c\circledast^{i,j}\nu$ as the neural network which, for $i \in \N \cap \lb 1,L-1\rb$, $j \in \N \cap \lb 1,l_i\rb$, is given by $c \circledast^{i,j} \nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2\rp, \hdots,\lp \tilde{W}_i,b_i \rp,\lp \tilde{W}_{i+1},b_{i+1}\rp,\hdots \lp W_L,b_L\rp\rp$ where it is the case that: + \begin{align} + \tilde{W}_i = \lp \mymathbb{k}^{j,j,c-1}_{l_i,l_{i}} + \mathbb{I}_{l_i}\rp W_i + \end{align} +\end{definition} +\begin{definition}[The Dropout Operator] + Let $\nu \in \neu$, $L\in \N$, $i_1,i_2,\hdots, i_k,j,k \in \N$, and $c_1,c_2,\hdots,c_k\in \R$. Assume also that $\lay \lp \nu\rp = \lp l_0,l_1,l_2,\hdots, l_L\rp$. Assume then that the neural network is given by $\nu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$. We will denote by $\dropout_n^{\unif}\lp \nu \rp$ the neural network that is given by: + \begin{align} + 0\circledast^{i_1,j_1} \lp 0 \circledast^{i_2,j_2}\lp \hdots 0\circledast^{i_n,j_n}\nu \hdots \rp\rp + \end{align} + Where for each $k \in \{1,2,\hdots,n \}$ it is the case that $i \sim \unif \{ 1,L-1\}$ and $j\sim \unif\{1,l_j\} $ +\end{definition} + +We will also define the dropout operator introduced in \cite{srivastava_dropout_2014}. + + +\begin{definition}[Realization with dropout] + Let $\nu \in \neu$, $L,n \in \N$, $p \in \lp 0,1\rp$, $\lay \lp \nu\rp = \lp l_0,l_1,\hdots, \l_L\rp$, and that $\neu = \lp \lp W_1,b_1\rp, \lp W_2,b_2\rp, \hdots , \lp W_L,b_L\rp \rp$. Let it be the case that for each $n\in \N$, $\rho_n = \{ x_1,x_2,\hdots,x_n\} \in \R^n$ where for each $i \in \{1,2,\hdots,n\}$ it is the case that $x_i \sim \bern(p)$. We will then denote $\real_{\rect}^{D} \lp \nu \rp \in C\lp \R^{\inn\lp \nu\rp},\R^{\out\lp \nu \rp}\rp$, the continuous function given by: + \begin{align} + \real_{\rect}^D\lp \nu \rp = \rho_{l_L}\odot \rect \lp W_l\lp \rho_{l_{L-1}} \odot \rect \lp W_{L-1}\lp \hdots\rp + b_{L-1}\rp\rp + b_L\rp + \end{align} +\end{definition} + + + + + + diff --git a/Dissertation/julia_sharktooth_mult.ipynb b/Dissertation/julia_sharktooth_mult.ipynb new file mode 100644 index 0000000..4f2c87b --- /dev/null +++ b/Dissertation/julia_sharktooth_mult.ipynb @@ -0,0 +1,7863 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "max_conv_operator (generic function with 1 method)" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "using Random\n", + "using Distributions\n", + "using Plots\n", + "using LinearAlgebra\n", + "# ===============================================================\n", + "\n", + "function max_conv_operator(samples, f_samples, x, L)\n", + " return maximum(f_samples .- L .* norm(x .- samples, 1))\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "sharktooth (generic function with 1 method)" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "function sharktooth(f, x_start, x_end, y_start, y_end, number_of_sharkteeth, L, plot_arg)\n", + " sample_x = rand(Uniform(x_start, x_end), number_of_sharkteeth)\n", + " sample_y = rand(Uniform(y_start, y_end), number_of_sharkteeth)\n", + " samples = hcat(sample_x,sample_y)\n", + " f_samples = f.(eachrow(samples))\n", + "\n", + " x = LinRange(x_start,x_end,1000)\n", + " y = LinRange(y_start,y_end,1000)\n", + " X = [xi for xi in x, yi in y]\n", + " Y = [yi for xi in x, yi in y]\n", + " meshgrid = hcat(reshape(X,:),reshape(Y,:))\n", + " \n", + " approximant = Float64[]\n", + " for i in 1:length(X)\n", + " push!(approximant, max_conv_operator(samples, f_samples,meshgrid[i,:]',L))\n", + " end\n", + " error = maximum(abs.(f.(eachrow(meshgrid)) - approximant))\n", + " if plot_arg ==1 \n", + " plot(x,y,approximant, st=:surface)\n", + " else \n", + " return error\n", + " end\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "f (generic function with 2 methods)" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "function f(x)\n", + " x[1]^3 + y[1]^3\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "ename": "UndefVarError", + "evalue": "UndefVarError: `y` not defined", + "output_type": "error", + "traceback": [ + "UndefVarError: `y` not defined\n", + "\n", + "Stacktrace:\n", + " [1] f(x::SubArray{Float64, 1, Matrix{Float64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true})\n", + " @ Main ~/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/julia_sharktooth_mult.ipynb:2\n", + " [2] _broadcast_getindex_evalf\n", + " @ ./broadcast.jl:683 [inlined]\n", + " [3] _broadcast_getindex\n", + " @ ./broadcast.jl:656 [inlined]\n", + " [4] getindex\n", + " @ ./broadcast.jl:610 [inlined]\n", + " [5] copy(bc::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{1}, Tuple{Base.OneTo{Int64}}, typeof(f), Tuple{RowSlices{Matrix{Float64}, Tuple{Base.OneTo{Int64}}, SubArray{Float64, 1, Matrix{Float64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}}})\n", + " @ Base.Broadcast ./broadcast.jl:912\n", + " [6] materialize\n", + " @ ./broadcast.jl:873 [inlined]\n", + " [7] sharktooth(f::Function, x_start::Int64, x_end::Int64, y_start::Int64, y_end::Int64, number_of_sharkteeth::Int64, L::Int64, plot_arg::Int64)\n", + " @ Main ~/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/julia_sharktooth_mult.ipynb:5\n", + " [8] top-level scope\n", + " @ ~/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/julia_sharktooth_mult.ipynb:1" + ] + } + ], + "source": [ + "sharktooth(f,-3,3,-2,2,300,4,1)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+\bibcite{Ito1946}{{17}{1942b}{{It\^o}}{{}}} +\bibcite{karatzas1991brownian}{{18}{1991}{{Karatzas and Shreve}}{{}}} +\bibcite{rio_moment_2009}{{19}{2009}{{Rio}}{{}}} +\bibstyle{apalike} +\@writefile{toc}{\contentsline {chapter}{Appendices}{180}{section*.3}\protected@file@percent } +\newlabel{pythoncode}{{1}{181}{Python Code}{lstlisting.a..1}{}} +\newlabel{pythoncode@cref}{{[lstlisting][1][0]1}{[1][181][]181}} +\@writefile{lol}{\contentsline {lstlisting}{\numberline {1}Python Code}{181}{lstlisting.a..1}\protected@file@percent } +\newlabel{pythoncode}{{2}{183}{Python Code}{lstlisting.a..2}{}} +\newlabel{pythoncode@cref}{{[lstlisting][2][0]2}{[1][183][]183}} +\@writefile{lol}{\contentsline {lstlisting}{\numberline {2}Python Code}{183}{lstlisting.a..2}\protected@file@percent } +\gdef \@abspage@last{185} diff --git a/Dissertation/main.bbl b/Dissertation/main.bbl new file mode 100644 index 0000000..07097c3 --- /dev/null +++ b/Dissertation/main.bbl @@ -0,0 +1,120 @@ +\begin{thebibliography}{} + +\bibitem[Bass, 2011]{bass_2011} +Bass, R.~F. (2011). +\newblock {\em Brownian Motion}, page 6–12. +\newblock Cambridge Series in Statistical and Probabilistic Mathematics. + Cambridge University Press. + +\bibitem[Beck et~al., 2021a]{Beck_2021} +Beck, C., Gonon, L., Hutzenthaler, M., and Jentzen, A. (2021a). +\newblock On existence and uniqueness properties for solutions of stochastic + fixed point equations. +\newblock {\em Discrete \& Continuous Dynamical Systems - B}, 26(9):4927. + +\bibitem[Beck et~al., 2021b]{BHJ21} +Beck, C., Hutzenthaler, M., and Jentzen, A. (2021b). +\newblock On nonlinear {Feynman}{\textendash}{Kac} formulas for viscosity + solutions of semilinear parabolic partial differential equations. +\newblock {\em Stochastics and Dynamics}, 21(08). + +\bibitem[Beck et~al., 2021c]{bhj20} +Beck, C., Hutzenthaler, M., and Jentzen, A. (2021c). +\newblock On nonlinear feynman–kac formulas for viscosity solutions of + semilinear parabolic partial differential equations. +\newblock {\em Stochastics and Dynamics}, 21(08):2150048. + +\bibitem[Crandall et~al., 1992]{crandall_lions} +Crandall, M.~G., Ishii, H., and Lions, P.-L. (1992). +\newblock User’s guide to viscosity solutions of second order partial + differential equations. +\newblock {\em Bull. Amer. Math. Soc.}, 27(1):1--67. + +\bibitem[Da~Prato and Zabczyk, 2002]{da_prato_zabczyk_2002} +Da~Prato, G. and Zabczyk, J. (2002). +\newblock {\em Second Order Partial Differential Equations in Hilbert Spaces}. +\newblock London Mathematical Society Lecture Note Series. Cambridge University + Press. + +\bibitem[Durrett, 2019]{durrett2019probability} +Durrett, R. (2019). +\newblock {\em Probability: Theory and Examples}. +\newblock Cambridge Series in Statistical and Probabilistic Mathematics. + Cambridge University Press. + +\bibitem[Golub and Van~Loan, 2013]{golub2013matrix} +Golub, G. and Van~Loan, C. (2013). +\newblock {\em Matrix Computations}. +\newblock Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins + University Press. + +\bibitem[Grohs et~al., 2018]{grohsetal} +Grohs, P., Hornung, F., Jentzen, A., and von Wurstemberger, P. (2018). +\newblock {A proof that artificial neural networks overcome the curse of + dimensionality in the numerical approximation of Black-Scholes partial + differential equations}. +\newblock Papers 1809.02362, arXiv.org. + +\bibitem[Grohs et~al., 2023]{grohs2019spacetime} +Grohs, P., Hornung, F., Jentzen, A., and Zimmermann, P. (2023). +\newblock Space-time error estimates for deep neural network approximations for + differential equations. +\newblock {\em Advances in Computational Mathematics}, 49(1):4. + +\bibitem[Grohs et~al., 2022]{Grohs_2022} +Grohs, P., Jentzen, A., and Salimova, D. (2022). +\newblock Deep neural network approximations for solutions of {PDEs} based on + monte carlo algorithms. +\newblock {\em Partial Differential Equations and Applications}, 3(4). + +\bibitem[Gy{\"o}ngy and Krylov, 1996]{Gyngy1996ExistenceOS} +Gy{\"o}ngy, I. and Krylov, N.~V. (1996). +\newblock Existence of strong solutions for {It\^o}'s stochastic equations via + approximations. +\newblock {\em Probability Theory and Related Fields}, 105:143--158. + +\bibitem[Hutzenthaler et~al., 2020a]{hutzenthaler_overcoming_2020} +Hutzenthaler, M., Jentzen, A., Kruse, T., Anh~Nguyen, T., and von + Wurstemberger, P. (2020a). +\newblock Overcoming the curse of dimensionality in the numerical approximation + of semilinear parabolic partial differential equations. +\newblock {\em Proceedings of the Royal Society A: Mathematical, Physical and + Engineering Sciences}, 476(2244):20190630. + +\bibitem[Hutzenthaler et~al., 2021]{hutzenthaler_strong_2021} +Hutzenthaler, M., Jentzen, A., Kuckuck, B., and Padgett, J.~L. (2021). +\newblock Strong {$L^p$}-error analysis of nonlinear {Monte} {Carlo} + approximations for high-dimensional semilinear partial differential + equations. +\newblock Technical Report arXiv:2110.08297, arXiv. +\newblock arXiv:2110.08297 [cs, math] type: article. + +\bibitem[Hutzenthaler et~al., 2020b]{hjw2020} +Hutzenthaler, M., Jentzen, A., and von Wurstemberger~Wurstemberger (2020b). +\newblock {Overcoming the curse of dimensionality in the approximative pricing + of financial derivatives with default risks}. +\newblock {\em Electronic Journal of Probability}, 25(none):1 -- 73. + +\bibitem[It\^o, 1942a]{Ito1942a} +It\^o, K. (1942a). +\newblock Differential equations determining {Markov} processes (original in + {Japanese}). +\newblock {\em Zenkoku Shijo Sugaku Danwakai}, 244(1077):1352--1400. + +\bibitem[It\^o, 1942b]{Ito1946} +It\^o, K. (1942b). +\newblock On a stochastic integral equation. +\newblock {\em Proc. Imperial Acad. Tokyo}, 244(1077):1352--1400. + +\bibitem[Karatzas and Shreve, 1991]{karatzas1991brownian} +Karatzas, I. and Shreve, S. (1991). +\newblock {\em Brownian Motion and Stochastic Calculus}. +\newblock Graduate Texts in Mathematics (113) (Book 113). Springer New York. + +\bibitem[Rio, 2009]{rio_moment_2009} +Rio, E. (2009). +\newblock Moment {Inequalities} for {Sums} of {Dependent} {Random} {Variables} + under {Projective} {Conditions}. +\newblock {\em J Theor Probab}, 22(1):146--163. + +\end{thebibliography} diff --git a/Dissertation/main.bib b/Dissertation/main.bib new file mode 100644 index 0000000..f635ff9 --- /dev/null +++ b/Dissertation/main.bib @@ -0,0 +1,664 @@ +@book{karatzas1991brownian, + title={Brownian Motion and Stochastic Calculus}, + author={Karatzas, I. and Shreve, S.E.}, + isbn={9780387976556}, + lccn={96167783}, + series={Graduate Texts in Mathematics (113) (Book 113)}, + url={https://books.google.com/books?id=ATNy\_Zg3PSsC}, + year={1991}, + publisher={Springer New York} +} + +@article{grohs2019spacetime, + abstract = {Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form {\$}{$[$}0,T{$]$}{$\backslash$}times {$\backslash$}mathbb {\{}R{\}}\^{}{\{}d{\}}{$\backslash$}ni (t,x){\{}{$\backslash$}kern -.5pt{\}}{$\backslash$}mapsto {\{}{$\backslash$}kern -.5pt{\}} tx{\{}{$\backslash$}kern -.5pt{\}}{$\backslash$}in {\{}{$\backslash$}kern -.5pt{\}} {$\backslash$}mathbb {\{}R{\}}\^{}{\{}d{\}}{\$}where {\$}T{\{}{$\backslash$}kern -.5pt{\}}{$\backslash$}in {\{}{$\backslash$}kern -.5pt{\}} (0,{$\backslash$}infty ){\$}, {\$}d{\{}{$\backslash$}kern -.5pt{\}}{$\backslash$}in {\{}{$\backslash$}kern -.5pt{\}} {$\backslash$}mathbb {\{}N{\}}{\$}, which we both develop within this article.}, + author = {Grohs, Philipp and Hornung, Fabian and Jentzen, Arnulf and Zimmermann, Philipp}, + date = {2023/01/11}, + date-added = {2023-09-08 14:49:03 -0500}, + date-modified = {2023-09-08 14:49:03 -0500}, + doi = {10.1007/s10444-022-09970-2}, + id = {Grohs2023}, + isbn = {1572-9044}, + journal = {Advances in Computational Mathematics}, + number = {1}, + pages = {4}, + title = {Space-time error estimates for deep neural network approximations for differential equations}, + url = {https://doi.org/10.1007/s10444-022-09970-2}, + volume = {49}, + year = {2023}, + bdsk-url-1 = {https://doi.org/10.1007/s10444-022-09970-2}} + +@article{Grohs_2022, + doi = {10.1007/s42985-021-00100-z}, + + url = {https://doi.org/10.1007%2Fs42985-021-00100-z}, + + year = 2022, + month = {jun}, + + publisher = {Springer Science and Business Media {LLC} +}, + + volume = {3}, + + number = {4}, + + author = {Philipp Grohs and Arnulf Jentzen and Diyora Salimova}, + + title = {Deep neural network approximations for solutions of {PDEs} based on Monte Carlo algorithms}, + + journal = {Partial Differential Equations and Applications} +} +@article{Ito1942a, +author={It\^o, K.}, +title={Differential Equations Determining {Markov} Processes (Original in {Japanese})}, +journal={Zenkoku Shijo Sugaku Danwakai}, +year="1942", +volume="244", +number="1077", +pages={1352-1400}, +URL="https://cir.nii.ac.jp/crid/1573105975386021120" +} +@article{Ito1946, +author={It\^o, K.}, +title={On a stochastic integral equation}, +journal={Proc. Imperial Acad. Tokyo}, +year="1942", +volume="244", +number="1077", +pages="1352-1400", +URL="https://cir.nii.ac.jp/crid/1573105975386021120" +} + +@inbook{bass_2011, place={Cambridge}, series={Cambridge Series in Statistical and Probabilistic Mathematics}, title={Brownian Motion}, DOI={10.1017/CBO9780511997044.004}, booktitle={Stochastic Processes}, publisher={Cambridge University Press}, author={Bass, Richard F.}, year={2011}, pages={6–12}, collection={Cambridge Series in Statistical and Probabilistic Mathematics}} + +@article{hutzenthaler_overcoming_2020, +author = {Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas and Anh Nguyen, Tuan and von Wurstemberger, Philippe }, +title = {Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations}, +journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, +volume = {476}, +number = {2244}, +pages = {20190630}, +year = {2020}, +doi = {10.1098/rspa.2019.0630}, + +URL = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2019.0630}, +eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0630} +, + abstract = { For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy. } +} + +@article{Beck_2021, + doi = {10.3934/dcdsb.2020320}, + + url = {https://doi.org/10.3934%2Fdcdsb.2020320}, + + year = {2021}, + publisher = {American Institute of Mathematical Sciences (AIMS)}, + + volume = {26}, + + number = {9}, + + pages = {4927}, + + author = {Christian Beck and Lukas Gonon and Martin Hutzenthaler and Arnulf Jentzen}, + + title = {On existence and uniqueness properties for solutions of stochastic fixed point equations}, + + journal = {Discrete \& Continuous Dynamical Systems - B} +} + +@article{BHJ21, + doi = {10.1142/s0219493721500489}, + + url = {https://doi.org/10.1142%2Fs0219493721500489}, + + year = 2021, + month = {jul}, + + publisher = {World Scientific Pub Co Pte Ltd}, + + volume = {21}, + + number = {08}, + + author = {Christian Beck and Martin Hutzenthaler and Arnulf Jentzen}, + + title = {On nonlinear {Feynman}{\textendash}{Kac} formulas for viscosity solutions of semilinear parabolic partial differential equations}, + + journal = {Stochastics and Dynamics} +} +@article{Gyngy1996ExistenceOS, + title={Existence of strong solutions for {It\^o}'s stochastic equations via approximations}, + author={Istv{\'a}n Gy{\"o}ngy and Nicolai V. Krylov}, + journal={Probability Theory and Related Fields}, + year={1996}, + volume={105}, + pages={143-158} +} + +@book{durrett2019probability, + title={Probability: Theory and Examples}, + author={Durrett, R.}, + isbn={9781108473682}, + lccn={2018047195}, + series={Cambridge Series in Statistical and Probabilistic Mathematics}, + url={https://books.google.com/books?id=b22MDwAAQBAJ}, + year={2019}, + publisher={Cambridge University Press} +} + +@techreport{hutzenthaler_strong_2021, + title = {Strong {$L^p$}-error analysis of nonlinear {Monte} {Carlo} approximations for high-dimensional semilinear partial differential equations}, + url = {http://arxiv.org/abs/2110.08297}, + + number = {arXiv:2110.08297}, + urldate = {2022-10-29}, + institution = {arXiv}, + author = {Hutzenthaler, Martin and Jentzen, Arnulf and Kuckuck, Benno and Padgett, Joshua Lee}, + month = oct, + year = {2021}, + doi = {10.48550/arXiv.2110.08297}, + note = {arXiv:2110.08297 [cs, math] +type: article}, + keywords = {Mathematics - Numerical Analysis, Mathematics - Probability}, + annote = {Comment: 42 pages.}, + file = {arXiv Fulltext PDF:files/6/Hutzenthaler et al. - 2021 - Strong \$L^p\$-error analysis of nonlinear Monte Car.pdf:application/pdf;arXiv.org Snapshot:files/7/2110.html:text/html}, +} + +@TechReport{grohsetal, + author={Philipp Grohs and Fabian Hornung and Arnulf Jentzen and Philippe von Wurstemberger}, + title={{A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations}}, + year=2018, + month=Sep, + institution={arXiv.org}, + type={Papers}, + url={https://ideas.repec.org/p/arx/papers/1809.02362.html}, + number={1809.02362}, + abstract={Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in \mathbb{N}$. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.}, + keywords={}, + doi={}, +} + + +@article{crandall_lions, + title = {User’s guide to viscosity solutions of second order partial differential equations}, + volume = {27}, + issn = {0273-0979, 1088-9485}, + url = {https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/}, + doi = {10.1090/S0273-0979-1992-00266-5}, + abstract = {Advancing research. Creating connections.}, + language = {en}, + number = {1}, + urldate = {2023-03-07}, + journal = {Bull. Amer. Math. Soc.}, + author = {Crandall, Michael G. and Ishii, Hitoshi and Lions, Pierre-Louis}, + year = {1992}, + keywords = {comparison theorems, dynamic programming, elliptic equations, fully nonlinear equations, generalized solutions, Hamilton-Jacobi equations, maximum principles, nonlinear boundary value problems, parabolic equations, partial differential equations, Perron’s method, Viscosity solutions}, + pages = {1--67}, + file = {Full Text PDF:files/129/Crandall et al. - 1992 - User’s guide to viscosity solutions of second orde.pdf:application/pdf}, +} + +@book{da_prato_zabczyk_2002, +place={Cambridge}, series={London Mathematical Society Lecture Note Series}, title={Second Order Partial Differential Equations in Hilbert Spaces}, DOI={10.1017/CBO9780511543210}, publisher={Cambridge University Press}, author={Da Prato, Giuseppe and Zabczyk, Jerzy}, year={2002}, collection={London Mathematical Society Lecture Note Series}} + +@article{rio_moment_2009, + title = {Moment {Inequalities} for {Sums} of {Dependent} {Random} {Variables} under {Projective} {Conditions}}, + volume = {22}, + issn = {1572-9230}, + url = {https://doi.org/10.1007/s10959-008-0155-9}, + doi = {10.1007/s10959-008-0155-9}, + abstract = {We obtain precise constants in the Marcinkiewicz-Zygmund inequality for martingales in \${\textbackslash}mathbb\{L\}{\textasciicircum}\{p\}\$for p{\textgreater}2 and a new Rosenthal type inequality for stationary martingale differences for p in ]2,3]. The Rosenthal inequality is then extended to stationary and adapted sequences. As in Peligrad et al. (Proc. Am. Math. Soc. 135:541–550, [2007]), the bounds are expressed in terms of \${\textbackslash}mathbb\{L\}{\textasciicircum}\{p\}\$-norms of conditional expectations with respect to an increasing field of sigma algebras. Some applications to a particular Markov chain are given.}, + language = {en}, + number = {1}, + urldate = {2023-01-06}, + journal = {J Theor Probab}, + author = {Rio, Emmanuel}, + month = mar, + year = {2009}, + keywords = {60 F 05, 60 F 17, Martingale, Moment inequality, Projective criteria, Rosenthal inequality, Stationary sequences}, + pages = {146--163}, +} +@book{golub2013matrix, + title={Matrix Computations}, + author={Golub, G.H. and Van Loan, C.F.}, + isbn={9781421407944}, + lccn={2012943449}, + series={Johns Hopkins Studies in the Mathematical Sciences}, + url={https://books.google.com/books?id=X5YfsuCWpxMC}, + year={2013}, + publisher={Johns Hopkins University Press} +} + +@article{hjw2020, +author = {Martin Hutzenthaler and Arnulf Jentzen and von Wurstemberger Wurstemberger}, +title = {{Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks}}, +volume = {25}, +journal = {Electronic Journal of Probability}, +number = {none}, +publisher = {Institute of Mathematical Statistics and Bernoulli Society}, +pages = {1 -- 73}, +keywords = {curse of dimensionality, high-dimensional PDEs, multilevel Picard method, semilinear KolmogorovPDEs, Semilinear PDEs}, +year = {2020}, +doi = {10.1214/20-EJP423}, +URL = {https://doi.org/10.1214/20-EJP423} +} + +@article{bhj20, +author = {Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf}, +title = {On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations}, +journal = {Stochastics and Dynamics}, +volume = {21}, +number = {08}, +pages = {2150048}, +year = {2021}, +doi = {10.1142/S0219493721500489}, + +URL = { + + https://doi.org/10.1142/S0219493721500489 + + + +}, +eprint = { + + https://doi.org/10.1142/S0219493721500489 + + + +} +, + abstract = { The classical Feynman–Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this paper, we extend the classical Feynman–Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic fixed point equations (SFPEs), which arise when the classical Feynman–Kac identity is formally applied to semilinear Kolmorogov PDEs, as viscosity solutions of the corresponding PDEs. This justifies, in particular, employing full-history recursive multilevel Picard (MLP) approximation algorithms, which have recently been shown to overcome the curse of dimensionality in the numerical approximation of solutions of SFPEs, in the numerical approximation of semilinear Kolmogorov PDEs. } +} + +@article{tsaban_harnessing_2022, + title = {Harnessing protein folding neural networks for peptide–protein docking}, + volume = {13}, + copyright = {2022 The Author(s)}, + issn = {2041-1723}, + url = {https://www.nature.com/articles/s41467-021-27838-9}, + doi = {10.1038/s41467-021-27838-9}, + abstract = {Highly accurate protein structure predictions by deep neural networks such as AlphaFold2 and RoseTTAFold have tremendous impact on structural biology and beyond. Here, we show that, although these deep learning approaches have originally been developed for the in silico folding of protein monomers, AlphaFold2 also enables quick and accurate modeling of peptide–protein interactions. Our simple implementation of AlphaFold2 generates peptide–protein complex models without requiring multiple sequence alignment information for the peptide partner, and can handle binding-induced conformational changes of the receptor. We explore what AlphaFold2 has memorized and learned, and describe specific examples that highlight differences compared to state-of-the-art peptide docking protocol PIPER-FlexPepDock. These results show that AlphaFold2 holds great promise for providing structural insight into a wide range of peptide–protein complexes, serving as a starting point for the detailed characterization and manipulation of these interactions.}, + language = {en}, + number = {1}, + urldate = {2023-11-15}, + journal = {Nat Commun}, + author = {Tsaban, Tomer and Varga, Julia K. and Avraham, Orly and Ben-Aharon, Ziv and Khramushin, Alisa and Schueler-Furman, Ora}, + month = jan, + year = {2022}, + note = {Number: 1 +Publisher: Nature Publishing Group}, + keywords = {Machine learning, Molecular modelling, Peptides, Protein structure predictions}, + pages = {176}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/EKLDKE65/Tsaban et al. - 2022 - Harnessing protein folding neural networks for pep.pdf:application/pdf}, +} + +@article{davies_signature_2021, + title={The signature and cusp geometry of hyperbolic knots}, + author={Alex Davies and Andr'as Juh'asz and Marc Lackenby and Nenad Tomasev}, + journal={ArXiv}, + year={2021}, + volume={abs/2111.15323}, + url={https://api.semanticscholar.org/CorpusID:244729717} +} + +@article{zhao_space-based_2023, + title = {Space-based gravitational wave signal detection and extraction with deep neural network}, + volume = {6}, + copyright = {2023 Springer Nature Limited}, + issn = {2399-3650}, + url = {https://www.nature.com/articles/s42005-023-01334-6}, + doi = {10.1038/s42005-023-01334-6}, + abstract = {Space-based gravitational wave (GW) detectors will be able to observe signals from sources that are otherwise nearly impossible from current ground-based detection. Consequently, the well established signal detection method, matched filtering, will require a complex template bank, leading to a computational cost that is too expensive in practice. Here, we develop a high-accuracy GW signal detection and extraction method for all space-based GW sources. As a proof of concept, we show that a science-driven and uniform multi-stage self-attention-based deep neural network can identify synthetic signals that are submerged in Gaussian noise. Our method exhibits a detection rate exceeding 99\% in identifying signals from various sources, with the signal-to-noise ratio at 50, at a false alarm rate of 1\%. while obtaining at least 95\% similarity compared with target signals. We further demonstrate the interpretability and strong generalization behavior for several extended scenarios.}, + language = {en}, + number = {1}, + urldate = {2023-11-15}, + journal = {Commun Phys}, + author = {Zhao, Tianyu and Lyu, Ruoxi and Wang, He and Cao, Zhoujian and Ren, Zhixiang}, + month = aug, + year = {2023}, + note = {Number: 1 +Publisher: Nature Publishing Group}, + keywords = {Astronomy and planetary science, Computational science}, + pages = {1--12}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/JCCM78TZ/Zhao et al. - 2023 - Space-based gravitational wave signal detection an.pdf:application/pdf}, +} +@misc{wu2022sustainable, + title={Sustainable AI: Environmental Implications, Challenges and Opportunities}, + author={Carole-Jean Wu and Ramya Raghavendra and Udit Gupta and Bilge Acun and Newsha Ardalani and Kiwan Maeng and Gloria Chang and Fiona Aga Behram and James Huang and Charles Bai and Michael Gschwind and Anurag Gupta and Myle Ott and Anastasia Melnikov and Salvatore Candido and David Brooks and Geeta Chauhan and Benjamin Lee and Hsien-Hsin S. Lee and Bugra Akyildiz and Maximilian Balandat and Joe Spisak and Ravi Jain and Mike Rabbat and Kim Hazelwood}, + year={2022}, + eprint={2111.00364}, + archivePrefix={arXiv}, + primaryClass={cs.LG} +} +@misc{strubell2019energy, + title={Energy and Policy Considerations for Deep Learning in NLP}, + author={Emma Strubell and Ananya Ganesh and Andrew McCallum}, + year={2019}, + eprint={1906.02243}, + archivePrefix={arXiv}, + primaryClass={cs.CL} +} + + +@article{e_multilevel_2021, + title = {Multilevel {Picard} iterations for solving smooth semilinear parabolic heat equations}, + volume = {2}, + issn = {2662-2971}, + url = {https://doi.org/10.1007/s42985-021-00089-5}, + doi = {10.1007/s42985-021-00089-5}, + abstract = {We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by \$\$O(d{\textbackslash},\{{\textbackslash}varepsilon \}{\textasciicircum}\{-(4+{\textbackslash}delta )\})\$\$for any \$\${\textbackslash}delta {\textbackslash}in (0,{\textbackslash}infty )\$\$under suitable assumptions, where \$\$d{\textbackslash}in \{\{{\textbackslash}mathbb \{N\}\}\}\$\$is the dimensionality of the problem and \$\$\{{\textbackslash}varepsilon \}{\textbackslash}in (0,{\textbackslash}infty )\$\$is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.}, + language = {en}, + number = {6}, + urldate = {2023-11-27}, + journal = {Partial Differ. Equ. Appl.}, + author = {E, Weinan and Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas}, + month = nov, + year = {2021}, + keywords = {65M75, Curse of dimensionality, High-dimensional PDEs, High-dimensional semilinear BSDEs, Multilevel Monte Carlo method, Multilevel Picard iteration}, + pages = {80}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/5ADX78DS/E et al. - 2021 - Multilevel Picard iterations for solving smooth se.pdf:application/pdf}, +} + + +@article{e_multilevel_2019, + title = {On {Multilevel} {Picard} {Numerical} {Approximations} for {High}-{Dimensional} {Nonlinear} {Parabolic} {Partial} {Differential} {Equations} and {High}-{Dimensional} {Nonlinear} {Backward} {Stochastic} {Differential} {Equations}}, + volume = {79}, + issn = {1573-7691}, + url = {https://doi.org/10.1007/s10915-018-00903-0}, + doi = {10.1007/s10915-018-00903-0}, + abstract = {Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in pricing and hedging models for financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article (E et al., Multilevel Picard iterations for solving smooth semilinear parabolic heat equations, arXiv:1607.03295) we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution of a semilinear heat equation that the computational complexity is bounded by \$\$O( d {\textbackslash}, \{{\textbackslash}varepsilon \}{\textasciicircum}\{-(4+{\textbackslash}delta )\})\$\$for any \$\${\textbackslash}delta {\textbackslash}in (0,{\textbackslash}infty )\$\$where d is the dimensionality of the problem and \$\$\{{\textbackslash}varepsilon \}{\textbackslash}in (0,{\textbackslash}infty )\$\$is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of 100-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for many of these 100-dimensional example PDEs are very satisfactory in terms of both accuracy and speed. Moreover, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the scientific literature.}, + language = {en}, + number = {3}, + urldate = {2023-11-27}, + journal = {J Sci Comput}, + author = {E, Weinan and Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas}, + month = jun, + year = {2019}, + keywords = {65M75, Curse of dimensionality, High-dimensional nonlinear BSDEs, High-dimensional PDEs, Multilevel Monte Carlo method, Multilevel Picard approximations}, + pages = {1534--1571}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/7KHG4238/E et al. - 2019 - On Multilevel Picard Numerical Approximations for .pdf:application/pdf}, +} + + +@inproceedings{vaswani_attention_2017, + title = {Attention is {All} you {Need}}, + volume = {30}, + url = {https://proceedings.neurips.cc/paper_files/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html}, + abstract = {The dominant sequence transduction models are based on complex recurrent orconvolutional neural networks in an encoder and decoder configuration. The best performing such models also connect the encoder and decoder through an attentionm echanisms. We propose a novel, simple network architecture based solely onan attention mechanism, dispensing with recurrence and convolutions entirely.Experiments on two machine translation tasks show these models to be superiorin quality while being more parallelizable and requiring significantly less timeto train. Our single model with 165 million parameters, achieves 27.5 BLEU onEnglish-to-German translation, improving over the existing best ensemble result by over 1 BLEU. On English-to-French translation, we outperform the previoussingle state-of-the-art with model by 0.7 BLEU, achieving a BLEU score of 41.1.}, + urldate = {2023-12-01}, + booktitle = {Advances in {Neural} {Information} {Processing} {Systems}}, + publisher = {Curran Associates, Inc.}, + author = {Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N and Kaiser, Łukasz and Polosukhin, Illia}, + year = {2017}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/T7R9QP6K/Vaswani et al. - 2017 - Attention is All you Need.pdf:application/pdf}, +} + + +@article{arik_tabnet_2021, + title = {{TabNet}: {Attentive} {Interpretable} {Tabular} {Learning}}, + volume = {35}, + copyright = {Copyright (c) 2021 Association for the Advancement of Artificial Intelligence}, + issn = {2374-3468}, + shorttitle = {{TabNet}}, + url = {https://ojs.aaai.org/index.php/AAAI/article/view/16826}, + doi = {10.1609/aaai.v35i8.16826}, + abstract = {We propose a novel high-performance and interpretable canonical deep tabular data learning architecture, TabNet. TabNet uses sequential attention to choose which features to reason from at each decision step, enabling interpretability and more efficient learning as the learning capacity is used for the most salient features. We demonstrate that TabNet outperforms other variants on a wide range of non-performance-saturated tabular datasets and yields interpretable feature attributions plus insights into its global behavior. Finally, we demonstrate self-supervised learning for tabular data, significantly improving performance when unlabeled data is abundant.}, + language = {en}, + number = {8}, + urldate = {2023-12-01}, + journal = {Proceedings of the AAAI Conference on Artificial Intelligence}, + author = {Arik, Sercan Ö and Pfister, Tomas}, + month = may, + year = {2021}, + note = {Number: 8}, + keywords = {Unsupervised \& Self-Supervised Learning}, + pages = {6679--6687}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/SHV66I4Y/Arik and Pfister - 2021 - TabNet Attentive Interpretable Tabular Learning.pdf:application/pdf}, +} + +@INPROCEEDINGS {8099678, +author = {F. Chollet}, +booktitle = {2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)}, +title = {Xception: Deep Learning with Depthwise Separable Convolutions}, +year = {2017}, +volume = {}, +issn = {1063-6919}, +pages = {1800-1807}, +abstract = {We present an interpretation of Inception modules in convolutional neural networks as being an intermediate step in-between regular convolution and the depthwise separable convolution operation (a depthwise convolution followed by a pointwise convolution). In this light, a depthwise separable convolution can be understood as an Inception module with a maximally large number of towers. This observation leads us to propose a novel deep convolutional neural network architecture inspired by Inception, where Inception modules have been replaced with depthwise separable convolutions. We show that this architecture, dubbed Xception, slightly outperforms Inception V3 on the ImageNet dataset (which Inception V3 was designed for), and significantly outperforms Inception V3 on a larger image classification dataset comprising 350 million images and 17,000 classes. Since the Xception architecture has the same number of parameters as Inception V3, the performance gains are not due to increased capacity but rather to a more efficient use of model parameters.}, +keywords = {computer architecture;correlation;convolutional codes;google;biological neural networks}, +doi = {10.1109/CVPR.2017.195}, +url = {https://doi.ieeecomputersociety.org/10.1109/CVPR.2017.195}, +publisher = {IEEE Computer Society}, +address = {Los Alamitos, CA, USA}, +month = {jul} +} + +@article{srivastava_dropout_2014, + title = {Dropout: a simple way to prevent neural networks from overfitting}, + volume = {15}, + issn = {1532-4435}, + shorttitle = {Dropout}, + abstract = {Deep neural nets with a large number of parameters are very powerful machine learning systems. However, overfitting is a serious problem in such networks. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. Dropout is a technique for addressing this problem. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much. During training, dropout samples from an exponential number of different "thinned" networks. At test time, it is easy to approximate the effect of averaging the predictions of all these thinned networks by simply using a single unthinned network that has smaller weights. This significantly reduces overfitting and gives major improvements over other regularization methods. We show that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets.}, + number = {1}, + journal = {J. Mach. Learn. Res.}, + author = {Srivastava, Nitish and Hinton, Geoffrey and Krizhevsky, Alex and Sutskever, Ilya and Salakhutdinov, Ruslan}, + month = jan, + year = {2014}, + keywords = {deep learning, model combination, neural networks, regularization}, + pages = {1929--1958}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/JK87IU3H/Srivastava et al. - 2014 - Dropout a simple way to prevent neural networks f.pdf:application/pdf}, +} + + +@article{petersen_optimal_2018, + title = {Optimal approximation of piecewise smooth functions using deep {ReLU} neural networks}, + volume = {108}, + issn = {1879-2782}, + doi = {10.1016/j.neunet.2018.08.019}, + abstract = {We study the necessary and sufficient complexity of ReLU neural networks - in terms of depth and number of weights - which is required for approximating classifier functions in an Lp-sense. As a model class, we consider the set Eβ(Rd) of possibly discontinuous piecewise Cβ functions f:[-12,12]d→R, where the different "smooth regions" of f are separated by Cβ hypersurfaces. For given dimension d≥2, regularity β{\textgreater}0, and accuracy ε{\textgreater}0, we construct artificial neural networks with ReLU activation function that approximate functions from Eβ(Rd) up to an L2 error of ε. The constructed networks have a fixed number of layers, depending only on d and β, and they have O(ε-2(d-1)∕β) many nonzero weights, which we prove to be optimal. For the proof of optimality, we establish a lower bound on the description complexity of the class Eβ(Rd). By showing that a family of approximating neural networks gives rise to an encoder for Eβ(Rd), we then prove that one cannot approximate a general function f∈Eβ(Rd) using neural networks that are less complex than those produced by our construction. In addition to the optimality in terms of the number of weights, we show that in order to achieve this optimal approximation rate, one needs ReLU networks of a certain minimal depth. Precisely, for piecewise Cβ(Rd) functions, this minimal depth is given - up to a multiplicative constant - by β∕d. Up to a log factor, our constructed networks match this bound. This partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions. Finally, we analyze approximation in high-dimensional spaces where the function f to be approximated can be factorized into a smooth dimension reducing feature map τ and classifier function g - defined on a low-dimensional feature space - as f=g∘τ. We show that in this case the approximation rate depends only on the dimension of the feature space and not the input dimension.}, + language = {eng}, + journal = {Neural Netw}, + author = {Petersen, Philipp and Voigtlaender, Felix}, + month = dec, + year = {2018}, + pmid = {30245431}, + keywords = {Curse of dimension, Deep neural networks, Function approximation, Metric entropy, Neural Networks, Computer, Piecewise smooth functions, Sparse connectivity}, + pages = {296--330}, + file = {Submitted Version:/Users/shakilrafi/Zotero/storage/UL4GLF59/Petersen and Voigtlaender - 2018 - Optimal approximation of piecewise smooth function.pdf:application/pdf}, +} + + + @misc{bigbook, + title={Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory}, + author={Arnulf Jentzen and Benno Kuckuck and Philippe von Wurstemberger}, + year={2023}, + eprint={2310.20360}, + archivePrefix={arXiv}, + primaryClass={cs.LG} +} + +@article{mcculloch_logical_1943, + title = {A logical calculus of the ideas immanent in nervous activity}, + volume = {5}, + issn = {1522-9602}, + url = {https://doi.org/10.1007/BF02478259}, + doi = {10.1007/BF02478259}, + abstract = {Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes. It is shown that many particular choices among possible neurophysiological assumptions are equivalent, in the sense that for every net behaving under one assumption, there exists another net which behaves under the other and gives the same results, although perhaps not in the same time. Various applications of the calculus are discussed.}, + number = {4}, + journal = {The bulletin of mathematical biophysics}, + author = {McCulloch, Warren S. and Pitts, Walter}, + month = dec, + year = {1943}, + pages = {115--133}, +} + +@article{Hornik1991ApproximationCO, + title={Approximation capabilities of multilayer feedforward networks}, + author={Kurt Hornik}, + journal={Neural Networks}, + year={1991}, + volume={4}, + pages={251-257}, + url={https://api.semanticscholar.org/CorpusID:7343126} +} + + +@article{cybenko_approximation_1989, + title = {Approximation by superpositions of a sigmoidal function}, + volume = {2}, + issn = {1435-568X}, + url = {https://doi.org/10.1007/BF02551274}, + doi = {10.1007/BF02551274}, + abstract = {In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real variables with support in the unit hypercube; only mild conditions are imposed on the univariate function. Our results settle an open question about representability in the class of single hidden layer neural networks. In particular, we show that arbitrary decision regions can be arbitrarily well approximated by continuous feedforward neural networks with only a single internal, hidden layer and any continuous sigmoidal nonlinearity. The paper discusses approximation properties of other possible types of nonlinearities that might be implemented by artificial neural networks.}, + number = {4}, + journal = {Mathematics of Control, Signals and Systems}, + author = {Cybenko, G.}, + month = dec, + year = {1989}, + pages = {303--314}, +} + +@article{KNOKE2021100035, +title = {Solving differential equations via artificial neural networks: Findings and failures in a model problem}, +journal = {Examples and Counterexamples}, +volume = {1}, +pages = {100035}, +year = {2021}, +issn = {2666-657X}, +doi = {https://doi.org/10.1016/j.exco.2021.100035}, +url = {https://www.sciencedirect.com/science/article/pii/S2666657X21000197}, +author = {Tobias Knoke and Thomas Wick}, +keywords = {Ordinary differential equation, Logistic equation, Feedforward neural network, numerical optimization, PyTorch}, +abstract = {In this work, we discuss some pitfalls when solving differential equations with neural networks. Due to the highly nonlinear cost functional, local minima might be approximated by which functions may be obtained, that do not solve the problem. The main reason for these failures is a sensitivity on initial guesses for the nonlinear iteration. We apply known algorithms and corresponding implementations, including code snippets, and present an example and counter example for the logistic differential equations. These findings are further substantiated with variations in collocation points and learning rates.} +} + +@article{Lagaris_1998, + title={Artificial neural networks for solving ordinary and partial differential equations}, + volume={9}, + ISSN={1045-9227}, + url={http://dx.doi.org/10.1109/72.712178}, + DOI={10.1109/72.712178}, + number={5}, + journal={IEEE Transactions on Neural Networks}, + publisher={Institute of Electrical and Electronics Engineers (IEEE)}, + author={Lagaris, I.E. and Likas, A. and Fotiadis, D.I.}, + year={1998}, + pages={987–1000} } + +@ARTICLE{gunnar_carlsson, + author = {{Carlsson}, Gunnar and {Br{\"u}el Gabrielsson}, Rickard}, + title = "{Topological Approaches to Deep Learning}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Machine Learning, Computer Science - Artificial Intelligence, Mathematics - Algebraic Topology, Statistics - Machine Learning, 68T05, 55N35, 62-07}, + year = 2018, + month = nov, + eid = {arXiv:1811.01122}, + pages = {arXiv:1811.01122}, + doi = {10.48550/arXiv.1811.01122}, +archivePrefix = {arXiv}, + eprint = {1811.01122}, + primaryClass = {cs.LG}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2018arXiv181101122C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@misc{shiebler2021category, + title={Category Theory in Machine Learning}, + author={Dan Shiebler and Bruno Gavranović and Paul Wilson}, + year={2021}, + eprint={2106.07032}, + archivePrefix={arXiv}, + primaryClass={cs.LG} +} + +@inproceedings{vaswani2, + author = {Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N and Kaiser, \L ukasz and Polosukhin, Illia}, + booktitle = {Advances in Neural Information Processing Systems}, + editor = {I. Guyon and U. Von Luxburg and S. Bengio and H. Wallach and R. Fergus and S. Vishwanathan and R. Garnett}, + pages = {}, + publisher = {Curran Associates, Inc.}, + title = {Attention is All you Need}, + url = {https://proceedings.neurips.cc/paper_files/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf}, + volume = {30}, + year = {2017} +} + +@article{arik2, + title = {{TabNet}: {Attentive} {Interpretable} {Tabular} {Learning}}, + volume = {35}, + copyright = {Copyright (c) 2021 Association for the Advancement of Artificial Intelligence}, + issn = {2374-3468}, + shorttitle = {{TabNet}}, + url = {https://ojs.aaai.org/index.php/AAAI/article/view/16826}, + doi = {10.1609/aaai.v35i8.16826}, + abstract = {We propose a novel high-performance and interpretable canonical deep tabular data learning architecture, TabNet. TabNet uses sequential attention to choose which features to reason from at each decision step, enabling interpretability and more efficient learning as the learning capacity is used for the most salient features. We demonstrate that TabNet outperforms other variants on a wide range of non-performance-saturated tabular datasets and yields interpretable feature attributions plus insights into its global behavior. Finally, we demonstrate self-supervised learning for tabular data, significantly improving performance when unlabeled data is abundant.}, + language = {en}, + number = {8}, + urldate = {2024-02-01}, + journal = {Proceedings of the AAAI Conference on Artificial Intelligence}, + author = {Arik, Sercan \"O and Pfister, Tomas}, + month = may, + year = {2021}, + note = {Number: 8}, + keywords = {Unsupervised \& Self-Supervised Learning}, + pages = {6679--6687}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/7MTMXR4G/Arik and Pfister - 2021 - TabNet Attentive Interpretable Tabular Learning.pdf:application/pdf}, +} + +@Manual{dplyr, + title = {dplyr: A Grammar of Data Manipulation}, + author = {Hadley Wickham and Romain François and Lionel Henry and Kirill Müller and Davis Vaughan}, + year = {2023}, + note = {R package version 1.1.4, https://github.com/tidyverse/dplyr}, + url = {https://dplyr.tidyverse.org}, +} + +@Book{ggplot2, + author = {Hadley Wickham}, + title = {ggplot2: Elegant Graphics for Data Analysis}, + publisher = {Springer-Verlag New York}, + year = {2016}, + isbn = {978-3-319-24277-4}, + url = {https://ggplot2.tidyverse.org}, + } + + @online{plotly, + author = {{Plotly Technologies Inc}}, + title = {Collaborative data science}, + publisher = {Plotly Technologies Inc.}, + address = {Montreal, QC}, + year = {2015}, + url = {https://plot.ly} +} + +@misc{rafi_towards_2024, + title = {Towards an {Algebraic} {Framework} {For} {Approximating} {Functions} {Using} {Neural} {Network} {Polynomials}}, + url = {https://arxiv.org/abs/2402.01058v1}, + abstract = {We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on {\textbackslash}cite\{bigbook\}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network polynomials, neural network exponentials, sine, and cosines in the sense that they do indeed approximate their real number counterparts subject to limitations on certain of their parameters, \$q\$, and \${\textbackslash}varepsilon\$. While doing this, we show that the parameter and depth growth are only polynomial on their desired accuracy (defined as a 1-norm difference over \${\textbackslash}mathbb\{R\}\$), thereby showing that this approach to approximating, where a neural network in some sense has the structural properties of the function it is approximating is not entire intractable.}, + language = {en}, + urldate = {2024-02-11}, + journal = {arXiv.org}, + author = {Rafi, Shakil and Padgett, Joshua Lee and Nakarmi, Ukash}, + month = feb, + year = {2024}, + file = {Full Text PDF:/Users/shakilrafi/Zotero/storage/A8LPKNZK/Rafi et al. - 2024 - Towards an Algebraic Framework For Approximating F.pdf:application/pdf}, +} + +@Manual{nnR-package, title = {nnR: Neural Networks Made Algebraic}, author = {Shakil Rafi and Joshua Lee Padgett}, year = {2024}, note = {R package version 0.1.0}, url = {https://github.com/2shakilrafi/nnR/}, } + + @software{Rafi_nnR_2024, +author = {Rafi, Shakil}, +license = {GPL-3.0}, +month = feb, +title = {{nnR}}, +url = {https://github.com/2shakilrafi/nnR}, +version = {0.10}, +year = {2024} +} + + + + + diff --git a/Dissertation/main.blg b/Dissertation/main.blg new file mode 100644 index 0000000..183e110 --- /dev/null +++ b/Dissertation/main.blg @@ -0,0 +1,55 @@ +This is BibTeX, Version 0.99d (TeX Live 2023) +Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 +The top-level auxiliary file: main.aux +A level-1 auxiliary file: Introduction.aux +A level-1 auxiliary file: Brownian_motion_monte_carlo.aux +A level-1 auxiliary file: u_visc_sol.aux +A level-1 auxiliary file: brownian_motion_monte_carlo_non_linear_case.aux +A level-1 auxiliary file: neural_network_introduction.aux +A level-1 auxiliary file: ann_product.aux +A level-1 auxiliary file: modified_mlp_associated_nn.aux +A level-1 auxiliary file: categorical_neural_network.aux +A level-1 auxiliary file: ann_rep_brownian_motion_monte_carlo.aux +The style file: apalike.bst +Database file #1: Dissertation.bib 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+] [6] [7] +Overfull \hbox (12.0844pt too wide) in paragraph at lines 104--105 +[]\OT1/cmr/m/it/10.95 A strong so-lu-tion to the stochas-tic dif-fer-en-tial eq +ua-tion ([]1.1.8[]) on prob-a-bil-ity space $\OT1/cmr/m/n/10.95 ( +\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 F\OML/cmm/m/it/10.95 ; \U/msb/m/n/10. +95 P\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 (\U/msb/m/n/10.95 F[]\OT1/cmr/m/n/ +10.95 )[])$\OT1/cmr/m/it/10.95 , + [] + +[8] +LaTeX Font Info: Trying to load font information for U+euf on input line 140 +. + (/usr/local/texlive/2023/texmf-dist/tex/latex/amsfonts/ueuf.fd +File: ueuf.fd 2013/01/14 v3.01 Euler Fraktur +) [9] +[10] [11] +LaTeX Font Info: Trying to load font information for U+BOONDOX-ds on input l +ine 218. + (/usr/local/texlive/2023/texmf-dist/tex/latex/boondox/uboondox-ds.fd +File: uboondox-ds.fd 2011/04/26 Fontinst v1.933 font definitions for U/BOONDOX- +ds. +) [12] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 267. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 267. + +[13] [14]) [15] +\openout2 = `Brownian_motion_monte_carlo.aux'. + + (./Brownian_motion_monte_carlo.tex +Chapter 2. +[16 + + + + +] [17] [18] + +LaTeX Warning: Reference `(1.16)' on page 19 undefined on input line 132. + +[19] [20] [21] [22] [23] [24] [25] [26] [27] [28]) [29] +\openout2 = `u_visc_sol.aux'. + + (./u_visc_sol.tex +Chapter 3. + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 1. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 1. + +[30 + + + + +] +LaTeX Font Info: Trying to load font information for U+bbm on input line 83. + + (/usr/local/texlive/2023/texmf-dist/tex/latex/bbm-macros/ubbm.fd +File: ubbm.fd 1999/03/15 V 1.2 Font definition for bbm font - TH +) [31] +[32] [33] [34] [35] + +LaTeX Warning: Reference `bigsum' on page 36 undefined on input line 209. + +[36] [37] [38] [39] +Overfull \hbox (0.39136pt too wide) in paragraph at lines 465--468 +[]\OT1/cmr/m/n/10.95 However note also that since $\OML/cmm/m/it/10.95 G[]$ \OT +1/cmr/m/n/10.95 is up-per semi-continuous, fur-ther the fact that, $\OML/cmm/m/ +it/10.95 ^^^[] \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/n/10.95 , + [] + + +Overfull \hbox (81.7409pt too wide) in paragraph at lines 465--468 +\OT1/cmr/m/n/10.95 and then $([]3\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 2\OML/ +cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 27[])$, and $([]3\OML/cmm/m/it/10.95 :\OT1/c +mr/m/n/10.95 2\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 16[])$, im-ply for all $\ +OML/cmm/m/it/10.95 " \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/1 +0.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ we have that: $[][] [] = + [] + +[40] [41] [42] [43] [44] [45] [46] [47] [48] +Overfull \hbox (86.18391pt too wide) in paragraph at lines 743--744 +\OT1/cmr/m/it/10.95 ery $\OML/cmm/m/it/10.95 r \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m +/n/10.95 (0\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ \OT +1/cmr/m/it/10.95 sat-isfy the con-di-tion that $\OML/cmm/m/it/10.95 O[] \OMS/cm +sy/m/n/10.95 ^^R O$\OT1/cmr/m/it/10.95 , where $\OML/cmm/m/it/10.95 O[] \OT1/cm +r/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2 +O \OT1/cmr/m/n/10.95 : []\OMS/cmsy/m/n/10.95 g$ + [] + +[49] [50] [51] [52] [53] +Overfull \hbox (2.98135pt too wide) in paragraph at lines 894--895 +[]\OT1/cmr/bx/n/10.95 Corollary 3.3.1.1. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/ +it/10.95 T \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OM +S/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$\OT1/cmr/m/it/10.95 , let $[]$ be a prob +-a-bil-ity space, let $\OML/cmm/m/it/10.95 u[] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m +/it/10.95 C[] []$\OT1/cmr/m/it/10.95 , + [] + +[54] [55] [56] [57]) [58] +\openout2 = `brownian_motion_monte_carlo_non_linear_case.aux'. + + (./brownian_motion_monte_carlo_non_linear_case.tex +Chapter 4. + +LaTeX Warning: Reference `3.3.21' on page 59 undefined on input line 3. + +[59 + + + + +]) [60] [61 + + + +] +\openout2 = `neural_network_introduction.aux'. + + (./neural_network_introduction.tex +Chapter 5. +[62 + + +] [63] [64] +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! +Missing character: There is no ; in font nullfont! 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Fur-ther-more let $[] \OMS/cmsy/m/n/10.95 2 []$ + [] + + +Overfull \hbox (3.52759pt too wide) in paragraph at lines 327--329 +\OT1/cmr/m/n/10.95 See Re-mark 5.3.2. Let $\OML/cmm/m/it/10.95 x[]; x[]; :::; x +[] \OMS/cmsy/m/n/10.95 2 []$ \OT1/cmr/m/n/10.95 sat-isfy for all $\OML/cmm/m/it +/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/ +cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$ + [] + +[71] [72] [73] +Overfull \hbox (20.19258pt too wide) detected at line 410 +[] + [] + +[74] [75] [76] [77] + +LaTeX Warning: Reference `5.4.4' on page 78 undefined on input line 546. + + +LaTeX Warning: Reference `5.4.5' on page 78 undefined on input line 546. + + +LaTeX Warning: Reference `5.4.4' on page 78 undefined on input line 559. + + +LaTeX Warning: Reference `5.4.5' on page 78 undefined on input line 610. + + +Overfull \hbox (24.45033pt too wide) detected at line 657 +[] + [] + +[78] [79] [80] [81] [82] [83] + +LaTeX Warning: Command \l invalid in math mode on input line 914. + +[84] [85] [86] [87] [88] + +LaTeX Warning: Reference `5.5.14' on page 89 undefined on input line 1095. + +[89] [90] [91] [92] [93]) [94] +\openout2 = `ann_product.aux'. + + (./ann_product.tex +Chapter 6. +[95 + + + + +] [96] [97] [98] [99] +Overfull \hbox (13.65341pt too wide) detected at line 188 +[] + [] + +[100] [101] +Overfull \hbox (7.17166pt too wide) in paragraph at lines 226--228 +\OT1/cmr/m/n/10.95 This and the as-sump-tion that $^^H \OMS/cmsy/m/n/10.95 2 \O +ML/cmm/m/it/10.95 C []$ \OT1/cmr/m/n/10.95 along with the as-sump-tion that $[] +[] \OMS/cmsy/m/n/10.95 j [] [] ^^@ + [] + + +Overfull \hbox (17.15163pt too wide) in paragraph at lines 237--239 +\OT1/cmr/m/n/10.95 The as-sump-tion that for all $[][] \OMS/cmsy/m/n/10.95 j [] + [] ^^@ [] [] j \OT1/cmr/m/n/10.95 = 0$ and the as-sump-tion that $[][] \OMS/cm +sy/m/n/10.95 j\OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@ + [] + +[102] [103] [104] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 317. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 317. + +[105] [106] [107] [108] [109] [110] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 500. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 500. + +[111] [112] [113] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 636. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 636. + +[114] +Underfull \hbox (badness 10000) in paragraph at lines 700--706 + + [] + +[115] +Underfull \hbox (badness 10000) in paragraph at lines 709--714 + + [] + +[116] [117] [118] [119] [120] [121] [122] +Overfull \hbox (7.31064pt too wide) in paragraph at lines 911--913 +\OT1/cmr/m/n/10.95 Note that since $\U/euf/m/n/10.95 p[] \OMS/cmsy/m/n/10.95 2 +O []$ \OT1/cmr/m/n/10.95 for $\OML/cmm/m/it/10.95 n \U/msa/m/n/10.95 > \OT1/cmr +/m/n/10.95 2$, it is the case for all $\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.9 +5 2 \U/msb/m/n/10.95 R$ \OT1/cmr/m/n/10.95 then that $[] \OMS/cmsy/m/n/10.95 2 + [] + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 915. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 915. + +[123] [124] +Overfull \hbox (34.92166pt too wide) in paragraph at lines 961--966 +[]$[] [] \U/msa/m/n/10.95 6 []$ + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 961--966 + + [] + + +Underfull \hbox (badness 10000) in paragraph at lines 969--971 + + [] + +[125] +Overfull \hbox (7.30336pt too wide) detected at line 1009 +[] + [] + +[126] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 1042. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `superscript' on input line 1042. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 1042. + +[127]) [128] +\openout2 = `modified_mlp_associated_nn.aux'. + + (./modified_mlp_associated_nn.tex +Chapter 7. +[129 + + + + +] [130] +Overfull \hbox (29.14606pt too wide) in paragraph at lines 54--55 +[]\OT1/cmr/m/it/10.95 that for all $\OML/cmm/m/it/10.95 ^^R \OMS/cmsy/m/n/10.95 + 2 \OT1/cmr/m/n/10.95 ^^B$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 n \OMS/cms +y/m/n/10.95 2 \U/msb/m/n/10.95 N[]$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 t + \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/it/10.95 , that $[][] \U/msa/m/n/10.95 6 [ +][] [] []$ + [] + +) [131] +\openout2 = `categorical_neural_network.aux'. + + (./categorical_neural_network.tex +Chapter 8. +) [132 + + + + +] +Chapter 9. + +Overfull \hbox (5.90657pt too wide) in paragraph at lines 169--170 +[]\OT1/cmr/bx/n/10.95 Definition 9.1.1 \OT1/cmr/m/n/10.95 (Ac-ti-va-tion ANN)\O +T1/cmr/bx/n/10.95 . []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 n \OMS/cmsy/ +m/n/10.95 2 \U/msb/m/n/10.95 N$\OT1/cmr/m/it/10.95 . We de-note by $\U/euf/m/n/ +10.95 i[] \OMS/cmsy/m/n/10.95 2 [] ^^R + [] + +[133 + + + +] [134] [135] +Overfull \hbox (4.2524pt too wide) in paragraph at lines 263--264 +[]\OT1/cmr/bx/n/10.95 Lemma 9.2.3. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/it/10. +95 ^^W \OMS/cmsy/m/n/10.95 2 []$ \OT1/cmr/m/it/10.95 with end-widths $\OML/cmm/ +m/it/10.95 d$\OT1/cmr/m/it/10.95 . It is then the case that $[][] [] [] \OT1/cm +r/m/n/10.95 = [][] [] = + [] + +[136] +Overfull \hbox (2.78886pt too wide) detected at line 337 +[] + [] + +[137] +Overfull \hbox (3.18243pt too wide) detected at line 411 +[] + [] + +[138] [139] +Overfull \hbox (7.72708pt too wide) detected at line 517 +[] + [] + +[140] [141] [142] [143] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 628. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 628. + +[144] +Overfull \hbox (32.59174pt too wide) in paragraph at lines 670--672 +[]\OT1/cmr/m/n/10.95 9.4.4[] then tells us that for all $\OML/cmm/m/it/10.95 i +\OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 0\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/1 +0.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; N\OM +S/cmsy/m/n/10.95 g$\OT1/cmr/m/n/10.95 , $[] [] = []$, $[][] [] \OMS/cmsy/m/n/10 +.95 2 + [] + +[145] [146] [147] [148] +Overfull \hbox (15.04857pt too wide) detected at line 761 +[] + [] + +[149] +Overfull \hbox (14.36722pt too wide) in paragraph at lines 797--798 +[]\OT1/cmr/bx/n/10.95 Corollary 9.5.2.1. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/ +it/10.95 " \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OM +S/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10. +95 L; a \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 R$\OT1/cmr/m/it/10.95 , $\OML/cm +m/m/it/10.95 b \OMS/cmsy/m/n/10.95 2 []$\OT1/cmr/m/it/10.95 , $\OML/cmm/m/it/10 +.95 N \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N[] \OMS/cmsy/m/n/10.95 \ []$\OT1/ +cmr/m/it/10.95 , $\OML/cmm/m/it/10.95 x[]; x[]; :::; x[] \OMS/cmsy/m/n/10.95 2 + [] + +[150] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `superscript' on input line 817. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 817. + +[151] [152] [153] + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `math shift' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathop' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathchardef' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\std@minus' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathcode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\leavevmode@ifvmode' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) \kern 2.43307pt +(hyperref) removed on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\mathgroup' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\symoperators' on input line 908. + + +Package hyperref Warning: Token not allowed in a PDF string (Unicode): +(hyperref) removing `\@ifnextchar' on input line 908. + + 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0000000..aa1bfb5 --- /dev/null +++ b/Dissertation/main.tex @@ -0,0 +1,44 @@ +\include{preamble} +\include{commands} + +\title{Artificial Neural Networks Applied to Stochastic Monte Carlo as a Way to Approximate Modified Heat Equations, and Their Associated Parameters.} +\author{Shakil Rafi} +\begin{document} +\maketitle + +\tableofcontents + +\part{On Convergence of Brownian Motion Monte Carlo} + +\include{Introduction} + +\include{Brownian_motion_monte_carlo} + +\include{u_visc_sol} + +\include{brownian_motion_monte_carlo_non_linear_case} + +\part{A Structural Description of Artificial Neural Networks} + +\include{neural_network_introduction} + +\include{ann_product} + +\include{modified_mlp_associated_nn} + +\include{ann_first_approximations} + +\part{A deep-learning solution for $u$ and Brownian motions} + +\include{ann_rep_brownian_motion_monte_carlo} + +\include{conclusions-further-research} + +\nocite{*} +\singlespacing +\bibliography{main.bib} +\bibliographystyle{apa} + +\include{appendices} + +\end{document} \ No newline at end of file diff --git a/Dissertation/main.toc b/Dissertation/main.toc new file mode 100644 index 0000000..0ec7bf7 --- /dev/null +++ b/Dissertation/main.toc @@ -0,0 +1,58 @@ +\contentsline {part}{I\hspace {1em}On Convergence of Brownian Motion Monte Carlo}{4}{part.1}% +\contentsline {chapter}{\numberline {1}Introduction.}{5}{chapter.1}% +\contentsline {section}{\numberline {1.1}Notation, Definitions \& Basic notions.}{5}{section.1.1}% +\contentsline {subsection}{\numberline {1.1.1}Norms and Inner Product}{5}{subsection.1.1.1}% +\contentsline {subsection}{\numberline {1.1.2}Probability Space and Brownian Motion}{6}{subsection.1.1.2}% +\contentsline {subsection}{\numberline {1.1.3}Lipschitz and Related Notions}{9}{subsection.1.1.3}% +\contentsline {subsection}{\numberline {1.1.4}Kolmogorov Equations}{10}{subsection.1.1.4}% +\contentsline {subsection}{\numberline {1.1.5}Linear Algebra Notation and Definitions}{12}{subsection.1.1.5}% +\contentsline {subsection}{\numberline {1.1.6}$O$-type notation and function growth}{13}{subsection.1.1.6}% +\contentsline {subsection}{\numberline {1.1.7}The Iverson Bracket}{15}{subsection.1.1.7}% +\contentsline {chapter}{\numberline {2}Brownian Motion Monte Carlo}{16}{chapter.2}% +\contentsline {section}{\numberline {2.1}Brownian Motion Preliminaries}{16}{section.2.1}% +\contentsline {section}{\numberline {2.2}Monte Carlo Approximations}{20}{section.2.2}% +\contentsline {section}{\numberline {2.3}Bounds and Covnvergence}{21}{section.2.3}% +\contentsline {chapter}{\numberline {3}That $u$ is a viscosity solution}{30}{chapter.3}% +\contentsline {section}{\numberline {3.1}Some Preliminaries}{30}{section.3.1}% +\contentsline {section}{\numberline {3.2}Viscosity Solutions}{34}{section.3.2}% +\contentsline {section}{\numberline {3.3}Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{53}{section.3.3}% +\contentsline {chapter}{\numberline {4}Brownian motion Monte Carlo of the non-linear case}{59}{chapter.4}% +\contentsline {part}{II\hspace {1em}A Structural Description of Artificial Neural Networks}{61}{part.2}% +\contentsline {chapter}{\numberline {5}Introduction and Basic Notions about Neural Networks}{62}{chapter.5}% +\contentsline {section}{\numberline {5.1}The Basic Definition of ANNs}{62}{section.5.1}% +\contentsline {section}{\numberline {5.2}Composition and extensions of ANNs}{66}{section.5.2}% +\contentsline {subsection}{\numberline {5.2.1}Composition}{66}{subsection.5.2.1}% +\contentsline {subsection}{\numberline {5.2.2}Extensions}{68}{subsection.5.2.2}% +\contentsline {section}{\numberline {5.3}Parallelization of ANNs}{68}{section.5.3}% +\contentsline {section}{\numberline {5.4}Affine Linear Transformations as ANNs}{72}{section.5.4}% +\contentsline {section}{\numberline {5.5}Sums of ANNs}{75}{section.5.5}% +\contentsline {subsection}{\numberline {5.5.1}Neural Network Sum Properties}{76}{subsection.5.5.1}% +\contentsline {section}{\numberline {5.6}Linear Combinations of ANNs}{83}{section.5.6}% +\contentsline {section}{\numberline {5.7}Neural Network Diagrams}{93}{section.5.7}% +\contentsline {chapter}{\numberline {6}ANN Product Approximations}{95}{chapter.6}% +\contentsline {section}{\numberline {6.1}Approximation for simple products}{95}{section.6.1}% +\contentsline {subsection}{\numberline {6.1.1}The $\prd $ network}{106}{subsection.6.1.1}% +\contentsline {section}{\numberline {6.2}Higher Approximations}{111}{section.6.2}% +\contentsline {subsection}{\numberline {6.2.1}The $\tun $ Neural Network}{112}{subsection.6.2.1}% +\contentsline {subsection}{\numberline {6.2.2}The $\pwr $ Neural Networks}{114}{subsection.6.2.2}% +\contentsline {subsection}{\numberline {6.2.3}The $\tay $ neural network}{123}{subsection.6.2.3}% +\contentsline {subsection}{\numberline {6.2.4}Neural network approximations for $e^x$.}{128}{subsection.6.2.4}% +\contentsline {chapter}{\numberline {7}A modified Multi-Level Picard and associated neural network}{129}{chapter.7}% +\contentsline {chapter}{\numberline {8}Some categorical ideas about neural networks}{132}{chapter.8}% +\contentsline {chapter}{\numberline {9}ANN first approximations}{133}{chapter.9}% +\contentsline {section}{\numberline {9.1}Activation Function as Neural Networks}{133}{section.9.1}% +\contentsline {section}{\numberline {9.2}ANN Representations for One-Dimensional Identity}{134}{section.9.2}% +\contentsline {section}{\numberline {9.3}Modulus of Continuity}{142}{section.9.3}% +\contentsline {section}{\numberline {9.4}Linear Interpolation of real-valued functions}{143}{section.9.4}% +\contentsline {subsection}{\numberline {9.4.1}The Linear Interpolation Operator}{143}{subsection.9.4.1}% +\contentsline {subsection}{\numberline {9.4.2}Neural Networks to approximate the $\lin $ operator}{144}{subsection.9.4.2}% +\contentsline {section}{\numberline {9.5}Neural network approximation of 1-dimensional functions.}{148}{section.9.5}% +\contentsline {section}{\numberline {9.6}$\trp ^h$ and neural network approximations for the trapezoidal rule.}{151}{section.9.6}% +\contentsline {section}{\numberline {9.7}Linear interpolation for multi-dimensional functions}{154}{section.9.7}% +\contentsline {subsection}{\numberline {9.7.1}The $\nrm ^d_1$ and $\mxm ^d$ networks}{154}{subsection.9.7.1}% +\contentsline {subsection}{\numberline {9.7.2}The $\mxm ^d$ neural network and maximum convolutions }{160}{subsection.9.7.2}% +\contentsline {subsection}{\numberline {9.7.3}Lipschitz function approximations}{164}{subsection.9.7.3}% +\contentsline {subsection}{\numberline {9.7.4}Explicit ANN approximations }{167}{subsection.9.7.4}% +\contentsline {part}{III\hspace {1em}A deep-learning solution for $u$ and Brownian motions}{169}{part.3}% +\contentsline {chapter}{\numberline {10}ANN representations of Brownian Motion Monte Carlo}{170}{chapter.10}% +\contentsline {chapter}{Appendices}{180}{section*.3}% diff --git 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\max \left\{ \beta, U_i\right\} + \mathbbm{1}_{\N} \lp i \rp \max \left\{ \beta, U_{\max \left\{i-1,0 \right\}} \right\} \rp + \end{align} + It is then also the case that for all $n\in \N$ that $U_n \les \lp 2M+1 \rp^n \max \left\{\alpha,\beta \right\}$. +\end{lemma} +\begin{proof} + Let: + \begin{align}\label{7.0.2} + S_n = M^n + \sum^{n-1}_{i=0} M^{n-i} \lb \lp 2M+1 \rp^i + \mathbbm{1}_\N \lp i \rp \lp 2M+1 \rp ^{\max \left\{i-1,0\right\}} \rb + \end{align} + We prove this by induction. The base case of $n=0$ already implies that $U_0 \les \alpha \les \max \left\{\alpha, \beta \right\}$. Next assume that $U_n \les \lp 2M+1 \rp^n \max \left\{ \alpha, \beta \right\}$ holds for all integers upto and including $n$, it is then the case that: + \begin{align} + U_{n+1} &\les \alpha M^{n+1} + \sum^n_{i=0} M^{n+1-i}\lp \max \left\{ \beta, U_i \right\} + \mathbbm{1}_\N \lp i \rp \max \left\{ \beta, U_{\max \left\{i-1,0 \right\}} \right\} \rp \nonumber \\ + &\les \alpha M^{n+1} + \sum^n_{i=0} M^{n+1-i} \lb \max \left\{ \beta, \lp 2M+1 \rp^k\max \left\{ \alpha,\beta \right\}\right\} \right. \nonumber\\&\left. + \mathbbm{1}_\N \lp i \rp \max\left\{ \beta, \lp 2M+1 \rp ^{\max \left\{ k-1,0 \right\}} \max \left\{ \alpha, \beta \right\}\right\} \rb \nonumber \\ + &\les \alpha M^{n+1} + \max \left\{ \alpha,\beta\right\} \sum^n_{i=0} M^{n+1-i} \lb \lp 2M+1 \rp^i + \mathbbm{1}_\N \lp i \rp \lp 2M+1\rp ^{\max\left\{i-1,0 \right\}} \rb \nonumber \\ + &\les \max \left\{\alpha,\beta \right\} S_{n+1} + \end{align} + Then (\ref{7.0.2}) and the assumption that $M\in \lb 0, \infty \rp$ tells us that: + \begin{align} + S_{n+1} &= M^{n+1} + \sum^n_{i=0} M^{n+1-i} \lb \lp 2M+1 \rp^i + \mathbbm{1}_\N \lp i \rp \lp 2M+1 \rp^{\max\left\{i-1,0 \right\}} \rb \nonumber \\ + &= M^{n+1} \sum^n_{i=0} M^{n+1-i} \lp 2M+1\rp^k + \sum^n_{i=1} M^{n+1-i} \lp 2M+1 \rp ^{i-1} \nonumber \\ + &=M^{n+1} + M \lb \frac{\lp 2M+1 \rp^{n+1} - M^{n+1}}{M+1} \rb + M \lb \frac{\lp 2M+1 \rp^n-M^n}{M+1} \rb \nonumber \\ + &= M^{n+1} + \frac{M\lp 2M+1\rp^{n+1}}{M+1} + \frac{\lp 2M+1 \rp^n}{M+1} - M \lb \frac{M^{n+1}+M^n}{M+1} \rb \nonumber \\ + &\les M^{n+1} + \frac{M \lp 2M+1 \rp ^{n+1}}{M+1} + \frac{\lp 2M+1\rp ^{n+1}}{M+1} - M^{n+1} \lb \frac{\cancel{M+1}}{\cancel{M+1}} \rb \nonumber \\ + &= \lp 2M+1\rp ^{n+1} + \end{align} + This completes the induction step proving (\ref{7.0.1}). +\end{proof} +\begin{lemma} + Let $\Theta = \lp \bigcup^{n\in \N} \Z^n \rp$, $d,M \in \N$, $T\in \lp 0,\infty \rp$, $f \in C \lp \R, \R \rp$, $g,\in C \lp \R^d, \R \rp$, $\mathsf{F}, \mathsf{G} \in \neu$ satisfy that $\real_{\rect} \lp \mathsf{F} \rp = f$ and $\real_{\rect} \lp \mathsf{G} \rp = g$, let $\mathfrak{u}^\theta \in \lb 0,1 \rb$, $\theta \in \Theta$, and $\mathcal{U}^\theta: \lb 0,T \rb \rightarrow \lb 0,T \rb$, $\theta \in \Theta$, satisfy for all $t \in \lb 0,T \rb$, $theta \in \Theta$ that $\mathcal{U}^\theta_t = t+(T-t)\mathfrak{u}^\theta$, let $\mathcal{W}^\theta: \lb 0,T \rb \rightarrow \R^d$, $\theta \in \Theta$, for every $\theta \in \Theta$, $t\in \lb 0,T\rb$, $s \in \lb t,T\rb$, let $\mathcal{Y}^\theta_{t,s} \in \R$ satisfy $\mathcal{Y}^\theta_{t,s} = \mathcal{W}^\theta_s - \mathcal{W}^\theta_t$ and let $\mathcal{U}^\theta_n: \lb 0,T\rb \times \R^d \rightarrow \R$, $n\in \N_0$, $\theta \in \Theta$, satisfy for all $\theta \in \Theta$, $n\in \N_0$, $t\in \lb 0,T\rb$, $x\in \R^d$ that: + \begin{align} + U^\theta_n \lp t,x\rp &= \frac{\mathbbm{1}_\N\lp n \rp}{M^n} \lb \sum^{M^n}_{k=1} g \lp x + \mathcal{Y}^{(\theta,0,-k)}_{t,T}\rp\rb \nonumber\\ + &+ \sum^{n-1}_{i=0} \frac{T-t}{M^{n-i}} \lb \sum^{M^{n-i}}_{k=1} \lp \lp f \circ U^{(\theta,i,k)}_i\rp - \mathbbm{1}_\N \lp i \rp \lp f \circ U^{(\theta,-i,k)}_{\max \{ i-1,0\}} \rp \rp \lp \mathcal{U}^{(\theta,i,k)}_t,x+ \mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}}\rp\rb + \end{align} + it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item there exists unique $\mathsf{U}^\theta_{n,t} \in \neu$, $t \in \lb 0,T \rb$, $n\in \N_0$, $\theta \in \Theta$, which satisfy for all $\theta_1,\theta_2 \in \Theta$, $n\in \N_0$, $t_1, t_2 \in \lb 0,T \rb$ that $\lay \lp \mathsf{U}^{\theta_1}_{n,t_1} \rp = \mathcal{L} \lp \mathsf{U}^{\theta_2}_{n,t_2} \rp$. + \item for all $\theta \in \Theta$, $t \in \lb 0,T\rb$ that $\mathsf{U}^\theta_{0,t} = \lb \lb 0 \quad 0 \quad \cdots \quad 0\rb,\lb 0 \rb \rp \in \R^{1 \times d}\times \R^1$ + \item for all $\theta \in \Theta$, $n\in \N$, $t \in \lb 0,T \rb$ that: +% \begin{align} +% \mathsf{U}^\theta_{n,t} = \lb \bigoplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathsf{G} \bullet \aff_{\mathbb{I}_d, \mathcal{Y}^{(\theta,0,-k)}_{t,T}} \rp \rp \rb \nonumber\\ +% \boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{T-t}{M^{n-i}} \rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}} \lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,i,k}_{i, \mathcal{U}_t^{(\theta,i,k)} \rp \rp \rp +% \end{align} + \begin{align} + \mathsf{U}^\theta_{n,t} &= \lb \bigoplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathsf{G}\bullet \aff_{\mathbb{I}_d, \mathcal{Y}^{(\theta,0,-k}_{t,T}} \rp \rp \rb \nonumber \\ + &\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{T-t}{M^{n-i}}\rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}}\lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,i,k)}_{i,\mathcal{U}_t^{(\theta,i,k)}} \rp \bullet \aff_{\mathbb{I}_d}, \mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}} \rp\rp \rb\rb \nonumber\\ + &\boxplus_{\mathbb{I}} \lb \boxplus^{n-1}_{i=0,\mathbb{I}} \lb \lp \frac{(t-T)\mathbbm{1}_\N}{M^{n-i}}\rp \circledast\lp \boxplus^{M^{n-i}}_{k=1,\mathbb{I}} \lp \lp \mathsf{F} \bullet \mathsf{U}^{(\theta,-i,k)}_{\max \{i-1,0\}, \mathcal{U}_t^{(\theta,i,k)}}\rp \bullet \aff_{\mathbb{I}_d,\mathcal{Y}^{(\theta,i,k)}_{t,\mathcal{U}_t^{(\theta,i,k)}}} \rp \rp\rb \rb + \end{align} + \item that for all $\theta \in \Theta$, $n\in \N_0$, $t\in \lb 0,T \rb$, that $\dep \lp \mathsf{U}^\theta_{n,t} \rp = n\cdot \hid \lp \mathsf{F} \rp + \max \left\{1,\mathbbm{1}_\N \lp n \rp \dep \lp \mathsf{G} \rp \right\}$ + \item that for all $\theta \in \Theta$, $n\in \N_0$, $t \in \lb 0,T \rb$, that $\left\| \lay \lp \mathsf{U^\theta_{n,t}} \rp\right\|_{\max} \les \lp 2M+1\rp ^n \max \left\{ 2, \left\| \lay \lp \mathsf{F} \rp\right\|_{\max}, \left\| \lay \lp \mathsf{G} \rp \right\|_F \right\}$ + \item it holds for all $\theta \in \Theta$, $n\in \N_0$, $t \in \lb 0,T \rb $, $x \in \R^d$ that $U^\theta_n \lp t,x \rp = \lp \real_{\rect} \lp \mathsf{U}^\theta_{n,t}\rp \rp \lp x \rp $, and + \item it holds for all $\theta \in \Theta$, $n \in \N_0$, $t\in \lb 0,T\rb$ that: + \begin{align} + \param \lp \mathsf{U}^\theta_{n,t} \rp \les 2n\hid \lp \mathsf{F} \rp + \max \left\{1,\mymathbb{1}_\N \lp n\rp \dep \lp \mathsf{G} \rp \right\} \lb \lp 2M+1\rp^n\max \left\{ 2, \left\| \lay\lp \mathsf{F}\rp\right\|_{\max}, \left\| \lay \lp \mathsf{G} \rp \right\|_{\max}\right\}\rb^2 + \end{align} + \end{enumerate} +\end{lemma} diff --git a/Dissertation/neural_network_introduction.aux b/Dissertation/neural_network_introduction.aux new file mode 100644 index 0000000..d189325 --- /dev/null +++ b/Dissertation/neural_network_introduction.aux @@ -0,0 +1,187 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\citation{grohsetal} +\@writefile{toc}{\contentsline {chapter}{\numberline {5}Introduction and Basic Notions about Neural Networks}{62}{chapter.5}\protected@file@percent } +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} 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+\setcounter{theorem}{0} +\setcounter{corollary}{0} +\setcounter{lstlisting}{0} +\setcounter{nn@layerindex@counter}{6} +\setcounter{nn@lastlayerstart@counter}{0} +\setcounter{nn@thislayerstart@counter}{1} +\setcounter{nn@lastlayercount@counter}{4} +\setcounter{nn@thislayercount@counter}{2} +\setcounter{nn@lastlayerindex@counter}{4} +\setcounter{nn@thislayerindex@counter}{5} +} diff --git a/Dissertation/neural_network_introduction.pdf b/Dissertation/neural_network_introduction.pdf new file mode 100644 index 0000000..a7c22c3 Binary files /dev/null and b/Dissertation/neural_network_introduction.pdf differ diff --git a/Dissertation/neural_network_introduction.tex b/Dissertation/neural_network_introduction.tex new file mode 100644 index 0000000..af88ea0 --- /dev/null +++ b/Dissertation/neural_network_introduction.tex @@ -0,0 +1,1536 @@ +\chapter{Introduction and Basic Notions About Neural Networks} +We seek here to introduce a unified framework for artificial neural networks. This framework borrows from the work presented in \cite{grohsetal} and work done by Joshua Padgett, Benno Kuckuk, and Arnulf Jentzen (unpublished). With this framework in place, we wish to study ANNs from the perspective of trying to see the number of parameters required to define a neural network to solve certain PDEs. The \textit{curse of dimensionality} here refers to the number of parameters necessary to model PDEs and their growth (exponential or otherwise) as dimensions $d$ increase. +\section{The Basic Definition of ANNs and instantiations of ANNs} + +\begin{definition}[Rectifier Function] + Let $d \in \N$ and $x \in \R^d$. We denote by $\rect: \R \rightarrow \R$ the function given by: + \begin{align} + \rect(x) = \max \left\{ 0,x\right\} + \end{align} +\end{definition} + +\begin{definition}[Artificial Neural Networks]\label{5.1.2}\label{def:nn_def} + Denote by $\neu$ the set given by: + \begin{align} + \neu = \bigcup_{L\in \N} \bigcup_{l_0,l_1,...,l_L \in \N} \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp + \end{align} + An artificial neural network is a tuple $\lp \nu, \param, \dep, \inn, \out, \hid, \lay, \wid \rp $ where $\nu \in \neu$ and is equipped with the following functions (referred to as auxiliary functions) satisfying for all $\nu \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$: + \begin{enumerate}[label = (\roman*)] + \item $\param: \neu \rightarrow \N$ denoting the number of parameters of $\nu$, given by: + \begin{align}\label{paramdef} + \param(\nu) = \sum^L_{k=1}l_k \lp l_{k-1}+1 \rp + \end{align} + \item $\dep: \neu \rightarrow \N$ denoting the number of layers of $\nu$ other than the input layer given by: + \begin{align} + \dep(\nu) = L + \end{align} + \item $\inn:\neu \rightarrow \N$ denoting the width of the input layer, given by: + \begin{align} + \inn(\nu) = l_0 + \end{align} + \item $\out: \neu \rightarrow \N$ denoting the width of the output layer, given by: + \begin{align} + \out(\nu) = l_L + \end{align} + \item $\hid: \neu \rightarrow \N_0$ denoting the number of hidden layers (i.e., layers other than the input and output), given by: + \begin{align} + \hid(\nu) = L-1 + \end{align} + \item $\lay: \neu \rightarrow \bigcup_{L \in \N} \N^L$ denoting the width of layers as an $(L+1)$-tuple, given by: + \begin{align} + \lay(\nu) = \lp l_0,l_1,l_2,...,l_L \rp + \end{align} + We sometimes refer to this as the layer configuration or layer architecture of $\nu$. + \item $\wid_i: \neu \rightarrow \N_0$ denoting the width of layer $i$, given by: + \begin{align} \label{widthdef} + \wid_i(\nu) = \begin{cases} + l_i & i \leqslant L \\ + 0 & i > L + \end{cases} + \end{align} + \end{enumerate} +\end{definition} + Note that this implies that that $\nu = ((W_1,b_1),(W_2,b_2),...(W_L,b_L)) \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$. Note that we also denote by $\we_{(\cdot ), \nu}: (\we_{n,\nu})_{n\in \{1,2,...,L\}}: \{1,2,...,L\} \rightarrow \lp \bigcup_{m,k \in \N}\R^{m \times k} \rp $ and also $\bi_{(\cdot),\nu}: \lp \bi_{n,\nu} \rp_{\{1,2,...,L\}}: \{1,2,...,L\} \rightarrow \lp \bigcup_{m \in \N}\R^m \rp$ the functions that satisfy for all $n \in \{1,2,...,L\}$ that $\we_{i,\nu} = W_i$ i.e. the weights matrix for neural network $\nu$ at layer $i$ and $\bi_{i,\nu} = b_i$, i.e. the bias vector for neural network $\nu$ at layer $i$. + + We will call $l_0$ the \textit{starting width} and $l_L$ the \textit{finishing width}. Together, they will be referred to as \textit{end-widths}. +\begin{remark} + Notice that our definition varies somewhat from the conventional ones found in \cite{petersen_optimal_2018} and \cite{grohs2019spacetime} in that whereas the former talk about auxiliary functions as existing within the set $\neu$ we will talk about these auxiliary functions as something elements of $\neu$ are endowed with. In other words, elements of $\neu$ may exist whose depths and parameter counts, for instance, are undefined and non-determinate. + + Note that we develop this definition to closely align to popular deep-learning frameworks such as \texttt{PyTorch}, \texttt{TensorFlow}, and \texttt{Flux}, where, in principle, it is always possible to know the parameter count, depth, number of layers, and other auxiliary information. + + We will often say let $\nu\in \neu$, and it is implied that the tuple $\nu$ with the auxiliary functions is what is being referred to. +\end{remark} + + +\begin{definition}[Instantiations of Artificial Neural Networks with Activation Functions]\label{def:rlz} + Let $\act \in C \lp \R, \R \rp$, we denote by $\real_{\act}: \neu \rightarrow \lp \bigcup_{k,l \in \N} C \lp \R^k, \R^l \rp \rp$ the function satisfying for all $L \in \N$, $l_0,l_1,...,l_L \in \N$, $\nu = \lp \lp W_1, b_1 \rp , \lp W_2, b_2\rp ,...,\lp W_L, b_L \rp \rp \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$, $x_0 \in \R^{l_0}, x_1 \in \R^{l_1},...,x_{L-1} \in \R^{l_L-1}$ and with $\forall k \in \N \cap (0,L):x_k = \act \lp \lb W_kx_k+b_k \rb_{*,*} \rp$such that: + \begin{align}\label{5.1.11} + \real_{\act}\lp \nu \rp \in C \lp \R^{l_0}, \R^{l_L} \rp & \text{ and } & \lp \real_{\act}\lp \nu\rp \rp \lp x_0 \rp = W_Lx_{L-1}+b_L + \end{align} +\end{definition} +We will often denote the instantiated neural network $\nu^{l_0,l_L}$ taking $\R^{l_0}$ to $\R^{l_L}$ as $\nu^{l_0,l_L}: \R^{l_0} \rightarrowtail \R^{l_L}$ or simply as $\R^{l_0} \overset{\nu}{\rightarrowtail} \R^{l_L}$ where $l_0$ and $l_L$ are obvious. + +\begin{center} + \begin{neuralnetwork}[height=4, title = {A neural network $\nu$ with $\lay(\nu) = \lp 4,5,4,2\rp$}, nodesize = 10pt, maintitleheight=1em] + \newcommand{\x}[2]{$x$} + \newcommand{\y}[2]{$x$} + \newcommand{\hfirst}[2]{\small $h$} + \newcommand{\hsecond}[2]{\small $h$} + \inputlayer[count=3, bias=true, title=, text=\x] + \hiddenlayer[count=4, bias=true, title=, text=\hfirst] \linklayers + \hiddenlayer[count=3, bias=true, title=, text=\hsecond] \linklayers + \outputlayer[count=2, title=, text=\y] \linklayers + \end{neuralnetwork} + +\end{center} +\begin{remark} + For an R implementation see Listings \ref{nn_creator}, \ref{aux_fun}, \ref{activations}, and \ref{instantiation} +\end{remark} + +\begin{lemma}\label{5.1.8} + Let $\nu \in \neu$, it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\lay(\nu) \in \N^{\dep(\nu)+1}$, and + \item for all $\act \in C \lp \R, \R \rp$, $\real_{\act} \in C \lp \R^{\inn(\nu)},\R^{\out(\nu)}\rp $ + \end{enumerate} +\end{lemma} +\begin{proof} + By assumption: + \begin{align} + \nu \in \neu = \bigcup_{L\in \N} \bigcup_{\lp l_0,l_1,...,l_L \rp \in \N^{L+1}} \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp + \end{align} + This ensures that there exist $l_0,l_1,...,l_L,L \in \N$ such that: + \begin{align} + \nu \in \lp \bigtimes^L_{j=1} \lb \R^{l_j \times l_{j-1}} \times \R^{\l_j} \rb \rp + \end{align} + This also ensures that $\lay(\nu) = \lp l_0,l_1,...,l_L \rp \in \N^{L+1} = \N^{\dep(\nu)+1}$ and further that $\inn(\nu) = l_0$, $\out(\nu) = l_L$, and that $\dep(\nu) = L$. Together with ($\ref{5.1.11}$), this proves the lemma. +\end{proof} +\section{Compositions of ANNs} +The first operation we want to be able to do is to compose neural networks. Note that the composition is not done in an obvious way; for instance, note that the last layer of the first component of the composition is superimposed with the first layer of the second component of the composition. +\subsection{Composition} +\begin{definition}[Compositions of ANNs]\label{5.2.1}\label{def:comp} + We denote by $\lp \cdot \rp \bullet \lp \cdot \rp: \{ \lp \nu_1,\nu_2 \rp \in \neu \times \neu: \inn(\nu_1) = \out (\nu_1) \} \rightarrow \neu$ the function satisfying for all $L,M \in \N, l_0,l_1,...,l_L, m_0, m_1,...,m_M \in \N$, $\nu_1 = \lp \lp W_1, b_1 \rp, \lp W_2, b_2 \rp,...,\lp W_L,b_L \rp \rp \in \lp \bigtimes^L_{k=1} \lb \R^{l_k \times l_{k-1}} \times \R^{l_k}\rb \rp$, and $\nu_2 = \lp \lp W'_1, b'_1 \rp, \lp W'_2, b'_2 \rp,... \lp W'_M, b'_M \rp \rp \in \lp \bigtimes^M_{k=1} \lb \R^{m_k \times m_{k-1}} \times \R^{m_k}\rb \rp$ with $l_0 = \inn(\nu_1)= \out(\nu_2) = m_M$ and : + \begin{align}\label{5.2.1} + \nu_1 \bullet \nu_2 = \begin{cases} + (( W'_1,b'_1 ), ( W'_2,b'_2 ), ...( W'_{M-1}, b'_{M-1}), ( W_1W'_M, W_1b'_{M} + b_1), (W_2, b_2 ),\\..., ( W_L,b_L )) & :( L> 1 ) \land ( M > 1 ) \\ + ((W_1W'_1,W_1b'_1+b_1),(W_2,b_2), (W_3,b_3),...,(W_Lb_L)) & :(L>1) \land (M=1) \\ + ((W'_1, b'_1),(W'_2,b'_2), ..., (W'_{M-1}, b'_{M-1})(W_1, b'_M + b_1)) &:(L=1) \land (M>1) \\ + ((W_1W'_1, W_1b'_1+b_1)) &:(L=1) \land (M=1) + \end{cases} + \end{align} + \end{definition} + + + + + + \begin{remark} + For an \texttt{R} implementation see Listing \ref{comp} + \end{remark} + \begin{lemma}\label{depthofcomposition} + Let $\nu, \mu \in \neu$ be such that $\out(\mu) = \inn(\nu)$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\dep( \nu \bullet \mu) = \dep(\nu) + \dep(\mu) - 1$ + \item For all $i \in \{1,2,...,\dep(\nu \bullet\mu)\}$ that: + \begin{align} + &\lp \we_{i,(\nu \bullet \mu)}, \bi_{i,(\nu \bullet \mu)} \rp \nonumber\\ &= \begin{cases} + \lp \we_{i,\mu}, \bi_{i, \mu} \rp & : i< \dep(\mu)\\ + \lp \we_{1,\nu}\we_{\dep(\mu),\mu}, \we_{1,\nu}\bi_{\dep(\mu),\mu} + \bi_{1,\nu}\rp & : i = \dep (\mu)\\ + \lp \we_{i-\dep(\mu)+1,\nu} \bi_{i-\dep(\mu)+1,\nu}\rp & :i> \dep(\mu) + \end{cases} \nonumber + \end{align} + \end{enumerate} + \end{lemma} + \begin{proof} + This is a consequence of (\ref{5.2.1}), which implies both (i) and (ii). + \end{proof} + \begin{lemma} \label{5.2.3} + Let $\nu_1,\nu_2,\nu_3 \in \neu$ satisfy that $\inn(\nu_1) = \out(\nu_2)$ and $\inn(\nu_2) = \out(\nu_3)$, it is then the case\\ that: + \begin{align} + \lp \nu_1 \bullet \nu_2 \rp \bullet \nu_3 = \nu_1 \bullet \lp \nu_2 \bullet \nu_3 \rp + \end{align} + \end{lemma} +\begin{proof} + This is a consequence of \cite[Lemma~2.8]{grohs2019spacetime} with $\Phi_1 \curvearrowleft \nu_1$, $\Phi_2 \curvearrowleft \nu_2$, and $\Phi_3 \curvearrowleft \nu_3$, and the functions $\mathcal{I} \curvearrowleft \inn$, $\mathcal{L} \curvearrowleft \dep$ and $\mathcal{O} \curvearrowleft \out$. +\end{proof} +The following Lemma will be important later on, referenced numerous times, and found in \cite[Proposition~2.6]{grohs2019spacetime}. For completion, we will include a simplified version of the proof here. +\begin{lemma}\label{comp_prop} + Let $\nu_1, \nu_2 \in \neu$. Let it also be that $\out\lp \nu_1\rp = \inn \lp \nu_2\rp$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\dep \lp \nu_1 \bullet \nu_2 \rp = \dep\lp \nu_1\rp + \dep \lp \nu_2\rp -1$ + \item $\lay\lp \nu_1 \bullet \nu_2\rp = \lp \wid_1\lp \nu_2\rp, \wid_2 \lp \nu_2\rp,\hdots, \wid_{\hid\lp \nu_2\rp},\wid_1\lp \nu_1\rp, \wid_2\lp \nu_1\rp,\hdots, \wid_{\dep\lp \phi_1\rp}\lp \nu_1\rp\rp$ + \item $\hid \lp \nu_1 \bullet \nu_2\rp = \hid \lp \nu_1\rp + \hid\lp \nu_2\rp$ + \item $\param \lp \nu_1 \bullet \nu_2\rp \les \param\lp \nu_1\rp + \param \lp \nu_2\rp + \wid_1 \lp \nu_1\rp\cdot \wid_{\hid\lp \nu_2\rp}\lp \nu_2\rp$ + \item for all $\act \in C \lp \R, \R\rp$ that $\real_{\act}\lp \nu_1 \bullet \nu_2\rp \lp x \rp \in C \lp \R^{\inn \lp \nu_2\rp},\R^{\out\lp \nu_1\rp}\rp$ and further: + \begin{align} + \real_{\act} \lp \nu_1 \bullet \nu_2\rp = \lb \real_{\act}\lp \nu_1\rp\rb \circ \lb \real_{\act}\lp \nu_2 \rp\rb + \end{align} + \end{enumerate} + \end{lemma} + \begin{proof} + Note that Items (i)---(iii) are a simple consequence of Definition \ref{5.2.1}. Specifically, given neural networks $\nu_1,\nu_2 \in \neu$, and $\dep\lp \nu_1\rp = n$ and $\dep \lp \nu_2\rp = m$, note that for all four cases, we have that the depth of the composed neural network $\nu_1 \bullet \nu_2$ is given by $n-1+m-1=n+m-1$ proving Item (i). Note that the outer neural network loses its last layer, yielding Item (ii) in all four cases. Finally since, for all $\nu \in \neu$ it is the case that $\hid \lp \nu\rp =\dep \lp \nu\rp-1$, Item (i) yields Item (iii). + + Now, suppose it is the case that $\nu_3 = \nu_1\bullet \nu_2$ and that: + \begin{align} + \nu_1 &= \lp \lp W_{1,1},b_{1,1}\rp, \lp W_{1,2},b_{1,2}\rp,\hdots, \lp W_{1,L_1},b_{1,L_1}\rp\rp \nonumber \\ + \nu_2 &= \lp \lp W_{2,1},b_{2,1}\rp, \lp W_{2,2},b_{2,2}\rp,\hdots, \lp W_{2,L_2},b_{2,L_2}\rp\rp \nonumber \\ + \nu_3 &= \lp \lp W_{3,1},b_{3,1}\rp, \lp W_{3,2},b_{3,2}\rp,\hdots, \lp W_{3,L_2},b_{3,L_2}\rp\rp \nonumber \\ + \end{align} + And that: + \begin{align} + &\lay \lp \nu_1\rp = \lp l_{1,1},l_{1,2},\hdots, l_{1,L_1}\rp \nonumber\\ + &\lay \lp \nu_2\rp = \lp l_{2,1},l_{2,2},\hdots, l_{2.L_2}\rp \nonumber \\ + &\lay \lp \nu_1 \bullet \nu_2\rp = \lp l_{3,1},l_{3,2}, \hdots, l_{3,L_3}\rp + \end{align} + and further let $x_0 \in \R^{l_{2,0}},x_1 \in \R^{l_{2,1}},\hdots,x_{L_2-1}\in \R^{l_{2,L_2-1}}$ satisfy the condition that: + \begin{align}\label{comp_x} + \forall k \in \N \cap \lp 0,L_2\rp: x_k = \act \lp \lb W_{2,k}x_{k-1} + b_{2,k}\rb_{*,*}\rp + \end{align} + also let $y_0 \in \R^{l_{1,0}}$, $y_1 \in \R^{l_{1,1}},\hdots, y_{L_1-1} \in \R^{l_{2,L_2-1}}$ satisfy: + \begin{align}\label{comp_y} + \forall k\in \N \cap \lp 0,L_1\rp:y_k = \act\lp \lb W_{1,k}y_{k-1}+b_{1,k}\rb_{*,*}\rp + \end{align} + and finally let $z_0 \in \R^{l_{3,0}}, z_1 \in \R^{l_{3,1}},\hdots, z_{L_3-1} \in \R^{l_{3,L_3-1}}$ satisfy: + \begin{align}\label{comp_z} + \forall k \in \N \cap \lp 0,L_3\rp: z_k = \act\lp \lb W_{3,k}z_{k-1} + b_{3,k}\rb_{*,*}\rp + \end{align} + + Note then that by Item (i) of Definition \ref{5.1.2} we have that: + \begin{align} + \param \lp \nu_1 \bullet \nu_2\rp &= \sum^{L_3}_{k=1} l_{3,k}\lp l_{3,k-1} +1\rp \nonumber \\ + &=\lb \sum^{L_2-1}_{k=1} l_{3,k} \lp l_{3,k-1} +1\rp\rb + l_{3,L_2}\lp l_{3,L_2-1}+1\rp+\lb \sum^{L_3}_{k=L_2+1} l_{3,k}\lp l_{3,k-1} +1\rp\rb \nonumber \\ + &= \lb \sum^{L_2-1}_{k=1}l_{2,j}\lp l_{2,j-1}+1\rp\rb + l_{1,1}\lp l_{2,L-1}+1\rp + \lb \sum^{L_3}_{k=L_2+1} l_{j-L_2+1}\lp l_{1,j-L_2}+1\rp\rb\nonumber \\ + &= \lb \sum^{L_2-1}_{k=1} l_{2,j} \lp l_{2,k-1}+1\rp\rb + \lb \sum^{L_1}_{k=2}l_{1,j} \lp l_{1,k-1} +1\rp\rb + l_{1,1}\lp l_{2,L_2-1} + 1\rp \nonumber \\ + &= \lb \sum^{L_2}_{k=1}l_{2,k}\lp l_{2,k-1}+1\rp \rb + \lb \sum_{k=1}^{L_1} l_{1,k}\lp l_{1,k-1}+1\rp\rb + l_{1,1}\lp l_{2,L_2-1} +1\rp \nonumber\\ + &- l_{2,L_2} \lp l_{2,L_2-1} +1\rp -l_{1,1}\lp l_{1,0}+1\rp \nonumber \\ + &= \param\lp \nu_1\rp + \param \lp \nu_2\rp + l_{1,1}\cdot l_{2,L_2-1} + \end{align} + Thus establishing Item (iv). + Note by Definition \ref{5.2.1}, and the fact that $\act \in C \lp \R, \R \rp$ it is the case that + \begin{align}\label{comp_cont} + \real_{\act}\lp \nu_1 \bullet \nu_2\rp \in C \lp \R^{\inn \lp \nu_1\rp},\R^{\out\lp \nu_2\rp} \rp + \end{align} + Next note that by definition, it is the case that: + \begin{align} + \lay \lp \nu_1 \bullet \nu_2\rp = \lp l_{2,0},l_{2,1},\hdots, l_{2,L_2-1},l_{1,1},l_{1,2},\hdots,l_{1,L_1}\rp + \end{align} + And further that: + \begin{align} + \forall k \in \N \cap \lp 0,L_2\rp : \lp W_{3,k},b_{3,k}\rp &= \lp W_{2,k},b_{2,k}\rp \nonumber \\ + \lp W_{3,L_2},b_{3,L_2} \rp &= \lp W_{1,1} \cdot W_{2,L_2}, W_{1,1}b_{2,L_2} + b_{1,1}\rp \nonumber\\ + \text{ and } \forall k \in \N \cap \lp L_2,L_1+L_2\rp: \lp W_{3,k},b_{3,k}\rp &= \lp W_{1,j+1-L_2},b_{1,j+1-L_2}\rp + \end{align} + Since for all $k\in \N \cap \lb 0,L_2\rp$ it is the case that $z_j = x_j$ and the fact that $y_0 = W_{2,l_2}x_{L_2-1} + b_{2,L_2}$ ensures us that: + \begin{align}\label{(5.2.12)} + W_{3,L_2} z_{L_2-1} + b_{3,L_2} &= W_{3,L_2}x_{L_2-1} + b_{3,L_2} \nonumber \\ + &=W_{1,1}W_{2,L_2}x_{L_2-1} + W_{1,1}b_{2,L_2} + b_{1,1} \nonumber \\ + &=W_{1,1} \lp W_{2,L_2}x_{L_2-1} + b_{2,L_2}\rp + b_{1,1} = W_{1,1}y_0 + b_{1,1} + \end{align} + We next claim that for all $k\in \N \cap \lb L_2, L_1+L_2\rp$ it is the case that: + \begin{align}\label{(5.2.13)} + W_{3,k}z_{k-1} + b_{3,k} = W_{1,k+1-L_2}y_{k-L_2} + b_{1,k+1-L_2} + \end{align} + This can be proved via induction on $k\in \N \cap \lb L_2, L_1+L_2\rp$. Consider that our base case of $k=L_2$ in (\ref{(5.2.13)}) is fulfilled by (\ref{(5.2.12)}). Now note that for all $k \in \N \cap \lb L_2,\infty\rp \cap \lp 0,L_1+L_2-1\rp$ with: + \begin{align} + W_{3,k}z_{k-1} +b_{3,k} = W_{1,k+1-L_2}y_{k-L_2} + b_{1,k+1-L_2} + \end{align} + it holds that: + \begin{align} + W_{3,k+1}z_k + b_{3,k+1} &= W_{3,k+1}\lp \lb W_{3,k}z_{k-1} + b_{3,k}\rb_{*,*}\rp + b_{3,k+1} \nonumber \\ + &= W_{1,k+2-L_2}\lp \lb W_{1,k+1-L_2}y_{k-L_2}\rb + b_{1,k+1-L_2}\rp + b_{1,k+2-L_2} \nonumber \\ + &= W_{1,k+2-L_2}y_{k+1-L_2} + b_{1,k+2-L_2} + \end{align} + Whence induction proves (\ref{(5.2.13)}). This, along with the fact that $L_3 = L_1+L_2-1$ then indicates that: + \begin{align} + W_{3,L_3}z_{L_3-1} + b_{3,L_3} = W_{3,L_1+L_2-1}z_{L_1+L_2-2} + b_{3,L_1+L_2-1} = W_{1,L_1}y_{L_1-1} + b_{1,L_1} + \end{align} + Finally, the fact that $\nu_3 = \nu_1 \bullet \nu_2$, in addition with (\ref{comp_x}),(\ref{comp_y}), and (\ref{comp_z}) then tells us that: + \begin{align} + \lb \real_{\act}\lp \nu_1 \bullet \nu_2\rp\rb \lp x_0\rp &= \lb \real_{\act}\lp \nu_3\rp\rb \lp x_0\rp = \lb \real_{\act}\lp \nu_3\rp\rb \lp z_0\rp = W_{3,L_3}z_{L_3-1} + b_{3,L_3} \nonumber \\ + &= W_{1,L_1}y_{L_1-1} + b_{1,L_1} = \lb \real_{\act}\lp \nu_1\rp\rb\lp y_0\rp \nonumber \\ + &=\lb \real_{\act}\lp \nu_1\rp\rb \lp \lb W_{2,L_2}x_{L_2-1} + b_{2,L_2}\rb_{*,*}\rp \nonumber \\ + &=\lb \real_{\act}\lp \nu_1\rp\rb \lp \lb \real_{\act}\lp \nu_2\rp\rb\lp x_0\rp\rp = \lb \real_{\act}\lp \nu_1\rp\rb \circ \lb \real_{\act}\lp \nu_2 \rp\rb \lp x_0\rp + \end{align} + This and (\ref{comp_cont}) then prove Item (v), hence proving the lemma. + \end{proof} +\section{Stacking of ANNs of Equal Depth} +\begin{definition}[Stacking of ANNs of same depth]\label{5.2.5}\label{def:stacking} + Let $L,n\in \N$, and let $\nu_1,\nu_2,\hdots, \nu_n \in \neu$, such that $\dep\lp \nu_1\rp= \dep \lp \nu_2\rp= \cdots = \dep\lp \nu_n\rp = L$. As such, for all $i \in \{1,\hdots,n\}$, let it also be the case that $\lay\lp \nu_i\rp = \lp \lp W_1^i,b^i_1\rp, \lp W^i_2,b^i_2\rp,\hdots, \lp W_L^i,b_L^i\rp \rp$. We then denote by $\boxminus^n_{i=1}\nu_i$, the neural network whose layer architecture is given by: + \begin{align*} + \lay \lp \boxminus^n_{i=1}\nu_i\rp = \lp \lp \diag\lp W_1^1,W_1^2,\hdots,W_1^n\rp , b_1^1 \frown b_1^2,\frown \cdots \frown b_1^n\rp,\right.\\ \left.\lp \diag\lp W_2^1,W_2^2,\hdots,W_2^n\rp , b_2^1 \frown b_2^2,\frown \cdots \frown b_2^n\rp, \right.\\ \left. \vdots \hspace{4cm}\right.\\ \left. \lp \diag\lp W_L^1,W_L^2,\hdots,W_L^n\rp , b_L^1 \frown b_L^2,\frown \cdots \frown b_L^n\rp\rp + \end{align*} + +\end{definition} + +\begin{remark} + For an \texttt{R} implementation see Listing \ref{par} +\end{remark} +\begin{lemma}\label{inst_of_stk} + Let $\nu_1,\nu_2\in \neu$, with $\dep\lp \nu_1\rp = \dep\lp \nu_2\rp$, $x_1 \in \R^{m_1}$, $x_2 \in \R^{m_2}$, and $\mathfrak{x} \in \R^{m_1+m_2}$. Let $\inst_{\rect}\lp \nu_1\rp: \R^{m_1} \rightarrow \R^{n_1}$, and $\inst_{\rect}:\R^{m_2} \rightarrow \R^{n_2}$. It is then the case that $\real_{\rect}\lp \nu_1\boxminus\nu_2\rp\lp \mathfrak{x}\rp = \inst_{\rect}\lp \nu_1\rp\lp x_1\rp \frown \inst_{\rect}\lp \nu_2\rp\lp x_2\rp$. +\end{lemma} +\begin{proof} + Let $\lay\lp \nu_1\rp = \lp \lp W_1,b_1 \rp,\lp W_2,b_2\rp,\hdots, \lp W_L,b_L\rp\rp$ and $\lay \lp \nu_2\rp = \lp \lp \fW_1, \fb_1\rp, \lp \fW_2,\fb_2\rp,\hdots, \lp \fW_L,\fb_L\rp\rp$, and as such it is the case according to Definition \ref{def:stacking} that: + \begin{align*} + \lay \lp \nu_1 \boxminus\nu_2\rp = \lp \lp \diag\lp W_1,\fW_1\rp , b_1 \frown \fb_1\rp,\right.\\ \left.\lp \diag\lp W_2,\fW_2\rp , b_2 \frown \fb_2\rp, \right.\\ \left. \vdots \hspace{2.5cm}\right.\\ \left. \lp \diag\lp W_L,\fW_L\rp , b_L^1 \frown \fb_L\rp\rp + \end{align*} + Note that for all, $\act \in \lp \R,\R\rp$, $j \in \{1,2,\hdots,L-1\}$ and for all $x \in \R^{\columns(W_j)+\columns(\fW_j)}$, $x_1 \in \R^{\columns\lp W_j\rp}$, $x_2 \in \R^{\columns\lp \fW_j \rp}$, $y \in \R^{\rows\lp W_j\rp + \rows \lp \fW_j\rp}$, $y_1 \in \R^{\rows\lp W_j\rp}$, $y_2 \in \R^{\rows \lp \fW_j\rp}$, where $y_1 = \act\lp \lb W_j \cdot x_1 + b_1\rb_{*,*}\rp$, $y_2 = \act \lp \lb \fW_j\cdot x_2+\fb_j\rb_{*,*}\rp$, $y=\act\lp\lb \diag\lp W_j, \fW_j\rp \cdot x + \lp b_j \frown \fb_j \rp\rb_{*,*}\rp$ it is the case that, Corollary \ref{concat_fun_fun_concat} tells us that: + \begin{align} + y=\act\lp\lb \diag\lp W_j, \fW_j\rp \cdot x + \lp b_j \frown \fb_j \rp\rb_{*,*}\rp &= \act \lp \lb \lp W_j \cdot x_1+ b_j \rp \frown \lp \fW_j \cdot x_2+\fb_j\rp \rb_{*,*}\rp \nonumber \\ + &= \act\lp \lb W_j\cdot x_1+b_j\rb_{*,*}\rp\frown \act \lp \lb \fW_j\cdot x_2+\fb_j\rb_{*,*}\rp \nonumber\\ + &= y_1 \frown y_2 + \end{align} + Note that this is repeated from one layer to the next, yielding that $\real_{\rect}\lp \nu_1\boxminus\nu_2\rp\lp \mathfrak{x}\rp = \inst_{\rect}\lp \nu_1\rp\lp x_1\rp \frown \inst_{\rect}\lp \nu_2\rp\lp x_2\rp$. +\end{proof} + \begin{remark}\label{5.3.2}\label{rem:stk_remark} + Given $n,L \in \N$, $\nu_1,\nu_2,...,\nu_n \in \neu$ such that $L = \dep (\nu_1) = \dep(\nu_2) =...= \dep(\nu_n)$ it is then the case, as seen from (\ref{5.4.2}) that: + \begin{align}\label{(5.3.3)} + \boxminus_{i=1}^n \nu_i \in \lp \bigtimes^L_{k=1} \lb \R^{\lp \sum^n_{j=1} \wid_k(\nu_j) \rp \times \lp \sum_{j=1}^n \wid_{k-1} \lp \nu_j \rp \rp} \times \R ^{\lp \sum^n_{j=1} \wid_k \lp \nu_j \rp \rp }\rb \rp + \end{align} + \end{remark} + \begin{lemma}\label{paramofparallel} + Let $n,L \in \N$, $\nu_1,\nu_2,\hdots, \nu_n \in \neu$ satisfty that $L = \dep \lp \nu_1 \rp = \dep \lp \nu_2\rp = \cdots = \dep \lp\nu_n \rp$. It is then the case that: + \begin{align} + \param \lp \lb \boxminus_{i=1}^n \nu_i \rb \rp \les \frac{1}{2}\lb \sum^n_{i=1} \param \lp \nu_i\rp\rb^2 + \end{align} + \begin{proof} + Note that by Remark \ref{5.3.2} we have that: + \begin{align} + \param \lp \lb \boxminus_{i=1}^n \nu_i\rb\rp &= \sum^L_{k=1} \lb \sum_{i=1}^n l_{i,k}\rb \lb \lp \sum^n_{i=1} l_{i,k-1} \rp +1 \rb \nonumber \\ + &= \sum^L_{k=1} \lb \sum^n_{i=1}l_{i,k}\rb \lb \lp \sum^n_{j=1} l_{j,k-1}\rp+1\rb \nonumber\\ + &\les \sum^n_{i=1} \sum^n_{j=1} \sum^L_{k=1} l_{i,k} \lp l_{j,k-1}+1 \rp \nonumber \\ + &\les \sum^n_{i=1} \sum^n_{j=1} \sum^L_{\ell=1} l_{i,k} \lp l_{j,\ell-1} +1\rp \nonumber \\ + &=\sum^n_{i=1} \sum^n_{j=1} \lb \sum^L_{k=1}l_{i,k}\rb \lb \sum^L_{\ell=1} \lp l_{j,\ell-1} +1\rp\rb \nonumber \\ + &\les \sum^n_{i=1} \sum^n_{j=1}\lb \sum^L_{k=1} \frac{1}{2} l_{i,k} \lp l_{i,k-1}+1\rp\rb \lb \sum^L_{\ell=1} l_{j,\ell}\lp l_{j,\ell-1}+1 \rp \rb \nonumber \\ + &= \sum^n_{i=1} \sum^n_{j=1} \frac{1}{2} \param \lp \nu_i \rp \param \lp \nu_j\rp = \frac{1}{2} \lb \sum^n_{i=1}\param \lp \nu_i\rp\rb^2 + \end{align} + This completes the proof of the lemma. + \end{proof} + \end{lemma} + + \begin{corollary}\label{cor:sameparal} + Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$ satisfy that $\lay \lp \nu_1\rp = \lay \lp \nu_2\rp= \cdots =\lay \lp \nu_n\rp$. It is then the case that: + \begin{align} + \param \lp \boxminus_{i=1}^n \nu_i\rp \les n^2\param \lp \nu_1\rp + \end{align} + \end{corollary} + \begin{proof} + Since it is the case that for all $j \in \{1,2,...,n \}$ that: $\lay\lp \nu_j\rp=\lp l_0,l_1,...,l_L \rp$, where $l_0,l_1,...,l_L,L \in \N$, we may say that: + \begin{align} + \param \lp \boxminus_{j=1}^n \nu_j \rp &= \sum^L_{j=1}\lp nl_j\rp \lb \lp nl_{j-1} \rp +1\rb \les \sum^L_{j=1}\lp nl_j \rp\lb \lp nl_{j-1}\rp + n \rb \nonumber \\ + &=n^2 \lb \sum^L_{j=1}l_j \lp l_{j-1}+1\rp\rb = n^2\param\lp \nu_1\rp + \end{align} + \end{proof} + \begin{lemma}\label{lem:paramparal_geq_param_sum} + Let $\nu_1,\nu_2 \in \neu$, such that $\dep \lp \nu_1\rp = \dep \lp \nu_2\rp = L$. It is then the case that $\param\lp \nu_1\rp + \param \lp \nu_2\rp \les \param \lp \nu_1 \boxminus \nu_2\rp$. + \end{lemma} + \begin{proof} + Remark \ref{rem:stk_remark} tells us that: + \begin{align} + \nu_1 \boxminus \nu_2 \in \lp \bigtimes^L_{k=1} \lb \R^{\lp \wid_k(\nu_1) + \wid_k\lp \nu_2\rp \rp \times \lp \wid_{k-1} \lp \nu_1\rp +\wid_{k-1}\lp\nu_2\rp \rp} \times \R^{ \wid_k \lp \nu_1 \rp + \wid_k\lp \nu_2\rp }\rb \rp + \end{align} + The definition of $\param()$ from Defition \ref{def:nn_def}, and the fact that $\wid_i \ges 1$ for all $i \in \{1,2,\hdots, L\}$ tells us then that: + \begin{align} + \param\lp \nu_1 \boxminus \nu_2\rp &= \sum_{k=1}^L \lb \lp \wid_k \lp \nu_1\rp + \wid_k\lp \nu_2\rp\rp \times \lp \wid_{k-1}\lp \nu_1\rp+ \wid_{k-1}\lp \nu_2\rp +1\rp \rb \nonumber \\ + &= \sum^L_{k=1} \lb \wid_k \lp \nu_1\rp \wid_{k-1}\lp \nu_1\rp+ \wid_k \lp \nu_1\rp\wid_{k-1}\lp \nu_2\rp \right. \nonumber\\ &\left. + \wid_k \lp \nu_1\rp + \wid_k\lp \nu_2\rp\wid_{k-1}\lp \nu_1\rp + \wid_k\lp \nu_2\rp\wid_{k-1}\lp \nu_2\rp + \wid_k\lp \nu_2\rp\rb \nonumber \\ + &\ges \sum_{k=1}^L \lb \wid_k \lp \nu_1\rp\wid_{k-1}\lp \nu_1\rp + \wid_k \lp \nu_1\rp + \wid_k\lp \nu_2\rp\wid_{k-1}\lp \nu_2\rp+ \wid_k\lp \nu_2 \rp\rb \nonumber \\ + &=\sum_{k=1}^L\lb \wid_k \lp \nu_1 \rp\lp \wid_{k-1}\lp \nu_1\rp+1\rp\rb + \sum_{k=1}^L\lb \wid_k\lp \nu_2 \rp\lp \wid_{k-1}\lp \nu_2\rp+1\rp\rb \nonumber \\ + &= \param \lp \nu_1\rp + \param \lp \nu_2\rp + \end{align} + \end{proof} + \begin{corollary}\label{cor:bigger_is_better} + Let $\nu_1,\nu_2,\nu_3 \in \neu$. Let $\param \lp \nu_2 \rp \les \param\lp \nu_3\rp$. It is then the case that $\param\lp \nu_1 \boxminus \nu_2\rp \les \param\lp \nu_1 \boxminus \nu_3\rp$. + \end{corollary} + \begin{proof} + Lemma \ref{lem:paramparal_geq_param_sum} tells us that: + \begin{align} + 0 &\les \param \lp \nu_1\rp + \param\lp \nu_3\rp \les \param \lp \nu_1 \boxminus \nu_3\rp \label{lin1} \\ + 0 &\les \param \lp \nu_1\rp + \param\lp \nu_2\rp \les \param \lp \nu_1 \boxminus \nu_2\rp \label{lin2} + \end{align} + Subtracting (\ref{lin1}) from (\ref{lin2}) gives us that: + \begin{align} + 0 \les \param\lp \nu_3\rp - \param\lp \nu_2\rp &\les \param\lp \nu_1 \boxminus\nu_3\rp - \param\lp \nu_1\boxminus\nu_2\rp \nonumber\\ + \param\lp \nu_1 \boxminus\nu_2\rp &\les \param\lp \nu_1 \boxminus \nu_2\rp \nonumber + \end{align} + \end{proof} + + + + \begin{lemma}\label{5.4.3} + Let $m_1,m_2,n_1,n_2\in\N$. Let $\nu_1,\nu_2 \in \neu$, such that $\real_{\rect} \lp \nu_1\rp \in C \lp \R^{m_1},\R^{n_1}\rp$ and $\real_{\rect} \lp \nu_2\rp \in C\lp \R^{m_2},\R^{n_2}\rp$. It is then the case that $\lp \real_{\act}(\nu_1 \boxminus \nu_2)\rp \lp \begin{bmatrix} + x \\ x' + \end{bmatrix}\rp = \lp \real_{\act}(\nu_2 \boxminus \nu _1) \rp \lp \begin{bmatrix} + x' \\ x + \end{bmatrix}\rp$ for $x \in \R^{m_1}, x' \in \R^{n_1}$, upto transposition. + \end{lemma} + \begin{proof} + Note that this is a consequence of the commutativity of summation in the exponents of (\ref{(5.3.3)}), and the fact that switching $\nu_1$ and $\nu_2$ with a transposition results in a transposed output for transposed input. + \end{proof} + \begin{lemma}\label{5.3.4} + Let $\act \in C \lp \R, \R \rp$, $n \in \N$, and $\nu = \boxminus_{i=1}^n \nu_i$ satisfy the condition that $\dep(\nu_1) = \dep(\nu_2) =...=\dep(\nu_n)$. It is then the case that $\real_{\act} \lp \nu \rp \in C \lp \R^{\sum_{i=1}^n \inn(\nu_i)}, \R^{\sum^n_{i=1}\out(\nu_i)} \rp $ + \end{lemma} + \begin{proof} + Let $L = \dep(\nu_1)$, and let $l_{i,0},l_{i,1}...l_{i,L} \in \N$ satisfy for all $i \in \{ 1,2,...,n\}$ that $\lay(\nu_i) = \lp l_{i,0}, l_{i,1},...,l_{i,L} \rp $. Furthermore let $\lp \lp W_{i,1},b_{i,1}\rp, \lp W_{i,2},b_{i,2} \rp , ..., \lp W_{i,L},b_{i,L} \rp \rp \in \lp \bigtimes^L_{j=1} \lb \R^{l_{i,j} \times l_{i,j-1}} \times \R^{l_{i,j}} \rb \rp $ satisfy for all $i \in \{ 1,2,...,n\}$ that: + \begin{align} + \nu_i = \lp \lp W_{i,1},b_{i,1} \rp , \lp W_{i,2}, b_{i,2}\rp ,...,\lp W_{i,L},b_{i,L} \rp \rp + \end{align} + Let $\alpha_j \in \N$ with $j \in \{0,1,...,L\}$ satisfy that $\alpha_j = \sum^n_{i=1} l_{i,j}$ and let $\lp \lp A_1,b_1 \rp, \lp A_2,b_2 \rp,...,\lp A_L,b_L \rp \rp \in \lp \bigtimes^L_{j=1} \lb \R^{\alpha_{j} \times \alpha_{j-1}} \times \R^{\alpha_{j}} \rb \rp $ satisfy that: + \begin{align}\label{5.3.5} + \boxminus_{i=1}^n \nu_i = \lp \lp A_1,b_1 \rp, \lp A_2,b_2 \rp,...,\lp A_L,b_L \rp \rp + \end{align} + See Remark 5.3.2. Let $x_{i,0},x_{i,1},...,x_{i,L-1} \in \lp \R^{l_{i,0}} \times \R^{l_{i,1}}\times \cdots \times \R^{l_{i,L-1}} \rp$ satisfy for all $i \in \{1,2,...,n\}$ $k \in \N \cap \lp 0,L \rp $ that: + \begin{align} + x_{i,j} = \mult^{l_{i,j}}_{\act} \lp W_{i,j}x_{i,j-1} + b_{i,j} \rp + \end{align} + Note that (\ref{5.3.5}) demonstrates that $\inn \lp \boxminus_{i=1}^n\nu_i \rp =\alpha_0$ and $\out \lp \boxminus^n_{i=1} \nu_i \rp = \alpha_L$. This and Item(ii) of Lemma \ref{5.1.8}, and the fact that for all $i \in \{1,2,...,n\}$it is the case that $\inn(\nu_i) = l_{i,0}$ and $\out(\nu_i) = l_{i,L}$ ensures that: + \begin{align} + \real_{\act} \lp \boxminus^n_{i=1} \rp \in C \lp \R^{\alpha_0}, \R^{\alpha_L} \rp &= C\lp \R^{\sum^n_{i=1}l_{i,0}}, \R^{\sum_{i=1}^n l_{i,L}} \rp \nonumber\\ + &= C \lp \R^{\sum^n_{i=1} \inn(\nu_i)}, \R^{\sum_{i=1}^n \out(\nu_i)} \rp \nonumber + \end{align} + This proves the lemma. + \end{proof} + +\section{Stacking of ANNs of Unequal Depth} +We will often encounter neural networks that we want to stack but have unequal depth. Definition \ref{5.2.5} only deals with neural networks of the same depth. We will facilitate this situation by introducing a form of ``padding" for our neural network. Hence, they come out to the same length before stacking them. This padding will be via the "tunneling" neural network, as shown below. +\begin{definition}[Identity Neural Network]\label{7.2.1} + We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that: + \begin{enumerate}[label = (\roman*)] + \item \begin{align} + \id_1 = \lp \lp \begin{bmatrix} + 1 \\ + -1 + \end{bmatrix}, \begin{bmatrix} + 0 \\ + 0 + \end{bmatrix}\rp \lp \begin{bmatrix} + 1 \quad -1 + \end{bmatrix},\begin{bmatrix} 0\end{bmatrix}\rp \rp \in \lp \lp \R^{2 \times 1} \times \R^2 \rp \times \lp \R^{1\times 2} \times \R^1 \rp \rp + \end{align} + \item \begin{align}\label{7.2.2} + \id_d = \boxminus^d_{i=1} \id_1 + \end{align} + For $d>1$. +\end{enumerate} +\begin{remark} + We will discuss some properties of $\id$ in Section \ref{sec_tun}. +\end{remark} +\end{definition} +\begin{definition}[The Tunneling Neural Network] + We define the tunneling neural network, denoted as $\tun_n$ for $n\in \N$ and $d\in \N$ by: + \begin{align} + \tun^d_n = \begin{cases} + \aff_{\mathbb{I}_d,0} &:n= 1 \\ + \id_d &: n=2 \\ + \bullet^{n-2} \id_d & n \in \N \cap [3,\infty) + \end{cases} + \end{align} + We will drop the requirement for $d$ and $\tun_n$ by itself will be used to denote $\tun_n^1$. +\end{definition} +\begin{remark} + We will discuss some properties of the $\tun^d_n$ network in Section \ref{sec_tun}. +\end{remark} +\begin{definition} + Let $n \in \N$, and $\nu_1,\nu_2,...,\nu_n \in \neu$. We will define the stacking of unequal length neural networks, denoted $\DDiamond^n_{i=1}\nu_i$ as the neural network given by: + \begin{align} + \DDiamond^n_{i=1}\nu_i = + \boxminus^n_{i=1} \lb \tun_{\max_i \left\{\dep \lp \nu_i \rp\right\} +1 - \dep \lp \nu_i\rp} \bullet \nu_i \rb + \end{align} +\end{definition} +Diagrammatically, this can be thought of as: +\begin{figure} +\begin{center} + + +\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt + +\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1] +%uncomment if require: \path (0,475); %set diagram left start at 0, and has height of 475 + +%Shape: Rectangle [id:dp6580978944544137] +\draw (199.5,139) -- (490,139) -- (490,179) -- (199.5,179) -- cycle ; +%Shape: Rectangle [id:dp353023162160477] +\draw (420,205) -- (490,205) -- (490,245) -- (420,245) -- cycle ; +%Shape: Rectangle [id:dp37062952177240926] +\draw (200.5,205) -- (403,205) -- (403,245) -- (200.5,245) -- cycle ; +%Straight Lines [id:da022591094656464583] +\draw (419,224) -- (404.5,224) ; +\draw [shift={(402.5,224)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da41864611287252906] +\draw (198.5,160) -- (101.5,160) ; +\draw [shift={(99.5,160)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da6288954656732593] +\draw (198.5,222) -- (101.5,222) ; +\draw [shift={(99.5,222)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da8590579460958981] +\draw (526,158) -- (493.5,158) ; +\draw [shift={(491.5,158)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; +%Straight Lines [id:da8647539484496881] +\draw (527,221) -- (494.5,221) ; +\draw [shift={(492.5,221)}, rotate = 360] [color={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.75] (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; + +% Text Node +\draw (337,155) node [anchor=north west][inner sep=0.75pt] {$\nu _{1}$}; +% Text Node +\draw (445,220) node [anchor=north west][inner sep=0.75pt] {$\nu _{2}$}; +% Text Node +\draw (296,220) node [anchor=north west][inner sep=0.75pt] {$\mathsf{Tun}$}; + + +\end{tikzpicture} +\end{center} +\caption{Diagrammmatic representation of the stacking of unequal depth neural networks} +\end{figure} +\begin{lemma} + Let $\nu_1,\nu_2 \in \neu$. It is then the case that: + \begin{align} + \param \lp \nu_1\DDiamond\nu_2 \rp \les 2\cdot \lp \max \left\{ \param\lp \nu_1\rp, \param\lp \nu_2\rp\right\}\rp^2 + \end{align} +\end{lemma} +\begin{proof} + This is a straightforward consequence of Lemma \ref{paramofparallel}. +\end{proof} + +\section{Affine Linear Transformations as ANNs and Their Properties.} +\begin{definition}\label{5.3.1}\label{def:aff} + Let $m,n \in \N$, $W \in \R^{m \times n}$, $b \in \R^m$.We denote by $\aff_{W,b} \in \lp \R^{m\times n} \times \R^m \rp \subseteq \neu$ the neural network given by $\aff_{W,b} = ((W,b))$. +\end{definition} +\begin{lemma}\label{5.3.2}\label{aff_prop} + Let $m,n \in \N$, $W \in \R^{m\times n}$, $b \in \R^m$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\lay (\aff_{W,b}) = (n,m) \in \N^2$. + \item for all $\act \in C ( \R,\R)$ it is the case that $\real_{\act} (\aff_{W,b}) \in C (\R^n, \R^m)$ + \item for all $\act \in C(\R,\R)$, $x \in \R^n$ we have $(\real_{\act}(\aff_{W,b}))(x) = Wx+b$ + \end{enumerate} +\end{lemma} +\begin{proof} + Note that $(i)$ is a consequence of Definition \ref{5.1.2} and \ref{5.3.1}. Note next that $\aff_{W,b} = (W,b) \in (\R^{m\times n} \times \R^m) \subseteq \neu$. Note that ($\ref{5.1.11}$) then tells us that $\real_{\act} (\aff_{W,b}) = Wx+b$ which in turn proves $(ii)$ and $(iii)$ +\end{proof} +\begin{remark}\label{remark:5.4.3}\label{param_of_aff} + Given $W\in \R^{m\times n}$, and $b \in \R^{m \times 1}$, it is the case that according to Definition (\ref{paramdef}) we have: $\param(\aff_{W,b})= m\times n + m$ +\end{remark} +\begin{remark} + For an \texttt{R} implementation see Listing \ref{affn} +\end{remark} +\begin{lemma}\label{5.3.3}\label{aff_effect_on_layer_architecture} + Let $\nu \in \neu$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item For all $m\in \N$, $W \in \R^{m\times \out(\nu)}$ + \begin{align} + \lay(\aff_{W,B} \bullet\nu) = \lp \wid_0(\nu), \wid_1(\nu),...,\wid_{\dep(\nu)-1}(\nu),m \rp \in \N^{\dep(\nu)+1} + \end{align} + \item For all $\act \in C(\R,\R)$, $m\in \N$, $W \in \R^{m \times \out(\nu)}$, $B \in \R^m$, we have that $\real_{\act} (\aff_{W,B} \bullet\nu) \in C\lp \R^{\inn(\nu)},\R^m\rp$. + \item For all $\act \in C(\R,\R)$, $m\in \N$, $W \in \R^{m \times \out(\nu)}$, $B \in \R^m$, $x \in \R^{\inn(\nu)}$ that: + \begin{align} + \lp \real \lp \aff_{W,b} \bullet \nu \rp \rp \lp x \rp= W \lp \real_{\act}\lp \nu \rp \rp \lp x \rp +b + \end{align} + \item For all $n\in \N$, $W \in \R^{\inn(\nu) \times n}$, $b \in \R^{\inn(\nu)}$ that: + \begin{align} + \lay(\nu \bullet \aff_{W,b}) = \lp n, \wid_1(\nu), \wid_2(\nu),...,\wid_{\dep(\nu)}(\nu) \rp \in \N^{\dep(\nu)+1} + \end{align} + \item For all $\act \in C(\R,\R)$, $n\in \N$, $W \in \R^{\inn(\nu) \times n}$, $b \in \R^{\inn(\nu)}$ that $\real_{\act} \lp \nu \bullet \aff_{W,b} \rp \in C \lp \R^n, \R^{\out(\nu)} \rp $ and, + \item For all $\act \in C(\R,\R)$, $n\in \N$, $W \in \R^{\inn(\nu) \times n}$, $b \in \R^{\inn (\nu)}$, $x \in \R^n$ that: + \begin{align} + \lp \real_{\act} \lp \nu \bullet \aff_{W,b} \rp \rp \lp x \rp = \lp \real_{\act} \lp \nu \rp \rp \lp Wx+b \rp + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + From Lemma \ref{5.3.2} we see that $\real_{\act}(\aff_{W,b}) \in C(\R^n,\R^m)$ given by $\real_{\act}(\aff_{W,b}) = Wx + b$. This and Lemma \ref{comp_prop} prove $(i)-(vi)$. +\end{proof} +\begin{corollary}\label{affcor} + Let $m,n \in \N$, and $W \in \R^{m \times n}$ and $b \in \R^m$. Let $\nu\in \neu$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item for all $\aff_{W,b} \in \neu$ with $\inn \lp \aff_{W,b} \rp = \out \lp \nu \rp$ that: + \begin{align} + \param \lp \aff_{W,b} \bullet \nu \rp \les \lb \max\left\{ 1, \frac{\out \lp \aff_{W,b}\rp}{l_L}\right\}\rb \param \lp \nu\rp + \end{align} + \item for all $\aff_{W,b} \in \neu$ with $\out\lp \aff_{W,b}\rp = \inn\lp \nu\rp$ that: + \begin{align} + \param \lp \nu \bullet \aff_{W,b}\rp \les \lb \max\left\{ 1, \frac{\inn \lp \aff_{W,b}\rp+1}{\inn\lp \nu\rp+1}\right\}\rb \param \lp \nu\rp + \end{align} + \end{enumerate} +\end{corollary} +\begin{proof} + Let it be the case that $\lay \lp \nu\rp = \lp l_0,l_1,...,l_L\rp$ for $l_0,l_1,...,l_L,L \in \N$. Lemma \ref{5.3.3}, Item (i), and Lemma \ref{comp_prop} then tells us that: + \begin{align} + \param \lp \aff_{W,b} \bullet \nu \rp &= \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1\rp\rb + \out \lp \aff_{W,b}\rp \lp l_{L-1}+1\rp \nonumber \\ + &= \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1 \rp\rb+ \lb \frac{\out\lp \aff_{W,b}\rp}{l_L}\rb l_L\lp l_{L-1}+1 \rp \nonumber \\ + &\les \lb \max \left\{ 1, \frac{\out(\aff_{W,b})}{l_L}\right\}\rb \lb \sum^{L-1}_{m=1} l_m \lp l_{m-1}+1\rp\rb + \lb \max\left\{ 1,\frac{\out\lp \aff_{W,b}\rp}{l_L}\right\}\rb l_L \lp l_{L-1}+1\rp \nonumber\\ + &= \lb \max\left\{ 1, \frac{\out \lp \aff_{W,b}\rp}{l_L}\right\}\rb \lb \sum^L_{m=1}l_m \lp l_{m-1} +1\rp\rb = \lb \max\left\{ 1, \frac{\out \lp \aff_{W,b}\rp}{l_L}\right\}\rb \param \lp \nu\rp \nonumber + \end{align} + and further that: + \begin{align} + \param \lp \nu \bullet\aff_{W,b} \rp &= \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1\rp\rb + l_{1}\lp \inn \lp \aff_{W,b}\rp+1\rp \nonumber \\ + &= \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1 \rp\rb+ \lb \frac{\inn \lp \aff_{W,b}\rp+1}{l_0+1}\rb l_1\lp l_{0}+1 \rp \nonumber \\ + &\les \lb \max \left\{ 1, \frac{\inn(\aff_{W,b})+1}{l_0+1}\right\}\rb \lb \sum^{L}_{m=2} l_m \lp l_{m-1}+1\rp\rb + \lb \max\left\{ 1,\frac{\inn\lp \aff_{W,b}\rp+1}{l_0+1}\right\}\rb l_1 \lp l_{0}+1\rp \nonumber\\ + &= \lb \max\left\{ 1, \frac{\inn \lp \aff_{W,b}\rp+1}{l_0+1}\right\}\rb \lb \sum^L_{m=1}l_m \lp l_{m-1} +1\rp\rb = \lb \max\left\{ 1, \frac{\inn \lp \aff_{W,b}\rp+1}{\inn\lp \nu\rp+1}\right\}\rb \param \lp \nu\rp \nonumber + \end{align} + This completes the proof of the lemma. +\end{proof} +\begin{lemma}\label{aff_stack_is_aff} + Let $\mathfrak{a}_1,\mathfrak{a}_2$ be two affine neural networks as defined in Definition \ref{def:aff}. It is then the case that $\mathfrak{a}_1 \boxminus \mathfrak{a}_2$ is also an affine neural network +\end{lemma} +\begin{proof} + This follows straightforwardly from Definition \ref{def:stacking}, where, given that $\mathfrak{a}_1 = \lp \lp W_1,b_1\rp \rp$, and $\mathfrak{a}_2 = \lp \lp W_2,b_2\rp \rp$, their stackings is the neural network $\lp \lp \diag\lp W_1,W_2\rp,b_1 \frown b_2\rp\rp$, which is clearly an affine neural network. +\end{proof} + +\section{Sums of ANNs of Same End-widths} + +\begin{definition}[The $\cpy$ Network]\label{def:cpy} + We define the neural network, $\cpy_{n,k} \in \neu$ for $n,k\in \N$ as the neural network given by: + \begin{align} + \cpy_{n,k} = \aff_{\underbrace{\lb \mathbb{I}_{k} \: \mathbb{I}_k \: \cdots \: \mathbb{I}_k \rb^T}_{n-\text{many}},\mymathbb{0}_{nk}} + \end{align} + Where $k$ represents the dimensions of the vectors being copied and $n$ is the number of copies of the vector being made. +\end{definition} + + \begin{remark} + See Listing \ref{affn} + \end{remark} +\begin{lemma}\label{dep_cpy}\label{lem:param_cpy} + Let $n,k \in \N$ and let $\cpy_{n,k} \in \neu$, it is then the case for all $n,k \in \N$ that: + \begin{enumerate}[label = (\roman*)] + \item $\dep \lp \cpy_{n,k} \rp = 1$ + \item $\param\lp \cpy_{n,k} \rp = nk^2+nk$ + \end{enumerate} + \begin{proof} + Note that $(i)$ is a consequence of Definition \ref{5.3.1} and (ii) follows from the structure of $\cpy_{n,k}$. + \end{proof} + +\end{lemma} +\begin{definition}[The $\sm$ Network]\label{def:sm} + We define the neural network $\sm_{n,k}$ for $n,k \in \N$ as the neural network given by: + \begin{align} + \sm_{n,k} = \aff_{\underbrace{\lb \mathbb{I}_k \: \mathbb{I}_k \: \cdots \: \mathbb{I}_k\rb}_{n-\text{many}}, \mymathbb{0}_{k}} + \end{align} + Where $k$ represents the dimensions of the vectors being added and $n$ is the number of vectors being added. +\end{definition} + +\begin{remark} + See again, Listing \ref{affn} +\end{remark} +\begin{lemma}\label{lem:5.5.4}\label{lem:param_sm} + Let $n,k \in \N$ and $\sm_{n,k} \in \neu$, it is then the case for all $n,k \in \N$ that: + \begin{enumerate}[label = (\roman*)] + \item $\dep \lp \sm_{n,k} \rp = 1$ + \item $\param\lp \sm_{n,k} \rp = nk^2+k$ + \end{enumerate} +\end{lemma} +\begin{proof} + (i) is a consequence of Definition $\ref{5.3.1}$ and (ii) follows from the structure of $\sm_{n,k}$. +\end{proof} +\begin{definition}[Sum of ANNs of the same depth and same end widths]\label{def:nn_sum} + Let $u,v \in \Z$ with $u \leqslant v$. Let $\nu_u,\nu_{u+1},...,\nu_v \in \neu$ satisfy for all $i \in \N \cap [u,v]$ that $\dep(\nu_i) = \dep(\nu_u)$, $\inn(\nu_i) = \inn(\nu_u)$, and $\out(\nu_i) = \out(\nu_u)$. We then denote by $\oplus^n_{i=u} \nu_i$ or alternatively $\nu_u \oplus\nu_{u+1} \oplus \hdots \oplus\nu_v$ the neural network given by: + \begin{align}\label{5.4.3} + \oplus^v_{i=u}\nu_i \coloneqq \lp \sm_{v-u+1,\out(\nu_2)} \bullet \lb \boxminus^v_{i=u}\nu_i \rb \bullet \cpy_{(v-u+1),\inn(\nu_1)} \rp + \end{align} +\end{definition} + +\begin{remark} + For an \texttt{R} implementation, see Listing \ref{nn_sum}. +\end{remark} + +\subsection{Neural Network Sum Properties} +\begin{lemma}\label{paramsum} + Let $\nu_1, \nu_2 \in \neu$ satisfy that $\dep(\nu_1) = \dep(\nu_2) = L$, $\inn(\nu_1) = \inn(\nu_2)$, and $\out(\nu_1) = \out(\nu_2)$, and $\lay(\nu_1) = \lp l_{1,1},l_{1,2},...l_{1,L} \rp$ and $\lay \lp \nu_2 \rp = \lp l_{2,1}, l_{2,2},...,l_{2,L} \rp $ it is then the case that: + \begin{align} + \param \lp \nu_1 \oplus \nu_2 \rp &= \param \lp \aff_{\lb \mathbb{I}_{\out(\nu_2)} \: \mathbb{I}_{\out(\nu_2)}\rb, \mymathbb{0}_{\out(\nu_2)} }\bullet \lb \nu_1 \boxminus \nu_2\rb \bullet \aff_{\lb\mathbb{I}_{\inn(\nu_1)}\: \mathbb{I}_{\inn(\nu_1)}\rb^T,\mymathbb{0}_{2\cdot\inn(\nu_1)}} \rp \\ + &\les \frac{1}{2}\lp \param \lp \nu_1\rp + \param \lp \nu_2\rp\rp^2\nonumber + \end{align} +\end{lemma} + +\begin{proof} + Note that by Lemma \ref{paramofparallel} we have that: + \begin{align} + \param \lp \nu_1 \boxminus \nu_2 \rp = \frac{1}{2}\lp \param \lp \nu_1\rp + \param \lp \nu_2\rp\rp^2 + \end{align} + Note also that since $\cpy$ and $\sm$ are affine neural networks, from Corollary \ref{affcor} we get that: + \begin{align}\label{(5.5.6)} + \param \lp \lb \nu_1 \boxminus \nu_2\rb \bullet \cpy_{2,\inn(\nu_1)}\rp &\les \max \left\{ 1, \frac{\inn\lp \nu_1\rp+1}{2\inn\lp \nu_1\rp+1}\right\} \frac{1}{2}\lp \param \lp \nu_1\rp + \param \lp \nu_2\rp\rp^2 \nonumber\\ + &= \frac{1}{2}\lp \param \lp \nu_1\rp + \param \lp \nu_2\rp\rp^2 + \end{align} + and further that: + \begin{align} + \param \lp \sm_{2,\out \lp \nu_1 \boxminus \nu_2\rp} \bullet \lb \nu_1 \boxminus \nu_2\rb \bullet \cpy_{2,\inn\lp \nu_1\rp} \rp &\les \lb \max\left\{ 1, \frac{\out \lp \aff_{W,b}\rp}{2\out\lp \nu_1\rp}\right\}\rb \frac{1}{2} \lp \param \lp \nu_1\rp + \param \lp \nu_2 \rp\rp^2 \nonumber \\ + &= \frac{1}{2}\lp \param \lp \nu_1\rp + \param \lp \nu_2\rp\rp^2 + \end{align} +\end{proof} +\begin{corollary}\label{corsum} + Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$ satisfy that $\lay \lp \nu_1\rp = \lay \lp \nu_2\rp= \cdots =\lay \lp \nu_n\rp$. It is then the case that: + \begin{align} + \param \lp \bigoplus_{i=1}^n \nu_i\rp \les n^2\param \lp \nu_1\rp + \end{align} +\end{corollary} +\begin{proof} + Let $\lay \lp \nu_1\rp = \lp l_0,l_1,...,l_L\rp$ where for all $i \in \{0,1,...,L \}$ it is the case that $l_i,L \in \N$. Corollary \ref{cor:sameparal} then tells us that: + \begin{align} + \param \lp \boxminus_{i=1}^n \nu_i\rp \les n^2 \param \lp \nu_i\rp + \end{align} + Then from Corollary \ref{affcor}, and (\ref{(5.5.6)}) we get that: + \begin{align} + \param \lp \lb \boxminus_{i=1}^n \nu_i \rb\bullet \cpy_{2,\inn\lp \nu_1\rp}\rp \les n^2\param \lp \nu_1\rp + \end{align} + And further that: + \begin{align} + \param \lp \sm_{2,\out \lp \boxminus_{i=1}^n \nu_i\rp} \bullet \lb \boxminus_{i=1}^n \nu_i\rb \bullet \cpy_{2,\inn\lp \nu_1\rp} \rp \les n^2\param \lp\nu_1 \rp + \end{align} +\end{proof} +\begin{lemma}\label{depth_prop} + Let $\nu_1, \nu_2 \in \neu$ satisfy that $\dep(\nu_1) = \dep(\nu_2) = L$, $\inn(\nu_1) = \inn(\nu_2)$, and $\out(\nu_1) = \out(\nu_2)$, and $\lay(\nu_1) = \lp l_{1,1},l_{1,2},...l_{1,L} \rp$ and $\lay \lp \nu_2 \rp = \lp l_{2,1}, l_{2,2},...,l_{2,L} \rp $ it is then the case that: + \begin{align} + \dep \lp \nu_1 \oplus \nu_2 \rp =L + \end{align} +\end{lemma} +\begin{proof} + Note that $\dep \lp \cpy_{n,k} \rp = 1 = \dep\lp \sm_{n,k} \rp$ for all $n,k \in \N$. Note also that $\dep \lp \nu_1 \boxminus \nu_2 \rp = \dep\lp \nu_1 \rp = \dep \lp \nu_2 \rp $ and that for $\nu,\mu \in \neu$ it is the case that $\dep \lp \nu \bullet \mu\rp = \dep\lp \nu \rp + \dep \lp \mu \rp-1$. Thus: + \begin{align} + \dep \lp \nu_1 \oplus \nu_1 \rp = \dep\lp \nu_1 \oplus \nu_2 \rp &= \dep \lp \aff_{\lb \mathbb{I}_{\out(\nu_2)} \: \mathbb{I}_{\out(\nu_2)}\rb, \mymathbb{0}_{\out(\nu_2)} }\bullet \lb \nu_1 \boxminus \nu_2\rb \bullet \aff_{\lb\mathbb{I}_{\inn(\nu_1)}\: \mathbb{I}_{\inn(\nu_1)}\rb^T,\mymathbb{0}_{2\cdot\inn(\nu_1)}} \rp\nonumber \\ + &= L\nonumber + \end{align} +\end{proof} + +\begin{lemma}\label{5.4.6} + Let $\nu_1,\nu_2 \in \neu$, such that $\dep(\nu_1) = \dep(\nu_2)= L$, $\inn(\nu_1) = \inn(\nu_2) = l_0$, and $\out(\nu_1) = \out(\nu_2) = l_L$. It is then the case that $\real(\nu_1 \oplus \nu_2) = \real(\nu_2 \oplus \nu_1)$, i.e., the instantiated sum of ANNs of the same depth and same end widths is commutative. +\end{lemma} +\begin{proof} + Let $\nu_1 = \lp (W_1,b_1),(W_2,b_2),...,(W_L,b_L) \rp$ and let $\nu_2 = \lp (W'_1,b'_1),(W'_2,b'_2),...,(W_L', b_L') \rp $. Note that Definition $\ref{5.2.5}$ then tells us that: + \begin{align} + \nu_1 \boxminus \nu_2 = \lp\lp + \begin{bmatrix} + W_1 & 0 \\ + 0 & W_1' + \end{bmatrix}, \begin{bmatrix} + b_1 \\ + b_1' + \end{bmatrix} + \rp,\lp \begin{bmatrix} + W_2 & 0 \\ + 0 & W_2' + \end{bmatrix}, \begin{bmatrix} + b_2\\ + b_2' + \end{bmatrix}\rp,..., \right. \nonumber\\ + \left. + \lp \begin{bmatrix} + W_L & 0 \\ + 0 & W_L' + \end{bmatrix}, \begin{bmatrix} + b_L \\ + b_L' + \end{bmatrix} \rp \rp \nonumber + \end{align} + Note also that by Claims $\ref{5.4.4}$ and $\ref{5.4.5}$ and Definition \ref{5.3.1} we know that: + \begin{align} + \aff_{\lb \mathbb{I}_{\inn \lp \nu_2 \rp } \: \mathbb{I}_{\inn \lp \nu_2 \rp} \rb^T,\mymathbb{0}_{2\inn(\nu_2),1}} = \lp \begin{bmatrix} + \mathbb{I}_{\inn(\nu_2)} \\ + \mathbb{I}_{\inn(\nu_2)} + \end{bmatrix},\mymathbb{0}_{2\inn(\nu_2),1}\rp + \end{align} + and: + \begin{align} + \aff_{\lb \mathbb{I}_{\out(\nu_1)} \: \mathbb{I}_{\out(\nu_1)} \rb,\mymathbb{0}_{2\out(\nu_1),1}} = \lp \begin{bmatrix} + \mathbb{I}_{\out(\nu_1)} \: \mathbb{I}_{\out(\nu_1)} + \end{bmatrix} ,\mymathbb{0}_{2\out(\nu_1),1} \rp + \end{align} + Applying Definition \ref{5.2.1}, specifically the second case, (\ref{5.4.3}) and ($\ref{5.4.4}$) yields that: + \begin{align} + &\lb \nu_1 \boxminus \nu_2 \rb \bullet \aff_{\lb \mathbb{I}_{\inn \lp \nu_2 \rp } \: \mathbb{I}_{\inn \lp \nu_2 \rp} \rb^T,\mymathbb{0}_{2\inn(\nu_2),1}} \nonumber \\ + &= \lp \lp \begin{bmatrix} + W_1 & 0 \\ + 0 & W'_1 + \end{bmatrix} \begin{bmatrix} + \mathbb{I}_{\inn(\nu_1)} \\ + \mathbb{I}_{\inn(\nu_1)} + \end{bmatrix}, \begin{bmatrix} + b_1 \\ + b_1' + \end{bmatrix} + \rp, + \lp \begin{bmatrix} + W_2 & 0 \\ + 0 & W'_2 + \end{bmatrix}, \begin{bmatrix} + b_2 \\ + b_2' + \end{bmatrix} \rp \right.,..., \nonumber + \left. \lp \begin{bmatrix} + W_L & 0 \\ + 0 & W'_L + \end{bmatrix}, \begin{bmatrix} + b_L \\ + b_L' + \end{bmatrix} \rp \rp \nonumber \\ + &= \lp \lp \begin{bmatrix} + W_1\\ + W'_1 + \end{bmatrix} ,\begin{bmatrix} + b_1 \\ + b_1' + \end{bmatrix} + \rp, + \lp \begin{bmatrix} + W_2 & 0\\ + 0 & W'_2 + \end{bmatrix}, \begin{bmatrix} + b_2 \\ + b_2' + \end{bmatrix} \rp \right.,..., \nonumber + \left. \lp \begin{bmatrix} + W_L & 0\\ + 0 & W'_L + \end{bmatrix}, \begin{bmatrix} + b_L \\ + b_L' + \end{bmatrix} \rp \rp \nonumber + \end{align} + Applying Claim \ref{5.4.5} and especially the third case of Definition \ref{5.2.1} to to the above then gives us: + \begin{align} + &\aff_{\lb \mathbb{I}_{\out(\nu_1)} \: \mathbb{I}_{\out(\nu_1)} \rb,0}\bullet \lb \nu_1 \boxminus \nu_2 \rb \bullet \aff_{\lb \mathbb{I}_{\inn \lp \nu_2 \rp } \: \mathbb{I}_{\inn \lp \nu_2 \rp} \rb^T,0} \nonumber\\ + &= \lp \lp \begin{bmatrix} + W_1 \\ + W'_1 + \end{bmatrix} ,\begin{bmatrix} + B_1 \\ + B_1' + \end{bmatrix} + \rp, + \lp \begin{bmatrix} + W_2 & 0\\ + 0 & W'_2 + \end{bmatrix} \begin{bmatrix} + b_2 \\ + b_2' + \end{bmatrix} \rp \right.,..., \nonumber + \left. \lp \begin{bmatrix} + \mathbb{I}_{\out(\nu_2)} \: \mathbb{I}_{\out(\nu_2)} + \end{bmatrix}\begin{bmatrix} + W_L & 0 \\ + 0 & W'_L + \end{bmatrix}, \begin{bmatrix} + \mathbb{I}_{\out(\nu_2)} \: \mathbb{I}_{\out(\nu_2)} + \end{bmatrix}\begin{bmatrix} + b_L \\ + b_L' + \end{bmatrix} \rp \rp \nonumber \\ + & =\lp \lp \begin{bmatrix} + W_1\\ + W'_1 + \end{bmatrix} ,\begin{bmatrix} + b_1 \\ + b_1' + \end{bmatrix} + \rp, + \lp \begin{bmatrix} + W_2 & 0\\ + 0 & W'_2 + \end{bmatrix}, \begin{bmatrix} + b_2 \\ + b_2' + \end{bmatrix} \rp \right.,..., + \left. \lp \begin{bmatrix} + W_L \quad W'_L \label{5.4.10} + \end{bmatrix}, b_L + b_L' \rp \rp + \end{align} + Now note that: + \begin{align} + \nu_2 \boxminus \nu_1 = \lp\lp + \begin{bmatrix} + W_1' & 0 \\ + 0 & W_1 + \end{bmatrix}, \begin{bmatrix} + b_1' \\ + b_1 + \end{bmatrix} + \rp,\lp \begin{bmatrix} + W_2' & 0 \\ + 0 & W_2 + \end{bmatrix}, \begin{bmatrix} + b_2'\\ + b_2 + \end{bmatrix}\rp,..., \right. \nonumber\\ + \left. + \lp \begin{bmatrix} + W_L' & 0 \\ + 0 & W_L + \end{bmatrix}, \begin{bmatrix} + b_L' \\ + b_L + \end{bmatrix} \rp \rp \nonumber + \end{align} + And thus: + \begin{align} + &\aff_{\lb \mathbb{I}_{\out(\nu_2)} \: \mathbb{I}_{\out(\nu_2)} \rb,0}\bullet \lb \nu_2 \boxminus \nu_1 \rb \bullet \aff_{\lb \mathbb{I}_{\inn \lp \nu_1 \rp } \: \mathbb{I}_{\inn \lp \nu_1 \rp} \rb^T,0} \nonumber\\ + &= \lp \lp \begin{bmatrix} + W'_1\\ + W_1 + \end{bmatrix} ,\begin{bmatrix} + b'_1 \\ + b_1 + \end{bmatrix} + \rp, + \lp \begin{bmatrix} + W'_2 & 0\\ + 0 & W_2 + \end{bmatrix}, \begin{bmatrix} + b'_2 \\ + b_2 + \end{bmatrix} \rp \right.,..., + \left. \lp \begin{bmatrix} + W'_L \quad W_L + \end{bmatrix}, \begin{bmatrix} + b_L' + b_L + \end{bmatrix} \rp \rp \label{5.4.11} + \end{align} + Let $x \in \R^{\inn(\nu_1)}$, note then that: + \begin{align} + \begin{bmatrix} + W_1 \\ + W'_1 + \end{bmatrix}x + \begin{bmatrix} + b_1\\ + b'_1 + \end{bmatrix} = \begin{bmatrix} + W_1x+b_1 \\ + W'_1x+b_1' + \end{bmatrix} \nonumber + \end{align} + The full instantiation of (\ref{5.4.10}) is then given by: + \begin{align} + \real \lp \begin{bmatrix} + W_L \quad W'_L + \end{bmatrix}\begin{bmatrix} + W_{L-1}(...(W_2\lp W_1x+b_1 \rp + b_2) + ... )+ b_{L-1} \\ + W'_{L-1}(...(W'_2 \lp W'_1x + b'_1 \rp + b'_2)+...)+b'_{L-1} + \end{bmatrix} + b_L+b'_L \rp \label{5.4.12} + \end{align} + The full instantiation of (\ref{5.4.11}) is then given by: + \begin{align} + \real \lp \begin{bmatrix} + W_L' \quad W_L + \end{bmatrix}\begin{bmatrix} + W'_{L-1}(...(W'_2\lp W'_1x+b'_1 \rp + b'_2) + ... )+ b'_{L-1} \\ + W_{L-1}(...(W_2 \lp W_1x + b_1 \rp + b_2)+...)+b_{L-1} + \end{bmatrix} + b_L+b'_L \rp \label{5.4.13} + \end{align} + Since (\ref{5.4.12}) and (\ref{5.4.13}) are the same this proves that $\nu_1 \oplus \nu_2 = \nu_2 \oplus \nu_1$. + \end{proof} + This is a special case of \cite[Lemma~3.28]{Grohs_2022}. + \begin{lemma}\label{5.4.7} + Let $ l_0,l_1,...,l_L \in \N$. Let $\nu \in \neu$ with $\lay(\nu) = \lp l_0,l_1,...,l_L \rp$. There then exists a neural network $\zero_{l_0,l_1,...,l_L} \in \neu$ such that $\real(\nu \oplus \zero_{l_0,l_1,...,l_L}) = \real(\zero_{l_0,l_1,...,l_L} \oplus \nu) = \nu $. + \end{lemma} +\begin{proof} + Let $\nu = \lp \lp W_1, b_1 \rp, \lp W_2, b_2 \rp,..., \lp W_L,b_L \rp \rp$, where $W_1 \in \R^{l_1\times l_0}$, $b_1 \in \R^{l_1}$, $W_2 \in \R^{l_2 \times l_1}$, $b_2 \in \R^{l_2},...,W_L \in \R^{l_L \times l_{L-1}}$, $b_L \in \R^{l_L}$. Denote by $\zero_{l_0,l_1,...,l_L}$ the neural network which for all $l_0,l_1,...,l_L \in \N$ is given by: + \begin{align} + \zero_{l_0,l_1,...,l_L} = \lp \lp \mymathbb{0}_{l_1, l_0}, \mymathbb{0}_{l_1} \rp, \lp \mymathbb{0}_{l_2,l_1},\mymathbb{0}_{l_2} \rp,...,\lp \mymathbb{0}_{l_{L},l_{L-1}}, \mymathbb{0}_{l_L} \rp \rp + \end{align} + Thus, by (\ref{5.4.12}), we have that: + \begin{align} + \real(\zero_{l_0,l_1,...,l_L} \oplus\nu) &= \begin{bmatrix} + 0 \quad W_L + \end{bmatrix} \begin{bmatrix} + 0 \nonumber \\ + W_{L-1}(...(W_2 \lp W_1x+b_1 \rp +b_2)+...)+b_{L-1} + \end{bmatrix} + b_L \\ + &= W_L(W_{L-1}(...W_2\lp W_1x+b_1 \rp +b_2)+...)+b_{L-1})+b_L + \end{align} + \begin{align} + \real(\nu \oplus \zero_{l_0,l_1,...,l_L}) &= \begin{bmatrix} + W_L \quad 0 + \end{bmatrix} \begin{bmatrix} + W_{L-1}(...(W_2 \lp W_1x+b_1 \rp +b_2)+...)+b_{L-1} \nonumber \\ + 0 + \end{bmatrix} + b_L \\ + &= W_L(W_{L-1}(...W_2\lp W_1x+b_1 \rp +b_2)+...)+b_{L-1})+b_L + \end{align} + And finally: + \begin{align}\label{5.4.17} + \real(\nu) = W_L(W_{L-1}(...W_2\lp W_1x+b_1 \rp +b_2)+...)+b_{L-1})+b_L + \end{align} + This completes the proof. +\end{proof} +\begin{lemma}\label{5.4.8} + Given neural networks $\nu_1,\nu_2,\nu_3 \in \neu$ with fixed depth $L$, fixed starting width of $l_0$ and fixed finishing width of $l_L$, it is then the case that $\real\lp \lp \nu_1 \oplus \nu_2 \rp \oplus \nu_3 \rp = \real \lp \nu_1 \oplus \lp \nu_2 \oplus \nu_3 \rp \rp$, i.e. the instantiation with a continuous activation function of $\oplus$ is associative. +\end{lemma} +\begin{proof} + Let $\nu_1 = \lp \lp W^1_1,b^1_1 \rp, \lp W^1_2,b^1_2 \rp, ..., \lp W^1_L,b^1_L \rp \rp$, $\nu_2 = \lp \lp W^2_1,b^2_1 \rp, \lp W^2_2,b^2_2 \rp,..., \lp W^2_L, b^2_L \rp \rp$, and $\nu_3 = \lp \lp W^3_1,b^3_1 \rp ,\lp W^3_2,b^3_2 \rp,..., \lp W^3_L,b^3_L \rp \rp$. Then (\ref{5.4.12}) tells us that: + \begin{align} + \real(\nu_1 \oplus \nu_2) =\begin{bmatrix} + W^1_L \quad W^2_L + \end{bmatrix}\begin{bmatrix} + W^1_{L-1}\lp...\lp W^1_2\lp W^1_1x+b^1_1 \rp + b^1_2 \rp + ... \rp + b^1_{L-1} \\ + W^2_{L-1}\lp...\lp W^2_2 \lp W^2_1x + b^2_1 \rp + b^2_2 \rp +...\rp+b^2_{L-1} + \end{bmatrix} + b^1_L+b^2_L \nonumber + \end{align} + And thus: + \begin{align}\label{5.4.18} + &\real \lp \lp \nu_1 \oplus \nu_2 \rp \oplus \nu_3 \rp \lp x \rp = \nonumber\\ + &\real \lp \begin{bmatrix} + \mathbb{I} \quad W^3_L + \end{bmatrix}\begin{bmatrix} + \begin{bmatrix} + W^1_L \quad W^2_L + \end{bmatrix}\begin{bmatrix} + W^1_{L-1}\lp...\lp W^1_2\lp W^1_1x+b^1_1 \rp + b^1_2 \rp + ... \rp + b^1_{L-1} \\ + W^2_{L-1}\lp...\lp W^2_2 \lp W^2_1x + b^2_1 \rp + b^2_2 \rp +...\rp+b^2_{L-1} + \end{bmatrix} + b^1_L+b^2_L \\ + W^3_{L-1}\lp...\lp W^3_2 \lp W^3_1x + b^3_1 \rp + b^3_2 \rp +...\rp+b^3_{L-1} + \end{bmatrix} +b^3_L \rp + \end{align} + Similarly, we have that: + \begin{align}\label{5.4.19} + &\real_{\act} \lp \nu_1 \oplus \lp \nu_2 \oplus \nu_3 \rp \rp \lp x \rp = \nonumber\\ + &\real \lp \begin{bmatrix} + W^1_L & \mathbb{I} + \end{bmatrix}\begin{bmatrix} + W^1_{L-1}\lp...\lp W^1_2\lp W^1_1x+b^1_1 \rp + b^1_2 \rp + ... \rp + b^1_{L-1} \\ + \begin{bmatrix} + W^2_L \quad W^3_L + \end{bmatrix}\begin{bmatrix} + W^2_{L-1}\lp...\lp W^2_2\lp W^2_1x+b^2_1 \rp + b^2_2 \rp + ... \rp + b^2_{L-1} \\ + W^3_{L-1}\lp...\lp W^3_2 \lp W^3_1x + b^3_1 \rp + b^3_2 \rp +...\rp+b^3_{L-1} + \end{bmatrix} + b^2_L+b^3_L + \end{bmatrix} +b^1_L \rp + \end{align} + Note that the associativity of matrix-vector multiplication ensures that (\ref{5.4.18}) and (\ref{5.4.19}) are the same. +\end{proof} +\begin{definition}[Commutative Semi-group] +%TODO: Modify the monoid definition; the additive identity is not unique. + A set $X$ equipped with a binary operation $*$ is called a monoid if: + \begin{enumerate}[label = (\roman*)] + \item for all $x,y,z \in X$ it is the case that $(x *y)*z = x*(y*z)$ and + \item for all $x,y \in X$ it is the case that $x*y=y*x$ + \end{enumerate} +\end{definition} +\begin{theorem} + For fixed depth and layer widths, the set of instantiated neural networks $\nu \in \neu$ form a commutative semi-group under the operation of $\oplus$. +\end{theorem} +\begin{proof} + This is a consequence of Lemmas \ref{5.4.6}, \ref{5.4.7}, and \ref{5.4.8}. +\end{proof} +\begin{lemma}\label{5.5.11}\label{nn_sum_is_sum_nn} + Let $\nu, \mu \in \neu$, with the same length and end-widths. It is then the case that $\real_{\act} \lp \nu \oplus \mu \rp = \real_{\act}\lp \nu \rp + \real_{\act}\lp \mu \rp$. +\end{lemma} +\begin{proof} + Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp,...,\lp W_L,b_L \rp \rp$ and $\mu = \lp \lp W'_1,b'_1 \rp, \lp W_2',b_2'\rp,...,\lp W_L',b_L' \rp \rp $. Note now that by (\ref{5.4.12}) we have that: + \begin{align}\label{5.5.20} + \real_{\act}\lp \nu \rp = W_L \act \lp W_{L-1}(...\act (W_2 \act \lp W_1x+b_1 \rp + b_2) + ... )+ b_{L-1}\rp + b_L + \end{align} + And: + \begin{align} + \real_{\act}\lp \mu \rp = W'_L\act \lp W'_{L-1}(...\act (W_2'\act \lp W_1'x+b_1' \rp + b_2') + ... )+ b_{L-1}'\rp + b_L' + \end{align} + In addition, because of the block matrix structure of the weights of our summands: + \begin{align} + \real_{\act}\lp \nu \oplus \mu \rp \lp x \rp&=\begin{bmatrix} + W_L \quad W'_L + \end{bmatrix}\begin{bmatrix} + \act \lp W_{L-1}(...\act ( W_2\act \lp W_1x+b_1 \rp + b_2) + ... )+ b_{L-1} \rp \\ + \act \lp W'_{L-1}(...\act( W'_2 \act \lp W'_1x + b'_1 \rp + b'_2)+...)+b'_{L-1} \rp + \end{bmatrix} + b_L+b'_L \label{5.4.12} \nonumber\\ + &= W_L \act \lp W_{L-1}(...\act (W_2 \act \lp W_1x+b_1 \rp + b_2) + ... )+ b_{L-1}\rp + b_L \nonumber \\ + &+ W'_L\act \lp W'_{L-1}(...\act (W_2'\act \lp W_1'x+b_1' \rp + b_2') + ... )+ b_{L-1}'\rp + b_L' \nonumber \\ + &=\real_{\act} \lp \nu\rp \lp x \rp + \real_{\act} \lp \mu\rp \lp x \rp + \end{align} + This proves the lemma. +\end{proof} +\begin{lemma}\label{nn_sum_cont} + Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$. It is then the case that: + \begin{align} + \real_{\act}\lp \bigoplus^n_{i=1} \nu_i \rp = \sum^n_{i=1} \real_{\rect} \lp \nu_i\rp + \end{align} +\end{lemma} + +\begin{proof} + This is the consequence of a finite number of applications of Lemma \ref{5.5.11}. +\end{proof} + + +\subsection{Sum of ANNs of Unequal Depth But Same End-widths} +\begin{definition}[Sum of ANNs of different depths but same end widths] + Let $n\in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$ such that they have the same end widths. We define the neural network $\dplus_{i=1}^n\nu_i \in \neu$, the neural network sum of neural networks of unequal depth as: + \begin{align} + \dplus^n_{i=1}\nu_i \coloneqq \lp \sm_{n,\out(\nu_2)} \bullet \lb \DDiamond^v_{i=u}\nu_i \rb \bullet \cpy_{n,\inn(\nu_1)} \rp + \end{align} +\end{definition} +\begin{lemma}\label{lem:diamondplus} + Let $n\in \N$. Let $\nu_1,\nu_2 \in \neu$ and assume also that they have the same end-widths. It is then the case that: + \begin{align} + \real_{\rect}\lp \nu_1 \dplus \nu_2\rp \lp x\rp = \real_{\rect}\lp \nu_1\rp + \real_{\rect}\lp \nu_2\rp + \end{align} +\end{lemma} +\begin{proof} + Note that Lemma \ref{6.2.2} tellls us that for all $n\in \N$ it is the case that $\real_{\rect} \lp \tun_n\rp \lp x\rp = x$. This combined with Lemma \ref{comp_prop} then tells us that for all $n\in \N$ it is the case for all $\nu \in \neu$ that: + \begin{align} + \real_{\rect} \lp \tun_n \bullet \nu \rp \lp x\rp = \real_{\rect} \lp \nu \rp \lp x\rp + \end{align} + Thus, this means that: + \begin{align} + \real_{\rect} \lp \nu_1 \dplus \nu_2\rp \lp x \rp &= \lp \sm_{n,\out(\nu_2)} \bullet \lb \nu_1 \DDiamond \nu_2\rb \bullet \cpy_{n,\inn(\nu_1)} \rp \nonumber\\ + &= \real_{\rect}\lp \nu_1\rp \lp x \rp + \real_{\rect} \lp \nu_2\rp \lp x \rp + \end{align} + This then proves the lemma. +\end{proof} +\begin{lemma} + Let $n \in \N$. Let $\nu_1,\nu_2,...,\nu_n \in \neu$. Let it also be the case that they have the same end-widths. It is then the case that: + \begin{align} + \real_{\rect}\lp \dplus^n_{i=1}\nu_i\rp \lp x\rp = \sum^n_{i=1}\real_{\rect}\lp \nu_i\rp \lp x \rp + \end{align} +\end{lemma} +\begin{proof} + This is a consequence of a finite number of applications of Lemma \ref{lem:diamondplus}. +\end{proof} + +\section{Linear Combinations of ANNs and Their Properties} +\begin{definition}[Scalar left-multiplication with an ANN]\label{slm} + Let $\lambda \in \R$. We will denote by $(\cdot ) \triangleright (\cdot ): \R \times \neu \rightarrow \neu$ the function that satisfy for all $\lambda \in \R$ and $\nu \in \neu$ that $\lambda \triangleright \nu = \aff_{\lambda \mathbb{I}_{\out(\nu)},0} \bullet \nu$. +\end{definition} +\begin{definition}[Scalar right-multiplication with an ANN] + Let $\lambda \in \R$. We will denote by $(\cdot) \triangleleft (\cdot): \neu \times \R \rightarrow \neu$ the function satisfying for all $\nu \in \neu$ and $\lambda \in \R$ that $\nu \triangleleft \lambda = \nu \bullet \aff_{\lambda \mathbb{I}_{\inn(\nu)},0}$. +\end{definition} +\begin{remark} + Note that whereas $\lambda \in \R$, the actual neural network in question, properly speaking, must always be referred to as $\lambda \triangleright$ or $\triangleleft\lambda$, and we shall do so whenever this comes up in any neural network diagrams. This is by analogy with, for example, $\log_\lambda$ or $\sqrt[\lambda ]{}$ for $\lambda \neq 0$, where the argument $\lambda$ is generally always written except for $\lambda = 10$ for the logarithm or $\lambda = 2$ for the root. +\end{remark} +\begin{remark} + For an \texttt{R} implementation, see Listing \ref{scalar_mult} +\end{remark} +\begin{lemma}\label{5.6.3} + Let $\lambda \in \R$ and $\nu \in \neu$. it is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\lay(\lambda \triangleright \nu) = \lay(\nu)$ + \item For all $\act \in C(\R, \R)$ that $\real_{\act}(\lambda \triangleright \nu) \in C \lp \R^{\inn(\nu)}, \R^{\out(\nu)} \rp $ + \item For all $\act \in C(\R,\R)$, and $x \in \R^{\inn(\nu)}$ that: + \begin{align} + \real_{\act} \lp \lambda \triangleright \nu \rp = \lambda \real_{\act}(\nu) + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + Let $\nu \in \neu$ such that $\lay(\nu) = \lp l_1,l_2,...,l_L \rp$ and $\dep(\nu) = L$ where $l_1,l_2,...,l_L,L \in \N$. Then Item (i) of Lemma $\ref{5.3.2}$ tells us that: + \begin{align} + \lay \lp \aff_{\mathbb{I}_{\out(\nu)},0}\rp = \lp \out(\nu), \out(\nu) \rp + \end{align} + This and Item (i) from Lemma \ref{5.3.3} gives us that: + \begin{align} + \lay \lp \lambda \triangleright \nu \rp = \lay \lp \aff_{\lambda \mathbb{I}_{\out(\nu)},0} \bullet \nu \rp = \lp l_0, l_1,...,l_{L-1}, \out(\nu) \rp = \lay(\nu) + \end{align} + Which proves $(i)$. Item $(ii)-(iii)$ of Lemma $\ref{5.3.2}$ then prove that for all $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu)}$, that $\real_{\act} \lp \lambda \triangleright \nu \rp \in C \lp \R^{\inn(\nu),\out(\nu)} \rp$ given by: + \begin{align} + \lp \real_{\act} \lp \lambda \triangleright \nu \rp \rp \lp x \rp &= \lp \real_{\act} \lp \aff_{\lambda \mathbb{I}_{\out(\nu),0}} \bullet \nu \rp \rp \lp x \rp \nonumber\\ + &= \lambda \mathbb{I}_{\out(\nu)} \lp \lp \real_{\act} \lp \nu \rp \rp \lp x \rp \rp = \lambda \lp \lp \real_{\act} \lp \nu \rp \rp \lp x \rp \rp + \end{align} + This establishes Items (ii)\textemdash(iii), completing the proof. +\end{proof} + +\begin{lemma}\label{5.6.4} + Let $\lambda \in \R$ and $\nu \in \neu$. It is then the case that: + \begin{enumerate}[label = (\roman*)] + \item $\lay(\nu \triangleleft \lambda) = \lay(\nu)$ + \item For all $\act \in C \lp \R, \R \rp$ that $\real_{\act}(\nu \triangleleft \lambda) \in C \lp \R^{\inn(\nu)}, \R^{\out(\nu)} \rp$ + \item For all $\act \in C \lp \R, \R \rp$, and $x \in \R^{\inn(\nu)}$ that: + \begin{align} + \real_{\act} \lp \nu \triangleleft \lambda \rp = \real_{\act}(\nu)\lp \lambda x \rp + \end{align} + \end{enumerate} +\end{lemma} + \begin{proof} + Let $\nu \in \neu$ such that $\lay(\nu) = \lp l_1,l_2,...,l_L \rp$ and $\dep(\nu) = L$ where $\l_1,l_2,...,l_L, L \in \N$. Then Item (i) of Lemma \ref{5.3.2} tells us that: + \begin{align} + \lay \lp \aff_{\mathbb{I}_{\inn(\nu)},0} \rp = \lp \inn(\nu), \inn(\nu) \rp + \end{align} + This and Item (iv) of Lemma \ref{5.3.3} tells us that: + \begin{align} + \lay(\nu \triangleleft\lambda) = \lay \lp \nu \bullet \aff_{\lambda \mathbb{I}_{\inn(\nu)}}\rp = \lp \inn(\nu), l_1,l_2,...,l_L \rp = \lay(\nu) + \end{align} + Which proves $(i)$. Item (v)--(vi) of Lemma \ref{5.3.3} then prove that for all $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu)}$ that $\real_{\act} \lp \nu \triangleleft \lambda \rp \in C\lp \R^{\inn(\nu),\out(\nu)} \rp$ given by: + \begin{align} + \lp \real_{\act} \lp \nu \triangleleft \lambda \rp \rp \lp x \rp &= \lp \real_{\act} \lp \nu \bullet \aff_{\lambda \mathbb{I}_{\inn(\nu),0}} \rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\act} \lp \nu \rp \rp \lp \aff_{\lambda \mathbb{I}_{\inn(\nu)}} \rp \lp x \rp \nonumber\\ + &= \lp \real_{\act} \lp \nu \rp \rp \lp \lambda x \rp + \end{align} + This completes the proof. + \end{proof} + +\begin{lemma}\label{scalar_right_mult_distribution} + Let $\nu,\mu \in \neu$ with the same length and end-widths, and $\lambda \in \R$. It is then the case, for all $\act \in C \lp \R, \R \rp$ that: + \begin{align} + \real_{\act} \lp \lp \nu \oplus \mu \rp \triangleleft \lambda \rp \lp x \rp &= \real_{\act} \lp \lp \nu \triangleleft \lambda \rp \oplus \lp \mu \triangleleft \lambda \rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\act}\lp \nu \rp \rp \lp \lambda x \rp + \lp \real_{\act} \lp \mu \rp \rp \lp \lambda x \rp \nonumber + \end{align} +\end{lemma} +\begin{proof} + Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp,...,\lp W_L,b_L \rp \rp$ and $\mu = \lp \lp W'_1,b'_1 \rp, \lp W'_2,b'_2 \rp,...,\lp W'_L,b'_L \rp \rp$. Then from Lemma \ref{5.6.4} and (\ref{5.4.12}) we have that: + \begin{align} + &\lp \real_{\act}\lp \nu \oplus \mu \rp \triangleleft \lambda \rp \lp x \rp \nonumber\\ &= \lp \real_{\act} \lp \nu \oplus \mu \rp \rp \lp \lambda x \rp \nonumber\\ + &= \begin{bmatrix} + W_L \quad W'_L + \end{bmatrix}\begin{bmatrix} + \inst_{\rect} \lp W_{L-1}(...(\inst_{\rect} \lp W_2\lp \inst_{\rect} \lp W_1\lambda x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp \\ + \inst_{\rect} \lp W'_{L-1}(...(\inst_{\rect} \lp W'_2\lp \inst_{\rect} \lp W'_1\lambda x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp \\ + \end{bmatrix} + b_L+b'_L \nonumber + \end{align} + Note that: + \begin{align} + \lp \real_{\act} \lp \nu \rp \rp \lp \lambda x \rp = W_L \cdot \inst_{\rect} \lp W_{L-1}(...(\inst_{\rect} \lp W_2\lp \inst_{\rect} \lp W_1\lambda x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp + b_L + \end{align} + and that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp \lambda x \rp = W'_L\cdot\inst_{\rect} \lp W'_{L-1}(...(\inst_{\rect} \lp W'_2\lp \inst_{\rect} \lp W'_1\lambda x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp + b'_L + \end{align} + This, together with Lemma \ref{5.5.11}, completes the proof. +\end{proof} +\begin{lemma}\label{scalar_left_mult_distribution} + Let $\nu,\mu \in \neu$ with the same length and end-widths, and $\lambda \in \R$. It is then the case, for all $\act \in C \lp \R, \R \rp$ that: + \begin{align} + \real_{\act} \lp \lambda \triangleright\lp \nu \oplus \mu \rp \rp \lp x \rp &= \real_{\act} \lp \lp \lambda \triangleright\nu \rp \oplus \lp \lambda \triangleright\mu \rp \rp \lp x \rp \nonumber\\ + &= \lambda \cdot \lp \real_{\act}\lp \nu \rp \rp \lp x \rp + \lambda \cdot \lp \real_{\act} \lp \mu \rp \rp \lp x \rp \nonumber + \end{align} +\end{lemma} +\begin{proof} + Let $\nu = \lp \lp W_1,b_1 \rp, \lp W_2,b_2 \rp,...,\lp W_L,b_L \rp \rp$ and $\mu = \lp \lp W'_1,b'_1 \rp, \lp W'_2,b'_2 \rp,...,\lp W'_L,b'_L \rp \rp$. Then from Lemma \ref{5.6.4} and (\ref{5.4.12}) we have that: + \begin{align} + & \real_{\act}\lp \lambda \lp \nu \oplus \mu \rp \rp \lp x \rp \nonumber\\ &= \real_{\act} \lp \lambda \triangleright \lp \nu \oplus \mu \rp\rp \lp \lambda x \rp \nonumber\\ + &= \lambda \cdot \begin{bmatrix} + W_L \quad W'_L + \end{bmatrix}\begin{bmatrix} + \inst_{\rect} \lp W_{L-1}(...(\inst_{\rect} \lp W_2\lp \inst_{\rect} \lp W_1 x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp \\ + \inst_{\rect} \lp W'_{L-1}(...(\inst_{\rect} \lp W'_2\lp \inst_{\rect} \lp W'_1x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp \\ + \end{bmatrix} + b_L+b'_L \nonumber + \end{align} + Note that: + \begin{align} + \lambda\cdot\lp \real_{\act} \lp \nu \rp \rp \lp x \rp = W_L \cdot \inst_{\rect} \lp W_{L-1}(...(\inst_{\rect} \lp W_2\lp \inst_{\rect} \lp W_1x+b_1 \rp \rp + b_2)\rp + ... )+ b_{L-1}\rp + b_L + \end{align} + and that: + \begin{align} + \lambda \cdot \lp \real_{\act} \lp \mu \rp \rp \lp x \rp = W'_L\cdot\inst_{\rect} \lp W'_{L-1}(...(\inst_{\rect} \lp W'_2\lp \inst_{\rect} \lp W'_1 x+b'_1 \rp \rp + b'_2)\rp + ... )+ b'_{L-1}\rp + b'_L + \end{align} + This, together with Lemma \ref{5.5.11}, completes the proof. +\end{proof} +\begin{lemma}\label{5.6.5} + Let $u,v \in \Z$ with $u \leqslant v$ and $n = v-u+1$. Let $\lambda_u,\lambda_{u+1},..., \lambda_v \in \R$. Let $\nu_u, \nu_{u+1},...,\nu_v, \mu \in \neu$, $B_{u}, B_{u+1},...,B_v \in \R^{\inn(\mu)}$ satisfy that $\lay (\nu_u) = \lay(\nu_{u+1}) = ...= \lay(\nu_v)$ and further that: + \begin{align} + \mu = \lb \oplus^v_{i=u} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_1)},B_i} \rp \rp \rb + \end{align} + It then holds: + \begin{enumerate}[label = (\roman*)] + \item That: + \begin{align} + \lay (\mu) &= \lp \inn(\nu_u), \sum^v_{i=u} \wid_1 \lp \nu_u \rp , \sum^v_{i=u} \wid_2 \lp \nu_u \rp,..., \sum^v_{i=u} \wid_{\dep(\nu_u)-1} \lp \nu_u \rp , \out(\nu_u) \rp \nonumber\\ + &= \lp \inn(\nu_u), n\wid_1(\nu_u), n\wid_2(\nu_u),...,n\wid_{\dep(\nu_u -1)}, \out(\nu_u) \rp \nonumber + \end{align} + \item that for all $\act \in C \lp \R ,\R \rp$, that $\real_{\act} (\mu) \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $, and + \item for all $\act \in C \lp \R, \R \rp $ and $x \in \R^{\inn(\nu_u)}$ that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp x \rp = \sum^v_{i=u} c_i \lp \real_{\act} \lp \nu_i \rp \rp \lp x + B_i \rp + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + Assume hypothesis that $\lay(\nu_u) = \lay(\nu_{u+1}) = ... = \lay(\nu_v)$. Note that Item (i) of Lemma \ref{5.3.2} gives us that for all $i \in \{u,u+1,...,v\}$ that: + \begin{align} + \lay \lp \aff_{\mathbb{I}_{\inn(\nu_i),B_i}} \rp = \lay \lp \aff_{\mathbb{I}_{\inn(\nu_u)}} \rp = \lp \inn \lp \nu_u \rp, \inn \lp \nu_u \rp \rp \in \N^2 + \end{align} + This together with Lemma \ref{comp_prop}, Item (i), assures us that for all $i \in \{ u,u+1,...,v\}$ it is the case that: + \begin{align}\label{5.3.15} + \lay\lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \rp = \lp \inn(\nu_u), \wid_1 \lp \nu_u \rp, \wid_2 \lp \nu_u \rp,..., \wid_{\dep(\nu_u)} \lp \nu_u \rp \rp + \end{align} + This and \cite[Lemma~3.14, Item~(i)]{Grohs_2022} tells us that for all $i \in \{u,u+1,...,v\}$ it is the case that: + \begin{align}\label{5.6.13} + \lay \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i), B_i}} \rp \rp = \lay \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i),B_i}} \rp + \end{align} + This, (\ref{5.3.15}), and \cite[Lemma~3.28, Item~(ii)]{Grohs_2022} then yield that: + \begin{align} + \lay(\mu) &= \lay \lp \oplus^v_{i=u} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i),B_i}} \rp \rp \rp \nonumber\\ + &= \lp \inn(\nu_u), \sum^v_{i=u} \wid_1 \lp \nu_u \rp,\sum^v_{i=u} \wid_2 \lp \nu_u \rp,..., \sum^v_{i=u} \wid_{\dep(\nu_u)-1} \lp \nu_u \rp , \out \lp \nu_u \rp \rp \nonumber \\ + &= \lp \inn(\nu_u), n\wid_1(\nu_u), n\wid_2 ( \nu_u),...,n\wid_{\dep(\nu_u)-1}(\nu_u), \out(\nu_u) \rp + \end{align} + This establishes item (i). Items (v) and (vi) from Lemma \ref{5.3.3} tells us that for all $i \in \{ u,u+1,...,v\}$, $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu_u)}$, it is the case that $\real_{\act}\lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \rp \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $ and further that: + \begin{align} + \lp \real_{\act} \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \lp x \rp = \lp \real_{\act} \lp \nu_i \rp \rp \lp x + b_i \rp + \end{align} + This along with \cite[Lemma~3.14]{Grohs_2022} ensures that for all $i \in \{u,u+1,...,v\}$, $\act \in C \lp \R, \R \rp$, $x \in \R^{\inn(\nu_u)}$, it is the case that: + \begin{align} + \real_{\act} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},B_i} \rp \rp \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)}\rp + \end{align} + and: + \begin{align} + \lp \real_{\act} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \lp x \rp = c_i \lp \real_{\act} \lp \nu_i \rp \rp \lp x + b_i \rp + \end{align} + Now observe that \cite[Lemma~3.28]{Grohs_2022} and (\ref{5.6.13}) ensure that for all $\act \in C \lp \R, \R \rp$, $x \in \R^{\inn(\nu_u)}$, it is the case that $\real_{\act} \lp \mu \rp \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp$ and that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp x \rp &= \lp \real_{\act} \lp \oplus^v_{i=u} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \rp \lp x \rp \nonumber\\ + &= \sum^v_{i=u} \lp \real_{\act} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \lp x \rp \nonumber\\ + &=\sum^v_{i=u} c_i \lp \real_{\act} \lp \nu_i \rp \rp \lp x+b_i \rp \nonumber + \end{align} + This establishes items (ii)--(iii); thus, the proof is complete. +\end{proof} + +\begin{lemma}\label{5.6.6} + Let $u,v \in \Z$ with $u \leqslant v$. Let $\lambda_u,\lambda_{u+1},..., \lambda_v \in \R$. Let $\nu_u, \nu_{u+1},...,\nu_v, \mu \in \neu$, $B_{u}, B_{u+1},...,B_v \in \R^{\inn(\mu)}$ satisfy that $\lay (\nu_u) = \lay(\nu_{u+1}) = ...= \lay(\nu_v)$ and further that: + \begin{align} + \mu = \lb \oplus^v_{i=u} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_1)},b_i} \bullet \nu \rp \triangleleft c_i \rp \rb + \end{align} + It then holds: + \begin{enumerate}[label = (\roman*)] + \item That: + \begin{align} + \lay (\mu) &= \lp \inn(\nu_u), \sum^v_{i=u} \wid_1 \lp \nu_u \rp , \sum^v_{i=u} \wid_2 \lp \nu_u \rp,..., \sum^v_{i=u} \wid_{\dep(\nu_u)-1} \lp \nu_u \rp , \out(\nu_u) \rp \nonumber\\ + &= \lp \inn(\nu_u), n\wid_1(\nu_u), n\wid_2(\nu_u),...,n\wid_{\dep(\nu_u -1)}, \out(\nu_u) \rp + \end{align} + \item that for all $\act \in C \lp \R ,\R \rp$, that $\real_{\act} (\mu) \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $, and + \item for all $\act \in C \lp \R, \R \rp $ and $x \in \R^{\inn(\nu_u)}$ that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp x \rp = \sum^v_{i=u} \lp \real_{\act} \lp \nu_i \rp \rp \lp c_ix + b_i \rp + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + Assume hypothesis that $\lay(\nu_u) = \lay(\nu_{u+1}) = ... = \lay(\nu_v)$. Note that Item (i) of Lemma \ref{5.3.2} gives us that for all $i \in \{u,u+1,...,v\}$ that: + \begin{align} + \lay \lp \aff_{\mathbb{I}_{\inn(\nu_i),B_i}} \rp = \lay \lp \aff_{\mathbb{I}_{\inn(\nu_u)}} \rp = \lp \inn \lp \nu_u \rp, \inn \lp \nu_u \rp \rp \in \N^2 + \end{align} + Note then that Lemma \ref{comp_prop}, Item (ii), tells us that for all $i \in \{u,u+1,...,v\}$ it is the case that: + \begin{align}\label{5.6.22} + \lay\lp \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \bullet \nu \rp = \lp \inn(\nu_u), \wid_1 \lp \nu_u \rp, \wid_2 \lp \nu_u \rp,..., \wid_{\dep(\nu_u)} \lp \nu_u \rp \rp + \end{align} + This and Item (i) of Lemma \ref{5.6.4} tells us that for all $i \in \{u,u+1,...,v\}$ it is the case that: + \begin{align}\label{5.6.23} + \lay \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i), b_i}} \bullet \nu \rp \triangleleft c_i \rp = \lay \lp \aff_{\mathbb{I}_{\inn(\nu_i),b_i} } \bullet\nu \rp + \end{align} + This, (\ref{5.6.22}), and \cite[Lemma~3.28, Item ~(ii)]{Grohs_2022} tell us that: + \begin{align} + \lay(\mu) &= \lay \lp \oplus^v_{i=u} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i),b_i}} \bullet \nu_i\rp \triangleleft c_i\rp \rp \nonumber\\ + &= \lp \inn(\nu_u), \sum^v_{i=u} \wid_1 \lp \nu_u \rp,\sum^v_{i=u} \wid_2 \lp \nu_u \rp,..., \sum^v_{i=u} \wid_{\dep(\nu_u)-1} \lp \nu_u \rp , \out \lp \nu_u \rp \rp \nonumber \\ + &= \lp \inn(\nu_u), n\wid_1(\nu_u), n\wid_2 ( \nu_u),...,n\wid_{\dep(\nu_u)-1}(\nu_u), \out(\nu_u) \rp + \end{align} + This establishes Item (i). Items (i) and (ii) from Lemma \ref{5.3.3} tells us that for all $i \in \{ u,u+1,...,v\}$, $\act \in C(\R,\R)$, $x \in \R^{\inn(\nu_u)}$, it is the case that $\real_{\act}\lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \rp \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $ and further that: + \begin{align} + \lp \real_{\act} \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \rp \lp x \rp = \lp \real_{\act} \lp \nu_i \rp \rp \lp x \rp + b_i + \end{align} + This along with Lemma \ref{5.6.4} ensures that for all $i \in \{u,u+1,...,v\}$, $\act \in C \lp \R, \R \rp$, $x \in \R^{\inn(\nu_u)}$, it is the case that: + \begin{align} + \real_{\act} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i\rp \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)}\rp + \end{align} + and: + \begin{align} + \lp \real_{\act} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i \rp \triangleleft c_i \rp \rp \lp x \rp = \lp \real_{\act} \lp \nu_i \rp \rp \lp c_i x + b_i \rp + \end{align} + Now observe that \cite[Lemma~3.28]{Grohs_2022} and (\ref{5.5.14}) ensure that for all $\act \in C \lp \R, \R \rp$, $x \in \R^{\inn(\nu_u)}$, it is the case that $\real_{\act} \lp \mu \rp \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp$ and that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp x \rp &= \lp \real_{\act} \lp \oplus^v_{i=u} \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i \rp \rp \triangleleft c_i \rp \lp x \rp \\ + &= \sum^v_{i=u} \lp \real_{\act} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i \rp \rp \lp x \rp\\ + &=\sum^v_{i=u} \lp \real_{\act} \lp \nu_i \rp \rp \lp c_i x+b_i \rp \nonumber + \end{align} + This establishes items (ii)--(iii); thus, the proof is complete. +\end{proof} +\begin{lemma}\label{5.6.9} + Let $L \in \N$, $u,v \in \Z$ with $u\leqslant v$. Let $c_u, c_{u+1},...,c_v \in \R$. $\nu_u, \nu_{u+1},...,\nu_v, \mu, \mathfrak{I} \in \neu$, $B_u, B_{u+1},...,B_v \in \R^{\inn(\nu_u)}$, $\act \in C\lp \R, \R \rp$, satisfy for all $j \in \N \cap [u,v]$ that $L = \max_{i\in \N \cap \lb u,v \rb} \dep(\nu_i)$, $\inn(\nu_j) = \inn(\nu_u)$, $\out(\nu_j) = \inn(\mathfrak{I})= \out(\mathfrak{I})$, $\hid(\mathfrak{I}) = 1$, $\real_{\act} (\mathfrak{I}) = \mathbb{I}_\R$, and that: + \begin{align} + \mu = \boxplus^v_{i = u, \mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i), },b_i} \rp \rp + \end{align} + We then have: + \begin{enumerate}[label = (\roman*)] + \item it holds that: + \begin{align} + \lay(\mu) = \lp \inn(\nu_u ), \sum^v_{i=u}\wid_1 \lp \ex_{L,\mathfrak{I}} \lp \nu_i \rp \rp ,\sum^v_{i=u}\wid_2 \lp \ex_{L,\mathfrak{I}} \lp \nu_i\rp\rp,...,\sum^v_{i=u} \wid_{L-1} \lp \ex_{I,\mathfrak{I}} \lp \nu_i \rp , \out \lp \nu_u \rp \rp \rp + \end{align} + \item it holds that $\real_{\act}(\mu) \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $, and that, + \item it holds for all $ x \in \R^{\inn(\nu_u)}$ that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp x \rp = \sum^v_{i=u} c_i \lp \real_{\act} \lp \nu_i \rp \rp \lp x + b_i\rp + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + Note that Item(i) from Lemma \ref{5.6.5} establish Item(i) and (\ref{5.5.20}); in addition, items (v) and (vi) from Lemma \ref{5.3.3} tell us that for all $i \in \N \cap [u,v]$, $x \in \R^{\inn(\nu_u}$, it holds that $\real_{\act} \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)}\rp \rp $ and further that: + \begin{align} + \lp \real_{\act} \lp \nu_i\bullet \aff_{\mathbb{I}_{\inn(\nu_i)},B_i} \rp \rp \lp x \rp = \lp \real_{\act} \lp \nu_i \rp \rp \lp x + b_k \rp + \end{align} + This, Lemma \ref{5.6.3} and \cite[Lemma~2.14, Item~(ii)]{grohs2019spacetime} show that for all $i \in \N \cap [u,v]$, $x \in \R^{\inn(\nu_u)}$, it holds that: + \begin{align} + \real_{\act} \lp \ex_{L,\mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp = \real_{\act}\lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \in C\lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp + \end{align} + and: + \begin{align} + \lp \real_{\act} \lp \ex_{L,\mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \rp \lp x \rp &= \lp \real_{\act} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \lp x \rp \nonumber\\ + &= c_i \lp \real_{\act} \lp \nu_i \rp \rp \lp x+b_i \rp + \end{align} + This combined with \cite[Lemma~3.28]{Grohs_2022} and (\ref{5.6.13}) demonstrate that for all $x \in \R^{\inn(\nu_u)}$ it holds that $\real_{\act}\lp \mu \rp \in C\lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $ and that: + \begin{align} + \lp \real_{\act}\lp \mu \rp \rp \lp x \rp &= \lp \real_{\act} \lp \boxplus^v_{i = u, \mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)}} \rp \rp \rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\act} \lp \oplus^v_{i=u} \ex_{L,\mathfrak{I}} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \rp\lp x \rp \nonumber \\ + &= \sum^v_{i=u} c_i \lp \real_{\act} \lp \nu_i \rp \rp \lp x+b_i \rp + \end{align} + This establishes Items(ii)--(iii), thus proving the lemma. +\end{proof} +\begin{lemma} + Let $L \in \N$, $u,v \in \Z$ with $u\leqslant v$. Let $c_u, c_{u+1},...,c_v \in \R$. $\nu_u, \nu_{u+1},...,\nu_v, \mu, \mathfrak{I} \in \neu$, $B_u, B_{u+1},...,B_v \in \R^{\inn(\nu_u)}$, $\act \in C\lp \R, \R \rp$, satisfy for all $j \in \N \cap [u,v]$ that $L = \max_{i\in \N \cap \lb u,v \rb} \dep(\nu_i)$, $\inn(\nu_j) = \inn(\nu_u)$, $\out(\nu_j) = \inn(\mathfrak{I})= \out(\mathfrak{I})$, $\hid(\mathfrak{I}) = 1$, $\real_{\act} (\mathfrak{I}) = \mathbb{I}_\R$, and that: + \begin{align} + \mu = \boxplus^v_{i = u, \mathfrak{I}} \lp \lp \aff_{\mathbb{I} _{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i \rp + \end{align} + We then have: + \begin{enumerate}[label = (\roman*)] + \item it holds that: + \begin{align} + \lay(\mu) = \lp \inn(\nu_u ), \sum^v_{i=u}\wid_1 \lp \ex_{L,\mathfrak{I}} \lp \nu_i \rp \rp ,\sum^v_{i=u}\wid_2 \lp \ex_{L,\mathfrak{I}} \lp \nu_i\rp\rp,...,\sum^v_{i=u} \wid_{L-1} \lp \ex_{L,\mathfrak{I}} \lp \nu_i \rp , \out \lp \nu_u \rp \rp \rp + \end{align} + \item it holds that $\real_{\act}(\mu) \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $, and that, + \item it holds for all $ x \in \R^{\inn(\nu_u)}$ that: + \begin{align} + \lp \real_{\act} \lp \mu \rp \rp \lp x \rp = \sum^v_{i=u} \lp \real_{\act} \lp \nu_i \rp \rp \lp c_ix + b_i\rp + \end{align} + \end{enumerate} +\end{lemma} +\begin{proof} + Note that Item(i) from Lemma \ref{5.6.6} establish Item(i) and (\ref{5.5.20}); in addition, items (ii) and (iii) from Lemma \ref{5.3.3} tell us that for all $i \in \N \cap [u,v]$, $x \in \R^{\inn(\nu_u}$, it holds that $\real_{\act} \lp \aff_{\mathbb{I}_{\inn(\nu_i)}, B_i} \bullet \nu_i \in C \lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)}\rp \rp $ and further that: + \begin{align} + \lp \real_{\act} \lp \aff_{\mathbb{I}_{\inn(\nu_i)},B_i} \bullet \nu_i \rp \rp \lp x \rp = \lp \real_{\act} \lp \nu_i \rp \rp \lp x \rp + b_k + \end{align} + This, Lemma \ref{5.6.4} and \cite[Lemma~2.14, Item~(ii)]{grohs2019spacetime} show that for all $i \in \N \cap [u,v]$, $x \in \R^{\inn(\nu_u)}$, it holds that: + \begin{align} + \real_{\act} \lp \ex_{L,\mathfrak{I}} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i \rp \rp = \real_{\act}\lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i \rp \triangleleft c_i\rp \in C\lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp + \end{align} + and: + \begin{align} + \lp \real_{\act} \lp \ex_{L,\mathfrak{I}} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i\rp \rp \rp \lp x \rp &= \lp \real_{\act} \lp c_i \triangleright \lp \nu_i \bullet \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \rp \rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\act} \lp \nu_i \rp \rp \lp c_ix+b_i \rp + \end{align} + This and \cite[Lemma~3.28]{Grohs_2022} and (\ref{5.6.23}) demonstrate that for all $x \in \R^{\inn(\nu_u)}$ it holds that $\real_{\act}\lp \mu \rp \in C\lp \R^{\inn(\nu_u)}, \R^{\out(\nu_u)} \rp $ and that: + \begin{align} + \lp \real_{\act}\lp \mu \rp \rp \lp x \rp &= \lp \real_{\act} \lp \boxplus^v_{i = u, \mathfrak{I}} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)}} \bullet \nu_i\rp \triangleleft c_i\rp \rp \rp \lp x \rp \nonumber\\ + &= \lp \real_{\act} \lp \oplus^v_{i=u} \ex_{L,\mathfrak{I}} \lp \lp \aff_{\mathbb{I}_{\inn(\nu_i)},b_i} \bullet \nu_i\rp \triangleleft c_i\rp \rp \rp\lp x \rp \nonumber \\ + &= \sum^v_{i=u} \lp \real_{\act} \lp \nu_i \rp \rp \lp c_ix+b_i \rp + \end{align} + This completes the proof. +\end{proof} + +\section{Neural Network Diagrams} + +Conceptually, it will be helpful to construct what are called ``neural network diagrams''. They take inspiration from diagrams typically seen in the literature, for instance, \cite{vaswani_attention_2017}, \cite{arik_tabnet_2021}, and \cite{8099678}. They are constructed as follows. +Lines with arrows indicate the flow of data: + +\begin{center} +\begin{tikzcd} +{} \arrow[rr, "x"] & & {} \\ +{} & & {} \arrow[ll, "x"] +\end{tikzcd} +\end{center} +Named neural networks are always enclosed in boxes with \textsf{serif} fonts: +\begin{center} + \begin{tikzpicture} + % Create a rectangular node with text inside + \node[draw, rectangle] at (0, 0) {$\aff_{a,b}$}; +\end{tikzpicture} +\end{center} +Where possible, we seek to label the arrows going in and going out of a boxed neural network with the appropriate operations that take place: +\begin{center} + \begin{tikzpicture} + % Create a rectangular node with text inside + \node[draw, rectangle] (box) at (0, 0) {$\aff_{a,b}$}; + + % Draw an arrow from left to right going into the box + \draw[<-] (-2, 0) -- (box.west) node[midway, above] {$ax+b$}; + + % Draw an arrow from left to right going out of the box + \draw[<-] (box.east) -- (2, 0) node[midway, above] {$x$}; +\end{tikzpicture} +\end{center} +It is often more helpful to draw the arrows from right to left, as above. + +Stacked neural networks are drawn in adjacent boxes. +\begin{center} + \begin{tikzpicture} + % Create the top box with text inside + \node[draw, rectangle] (topbox) at (0, 1) {$\aff_{a,b}$}; + + % Create the bottom box with text inside + \node[draw, rectangle] (bottombox) at (0, -1) {$\aff_{c,d}$}; + + % Draw an arrow from left to right going into the top box + \draw[<-] (-2, 1) -- (topbox.west) node[midway, above] {$ax+b$}; + + % Draw an arrow from left to right going out of the top box + \draw[<-] (topbox.east) -- (2, 1) node[midway, above] {$x$}; + + % Draw an arrow from left to right going into the bottom box + \draw[<-] (-2, -1) -- (bottombox.west) node[midway, below] {$cx+d$}; + + % Draw an arrow from left to right going out of the bottom box + \draw[<-] (bottombox.east) -- (2, -1) node[midway, below] {$x$}; +\end{tikzpicture} +\end{center} + +For neural networks that take in two inputs and give out one output, we use two arrows going in and one arrow going out: +\begin{center} +\begin{tikzpicture} + % Create the rectangular node with text inside + \node[draw, rectangle] (box) at (0, 0) {$\sm_{2,1}$}; + + % Draw arrow hitting the top right corner of the box + \draw[->] (2, 1) -- (box.north east) node[midway, above right] {$x$}; + + % Draw arrow hitting the bottom right corner of the box + \draw[->] (2, -1) -- (box.south east) node[midway, below right] {$y$}; + + % Draw an arrow going out to the left + \draw[->] (box.west) -- (-2, 0) node[midway, above] {$x+y$}; +\end{tikzpicture} +\end{center} + +For neural networks that take in one input and give out two outputs, we use one arrow going in and two arrows going out: + +\begin{center} +\begin{tikzpicture} + % Create the rectangular node with text inside + \node[draw, rectangle] (box) at (0, 0) {$\cpy_{1,2}$}; + + % Draw arrow hitting the top right corner of the box + \draw[->] (box.north west) -- (-2,1) node[midway, above right] {$x$}; + + % Draw arrow hitting the bottom right corner of the box + \draw[->] (box.south west) -- (-2,-1) node[midway, below right] {$x$}; + + % Draw an arrow going out to the left + \draw[->] (2,0) -- (box.east) node[midway, above] {$x$}; +\end{tikzpicture} +\end{center} + +Thus taking this all together the sum of neural networks $\aff_{a,b},\aff_{c,d} \in \neu$ is given by: + +\begin{center} + \begin{tikzpicture} + % Define nodes + \node[draw, rectangle] (top) at (0, 2) {$\aff_{a,b}$}; + \node[draw, rectangle] (right) at (2, 0) {$\cpy$}; + \node[draw, rectangle] (bottom) at (0, -2) {$\aff_{c,d}$}; + \node[draw, rectangle] (left) at (-2, 0) {$\sm$}; + + % Arrows with labels + \draw[->] (right) -- node[midway, above] {$x$} (top); + \draw[<-] (right) -- node[midway, above] {$x$} (4,0)(right); + \draw[->] (right) -- node[midway, right] {$x$} (bottom); + \draw[->] (top) -- node[midway, left] {$ax+b$} (left); + \draw[->] (bottom) -- node[midway, left] {$cx+d$} (left); + \draw[->] (left) -- node[midway, above] {$ax+b+cx+d$} (-5.5,0); +% \draw[->] (-3,0) -- node[midway, above] {Arrow 6} (left); +\end{tikzpicture} +\end{center} + + + + + + + + + + + + + + + + + + + + diff --git a/Dissertation/preamble.tex b/Dissertation/preamble.tex new file mode 100644 index 0000000..4421e22 --- /dev/null +++ b/Dissertation/preamble.tex @@ -0,0 +1,215 @@ +\documentclass[11pt]{report} +\usepackage{setspace} +\doublespacing +\usepackage[toc,page]{appendix} + +\usepackage{pdfpages} +\usepackage[]{amsmath} +\usepackage[]{amsthm} +\usepackage{mathtools} +\numberwithin{equation}{section} +\usepackage[]{amssymb} +\usepackage[margin=1in]{geometry} +\usepackage[]{soul} +\usepackage[]{bbm} +\usepackage[]{cancel} +\usepackage[]{xcolor} +\usepackage[]{enumitem} +\usepackage{mathrsfs} + +\usepackage{graphicx} +\usepackage{subfigure} + +\usepackage{booktabs} + +\usepackage{hyperref} +\hypersetup{ + pdfauthor={Shakil Rafi}, + pdftitle={Dissertation}, + pdfkeywords={neural-networks, stochastic-processes}, + colorlinks = true, + filecolor = magenta, + urlcolor = cyan +} + + +\usepackage[capitalise]{cleveref} +\usepackage{natbib} +\usepackage{neuralnetwork} +\usepackage{witharrows} +\usepackage{stmaryrd} +\usepackage{stackengine,amssymb,graphicx} + +\usepackage{listings} + +\usepackage{xcolor} + +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolor}{rgb}{0.95,0.95,0.92} + +% Define R language settings +\lstdefinestyle{rstyle}{ + language=R, + basicstyle=\ttfamily\footnotesize, + backgroundcolor = \color{backcolor}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\ttfamily\tiny\color{codegray}, + numbers=left, + stepnumber=1, + frame=single, + breaklines=true, + numbersep = 5pt, + breakatwhitespace=false, + tabsize=4, + morekeywords={install.packages, library, ggplot, aes, geom_bar} +} + +\usepackage{fontspec} +\setmonofont{Fira Code} + + + + 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+\DeclareMathOperator{\nrm}{\mathsf{Nrm}} +\DeclareMathOperator{\mxm}{\mathsf{Mxm}} +\DeclareMathOperator{\trp}{\mathsf{Trp}} +\DeclareMathOperator{\etr}{\mathsf{Etr}} +\DeclareMathOperator{\lc}{\left\lceil} +\DeclareMathOperator{\rc}{\right\rceil} +\DeclareMathOperator{\csn}{\mathsf{Csn}} +\DeclareMathOperator{\sne}{\mathsf{Sne}} +\DeclareMathOperator{\pln}{\mathsf{Pln}} +\DeclareMathOperator{\pnm}{\mathsf{Pnm}} +\DeclareMathOperator{\inst}{\mathfrak{I}} +\DeclareMathOperator{\rows}{rows} +\DeclareMathOperator{\columns}{columns} + +\DeclareMathOperator{\obj}{obj} +\DeclareMathOperator{\dom}{dom} +\DeclareMathOperator{\cod}{cod} + +\newcommand{\bbP}{\mathbb{P}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\mft}{\mathfrak{t}} +\newcommand{\f}{\mathfrak{f}} +\newcommand{\C}{\mathfrak{C}} +\newcommand{\n}{\mathscr{N}} +\newcommand{\lp}{\left(} +\newcommand{\rp}{\right)} +\newcommand{\rb}{\right]} +\newcommand{\lb}{\left[} +\newcommand{\lv}{\left|} +\newcommand{\rv}{\right|} +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\ve}{\varepsilon} +\newcommand{\les}{\leqslant} +\newcommand{\ges}{\geqslant} + + + +\newcommand\dplus{\ensurestackMath{% + \stackengine{.5pt}{\Diamond}{\scalebox{.75}[1]{$+$}}{O}{c}{F}{F}{L}}} + +\newcommand{\DDiamond}{% + \begin{tikzpicture}[baseline={(0,-0.2ex)}] + \draw[rotate=45] (0,0) rectangle (0.2,0.2); + \draw[rotate=45] (0,0.2) -- (0.2,0); + \end{tikzpicture}% +} + +% Input layer neurons number +\newcommand{\inputnum}{3} + +% Hidden layer neurons number +\newcommand{\hiddennum}{5} + +% Output layer neurons number +\newcommand{\outputnum}{2} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{corollary}{Corollary}[theorem] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{claim}[theorem]{Claim} + + + + + + + + + + + + + diff --git a/Dissertation/sharkteeth.png b/Dissertation/sharkteeth.png new file mode 100644 index 0000000..0bbb1a6 Binary files /dev/null and b/Dissertation/sharkteeth.png differ diff --git a/Dissertation/sharktooth.jl b/Dissertation/sharktooth.jl new file mode 100644 index 0000000..e7c0d13 --- /dev/null +++ b/Dissertation/sharktooth.jl @@ -0,0 +1,47 @@ +using Random +using Distributions +using Plots + +# =============================================================== + +function max_conv_operator(samples, f_samples, input, L) # max convolution operator, with random uniform sampling + return maximum(f_samples .- L .* abs.(input .- samples)) +end + +function sharktooth(f, domain, number_of_sharkteeth, L, plot_arg) + samples = rand(Uniform(domain[1], domain[2]), number_of_sharkteeth) #samples are uniformly taken from domain + f_samples = f.(samples) #y_i are f applied to samples componentwise + x = LinRange(domain[1], domain[2], 10000) #approximant is plotted over a mesh of resolution 10000 + approximant = broadcast(x -> max_conv_operator(samples, f_samples, x, L), x) + error = maximum(abs.(f.(x) - approximant)) + if plot_arg == 1 + plot(x, approximant) + else + return error + end +end + + +function sharktooth_error_plot(L_values, sharktooth_upper_limit, f, domain) + # Initialize an empty plot + p = plot(legend=false, xlabel="Number of Sharkteeth", ylabel="Error") + + for L in L_values + errors = Float64[] # Initialize an empty array to store errors + for i in 1:sharktooth_upper_limit + push!(errors, sharktooth(f, domain[1], domain[2], i, L, 0)) + end + plot!(1:sharktooth_upper_limit, errors, label="L = $L") # Add the plot to the existing figure + end + + # Show the legend + plot!(legend=true) + + # Display the final plot + display(p) +end + +sharktooth_error_plot([1, 2, 3, 4], 150, sin, [0, 30]) + +sharktooth(tanh, [-5, 5], 1500, 1, 1) + diff --git a/Dissertation/sharktooth.nb b/Dissertation/sharktooth.nb new file mode 100644 index 0000000..d32c942 --- /dev/null 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+ +
+ +
+
+

Sharktooth Functions in 1-dimension

+
+ + + +
+ + + + +
+ + +
+ +
+
import numpy as np 
+import matplotlib.pyplot as plt
+
+
+
def max_conv_operator(samples, f_samples, input, L):
+    return np.max(f_samples - L * np.abs(input - samples))
+
+def sharktooth_function(function, x_start, x_stop, number_of_sharkteeth, L, plot_arg):
+    samples = np.random.uniform(x_start,x_stop, number_of_sharkteeth)
+    f_samples = function(samples)
+    x = np.linspace(x_start, x_stop, 10000)
+    approximate_y = []
+    for i in range(len(x)):
+        approximate_y.append(max_conv_operator(samples, f_samples, x[i], L))
+    error = np.max(np.abs(f(x) - approximate_y))
+    if plot_arg == 1:
+        plt.plot(x,approximate_y)
+    else:
+        return error
+
+
+
def f(x):
+    return x/(x+2)
+
+
+
L = [1,4,8,16]
+plt.figure(figsize=(17,5))
+for L in L:
+    errors = []
+    for i in range(1,50):
+        errors.append(sharktooth_function(f,0,1,i,L,0))
+    import matplotlib.pyplot as plt
+    plt.plot(errors, label =L)
+
+plt.xlabel("Number of teeth")
+plt.ylabel(r"$\sup_{x\in x_i} | f(x)- \mathfrak{R}_{\mathfrak{r}} ( \mathsf{P})|$")
+plt.legend()
+plt.title("Sup of deviance from  f(x) as the number of teeth increase and as we get closer to L")
+
+
Text(0.5, 1.0, 'Sup of deviance from  f(x) as the number of teeth increase and as we get closer to L')
+
+
+

+
+
+ +
+ + +
+ + + + \ No newline at end of file diff --git a/Dissertation/sharktooth_functions.ipynb b/Dissertation/sharktooth_functions.ipynb new file mode 100644 index 0000000..e9c08fe --- /dev/null +++ b/Dissertation/sharktooth_functions.ipynb @@ -0,0 +1,145 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Sharktooth Functions in 1-dimension" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np \n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "def max_conv_operator(samples, f_samples, input, L):\n", + " return np.max(f_samples - L * np.abs(input - samples))\n", + "\n", + "def sharktooth_function(function, x_start, x_stop, number_of_sharkteeth, L, plot_arg):\n", + " samples = np.random.uniform(x_start,x_stop, number_of_sharkteeth)\n", + " f_samples = function(samples)\n", + " x = np.linspace(x_start, x_stop, 10000)\n", + " approximate_y = []\n", + " for i in range(len(x)):\n", + " approximate_y.append(max_conv_operator(samples, f_samples, x[i], L))\n", + " error = np.max(np.abs(f(x) - approximate_y))\n", + " if plot_arg == 1:\n", + " plt.plot(x,approximate_y)\n", + " else:\n", + " return error" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "def f(x):\n", + " return np.tanh(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Text(0.5, 1.0, 'Sup of deviance from tanh(x) as the number of teeth increase and as we get closer to L, for tanh that is 1')" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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+ "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "L = [1,4,8,16]\n", + "plt.figure(figsize=(17,5))\n", + "for L in L:\n", + " errors = []\n", + " for i in range(1,50):\n", + " errors.append(sharktooth_function(f,0,1,i,L,0))\n", + " import matplotlib.pyplot as plt\n", + " plt.plot(errors, label =L)\n", + "\n", + "plt.xlabel(\"Number of teeth\")\n", + "plt.ylabel(r\"$\\sup_{x\\in x_i} | \\tanh(x)- \\mathfrak{I}_{\\mathsf{ReLU}} ( \\mathsf{MC})|$\")\n", + "plt.legend()\n", + "plt.title(\"Sup of deviance from tanh(x) as the number of teeth increase and as we get closer to L, for tanh that is 1\")" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", 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this._config.reference&&(e=this._config.reference);const i=this._getPopperConfig(),n=i.modifiers.find((t=>"applyStyles"===t.name&&!1===t.enabled));this._popper=qe(e,this._menu,i),n&&U.setDataAttribute(this._menu,"popper","static")}_isShown(t=this._element){return t.classList.contains(Je)}_getMenuElement(){return V.next(this._element,ei)[0]}_getPlacement(){const t=this._element.parentNode;if(t.classList.contains("dropend"))return ri;if(t.classList.contains("dropstart"))return ai;const e="end"===getComputedStyle(this._menu).getPropertyValue("--bs-position").trim();return t.classList.contains("dropup")?e?ni:ii:e?oi:si}_detectNavbar(){return null!==this._element.closest(".navbar")}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return"static"===this._config.display&&(t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,..."function"==typeof this._config.popperConfig?this._config.popperConfig(t):this._config.popperConfig}}_selectMenuItem({key:t,target:e}){const i=V.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter(l);i.length&&v(i,e,t===Ye,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(t&&(2===t.button||"keyup"===t.type&&"Tab"!==t.key))return;const e=V.find(ti);for(let i=0,n=e.length;ie+t)),this._setElementAttributes(di,"paddingRight",(e=>e+t)),this._setElementAttributes(ui,"marginRight",(e=>e-t))}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t)[e];t.style[e]=`${i(Number.parseFloat(s))}px`}))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,"paddingRight"),this._resetElementAttributes(di,"paddingRight"),this._resetElementAttributes(ui,"marginRight")}_saveInitialAttribute(t,e){const i=t.style[e];i&&U.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=U.getDataAttribute(t,e);void 0===i?t.style.removeProperty(e):(U.removeDataAttribute(t,e),t.style[e]=i)}))}_applyManipulationCallback(t,e){o(t)?e(t):V.find(t,this._element).forEach(e)}isOverflowing(){return this.getWidth()>0}}const pi={className:"modal-backdrop",isVisible:!0,isAnimated:!1,rootElement:"body",clickCallback:null},mi={className:"string",isVisible:"boolean",isAnimated:"boolean",rootElement:"(element|string)",clickCallback:"(function|null)"},gi="show",_i="mousedown.bs.backdrop";class bi{constructor(t){this._config=this._getConfig(t),this._isAppended=!1,this._element=null}show(t){this._config.isVisible?(this._append(),this._config.isAnimated&&u(this._getElement()),this._getElement().classList.add(gi),this._emulateAnimation((()=>{_(t)}))):_(t)}hide(t){this._config.isVisible?(this._getElement().classList.remove(gi),this._emulateAnimation((()=>{this.dispose(),_(t)}))):_(t)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_getConfig(t){return(t={...pi,..."object"==typeof t?t:{}}).rootElement=r(t.rootElement),a("backdrop",t,mi),t}_append(){this._isAppended||(this._config.rootElement.append(this._getElement()),j.on(this._getElement(),_i,(()=>{_(this._config.clickCallback)})),this._isAppended=!0)}dispose(){this._isAppended&&(j.off(this._element,_i),this._element.remove(),this._isAppended=!1)}_emulateAnimation(t){b(t,this._getElement(),this._config.isAnimated)}}const vi={trapElement:null,autofocus:!0},yi={trapElement:"element",autofocus:"boolean"},wi=".bs.focustrap",Ei="backward";class Ai{constructor(t){this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}activate(){const{trapElement:t,autofocus:e}=this._config;this._isActive||(e&&t.focus(),j.off(document,wi),j.on(document,"focusin.bs.focustrap",(t=>this._handleFocusin(t))),j.on(document,"keydown.tab.bs.focustrap",(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,j.off(document,wi))}_handleFocusin(t){const{target:e}=t,{trapElement:i}=this._config;if(e===document||e===i||i.contains(e))return;const n=V.focusableChildren(i);0===n.length?i.focus():this._lastTabNavDirection===Ei?n[n.length-1].focus():n[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?Ei:"forward")}_getConfig(t){return t={...vi,..."object"==typeof t?t:{}},a("focustrap",t,yi),t}}const Ti="modal",Oi="Escape",Ci={backdrop:!0,keyboard:!0,focus:!0},ki={backdrop:"(boolean|string)",keyboard:"boolean",focus:"boolean"},Li="hidden.bs.modal",xi="show.bs.modal",Di="resize.bs.modal",Si="click.dismiss.bs.modal",Ni="keydown.dismiss.bs.modal",Ii="mousedown.dismiss.bs.modal",Pi="modal-open",ji="show",Mi="modal-static";class Hi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._dialog=V.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._ignoreBackdropClick=!1,this._isTransitioning=!1,this._scrollBar=new fi}static get Default(){return Ci}static get NAME(){return Ti}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||j.trigger(this._element,xi,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isAnimated()&&(this._isTransitioning=!0),this._scrollBar.hide(),document.body.classList.add(Pi),this._adjustDialog(),this._setEscapeEvent(),this._setResizeEvent(),j.on(this._dialog,Ii,(()=>{j.one(this._element,"mouseup.dismiss.bs.modal",(t=>{t.target===this._element&&(this._ignoreBackdropClick=!0)}))})),this._showBackdrop((()=>this._showElement(t))))}hide(){if(!this._isShown||this._isTransitioning)return;if(j.trigger(this._element,"hide.bs.modal").defaultPrevented)return;this._isShown=!1;const t=this._isAnimated();t&&(this._isTransitioning=!0),this._setEscapeEvent(),this._setResizeEvent(),this._focustrap.deactivate(),this._element.classList.remove(ji),j.off(this._element,Si),j.off(this._dialog,Ii),this._queueCallback((()=>this._hideModal()),this._element,t)}dispose(){[window,this._dialog].forEach((t=>j.off(t,".bs.modal"))),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new bi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_getConfig(t){return t={...Ci,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Ti,t,ki),t}_showElement(t){const e=this._isAnimated(),i=V.findOne(".modal-body",this._dialog);this._element.parentNode&&this._element.parentNode.nodeType===Node.ELEMENT_NODE||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0,i&&(i.scrollTop=0),e&&u(this._element),this._element.classList.add(ji),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,j.trigger(this._element,"shown.bs.modal",{relatedTarget:t})}),this._dialog,e)}_setEscapeEvent(){this._isShown?j.on(this._element,Ni,(t=>{this._config.keyboard&&t.key===Oi?(t.preventDefault(),this.hide()):this._config.keyboard||t.key!==Oi||this._triggerBackdropTransition()})):j.off(this._element,Ni)}_setResizeEvent(){this._isShown?j.on(window,Di,(()=>this._adjustDialog())):j.off(window,Di)}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(Pi),this._resetAdjustments(),this._scrollBar.reset(),j.trigger(this._element,Li)}))}_showBackdrop(t){j.on(this._element,Si,(t=>{this._ignoreBackdropClick?this._ignoreBackdropClick=!1:t.target===t.currentTarget&&(!0===this._config.backdrop?this.hide():"static"===this._config.backdrop&&this._triggerBackdropTransition())})),this._backdrop.show(t)}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(j.trigger(this._element,"hidePrevented.bs.modal").defaultPrevented)return;const{classList:t,scrollHeight:e,style:i}=this._element,n=e>document.documentElement.clientHeight;!n&&"hidden"===i.overflowY||t.contains(Mi)||(n||(i.overflowY="hidden"),t.add(Mi),this._queueCallback((()=>{t.remove(Mi),n||this._queueCallback((()=>{i.overflowY=""}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;(!i&&t&&!m()||i&&!t&&m())&&(this._element.style.paddingLeft=`${e}px`),(i&&!t&&!m()||!i&&t&&m())&&(this._element.style.paddingRight=`${e}px`)}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=Hi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}j.on(document,"click.bs.modal.data-api",'[data-bs-toggle="modal"]',(function(t){const e=n(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),j.one(e,xi,(t=>{t.defaultPrevented||j.one(e,Li,(()=>{l(this)&&this.focus()}))}));const i=V.findOne(".modal.show");i&&Hi.getInstance(i).hide(),Hi.getOrCreateInstance(e).toggle(this)})),R(Hi),g(Hi);const Bi="offcanvas",Ri={backdrop:!0,keyboard:!0,scroll:!1},Wi={backdrop:"boolean",keyboard:"boolean",scroll:"boolean"},$i="show",zi=".offcanvas.show",qi="hidden.bs.offcanvas";class Fi extends B{constructor(t,e){super(t),this._config=this._getConfig(e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get NAME(){return Bi}static get Default(){return Ri}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||j.trigger(this._element,"show.bs.offcanvas",{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._element.style.visibility="visible",this._backdrop.show(),this._config.scroll||(new fi).hide(),this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add($i),this._queueCallback((()=>{this._config.scroll||this._focustrap.activate(),j.trigger(this._element,"shown.bs.offcanvas",{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(j.trigger(this._element,"hide.bs.offcanvas").defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.remove($i),this._backdrop.hide(),this._queueCallback((()=>{this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._element.style.visibility="hidden",this._config.scroll||(new fi).reset(),j.trigger(this._element,qi)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_getConfig(t){return t={...Ri,...U.getDataAttributes(this._element),..."object"==typeof t?t:{}},a(Bi,t,Wi),t}_initializeBackDrop(){return new bi({className:"offcanvas-backdrop",isVisible:this._config.backdrop,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:()=>this.hide()})}_initializeFocusTrap(){return new Ai({trapElement:this._element})}_addEventListeners(){j.on(this._element,"keydown.dismiss.bs.offcanvas",(t=>{this._config.keyboard&&"Escape"===t.key&&this.hide()}))}static jQueryInterface(t){return this.each((function(){const e=Fi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}j.on(document,"click.bs.offcanvas.data-api",'[data-bs-toggle="offcanvas"]',(function(t){const e=n(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),c(this))return;j.one(e,qi,(()=>{l(this)&&this.focus()}));const i=V.findOne(zi);i&&i!==e&&Fi.getInstance(i).hide(),Fi.getOrCreateInstance(e).toggle(this)})),j.on(window,"load.bs.offcanvas.data-api",(()=>V.find(zi).forEach((t=>Fi.getOrCreateInstance(t).show())))),R(Fi),g(Fi);const Ui=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Vi=/^(?:(?:https?|mailto|ftp|tel|file|sms):|[^#&/:?]*(?:[#/?]|$))/i,Ki=/^data:(?:image\/(?:bmp|gif|jpeg|jpg|png|tiff|webp)|video\/(?:mpeg|mp4|ogg|webm)|audio\/(?:mp3|oga|ogg|opus));base64,[\d+/a-z]+=*$/i,Xi=(t,e)=>{const i=t.nodeName.toLowerCase();if(e.includes(i))return!Ui.has(i)||Boolean(Vi.test(t.nodeValue)||Ki.test(t.nodeValue));const n=e.filter((t=>t instanceof RegExp));for(let t=0,e=n.length;t{Xi(t,r)||i.removeAttribute(t.nodeName)}))}return n.body.innerHTML}const Qi="tooltip",Gi=new Set(["sanitize","allowList","sanitizeFn"]),Zi={animation:"boolean",template:"string",title:"(string|element|function)",trigger:"string",delay:"(number|object)",html:"boolean",selector:"(string|boolean)",placement:"(string|function)",offset:"(array|string|function)",container:"(string|element|boolean)",fallbackPlacements:"array",boundary:"(string|element)",customClass:"(string|function)",sanitize:"boolean",sanitizeFn:"(null|function)",allowList:"object",popperConfig:"(null|object|function)"},Ji={AUTO:"auto",TOP:"top",RIGHT:m()?"left":"right",BOTTOM:"bottom",LEFT:m()?"right":"left"},tn={animation:!0,template:'',trigger:"hover focus",title:"",delay:0,html:!1,selector:!1,placement:"top",offset:[0,0],container:!1,fallbackPlacements:["top","right","bottom","left"],boundary:"clippingParents",customClass:"",sanitize:!0,sanitizeFn:null,allowList:{"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},popperConfig:null},en={HIDE:"hide.bs.tooltip",HIDDEN:"hidden.bs.tooltip",SHOW:"show.bs.tooltip",SHOWN:"shown.bs.tooltip",INSERTED:"inserted.bs.tooltip",CLICK:"click.bs.tooltip",FOCUSIN:"focusin.bs.tooltip",FOCUSOUT:"focusout.bs.tooltip",MOUSEENTER:"mouseenter.bs.tooltip",MOUSELEAVE:"mouseleave.bs.tooltip"},nn="fade",sn="show",on="show",rn="out",an=".tooltip-inner",ln=".modal",cn="hide.bs.modal",hn="hover",dn="focus";class un extends B{constructor(t,e){if(void 0===Fe)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t),this._isEnabled=!0,this._timeout=0,this._hoverState="",this._activeTrigger={},this._popper=null,this._config=this._getConfig(e),this.tip=null,this._setListeners()}static get Default(){return tn}static get NAME(){return Qi}static get Event(){return en}static get DefaultType(){return Zi}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(t){if(this._isEnabled)if(t){const e=this._initializeOnDelegatedTarget(t);e._activeTrigger.click=!e._activeTrigger.click,e._isWithActiveTrigger()?e._enter(null,e):e._leave(null,e)}else{if(this.getTipElement().classList.contains(sn))return void this._leave(null,this);this._enter(null,this)}}dispose(){clearTimeout(this._timeout),j.off(this._element.closest(ln),cn,this._hideModalHandler),this.tip&&this.tip.remove(),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this.isWithContent()||!this._isEnabled)return;const t=j.trigger(this._element,this.constructor.Event.SHOW),e=h(this._element),i=null===e?this._element.ownerDocument.documentElement.contains(this._element):e.contains(this._element);if(t.defaultPrevented||!i)return;"tooltip"===this.constructor.NAME&&this.tip&&this.getTitle()!==this.tip.querySelector(an).innerHTML&&(this._disposePopper(),this.tip.remove(),this.tip=null);const n=this.getTipElement(),s=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME);n.setAttribute("id",s),this._element.setAttribute("aria-describedby",s),this._config.animation&&n.classList.add(nn);const o="function"==typeof this._config.placement?this._config.placement.call(this,n,this._element):this._config.placement,r=this._getAttachment(o);this._addAttachmentClass(r);const{container:a}=this._config;H.set(n,this.constructor.DATA_KEY,this),this._element.ownerDocument.documentElement.contains(this.tip)||(a.append(n),j.trigger(this._element,this.constructor.Event.INSERTED)),this._popper?this._popper.update():this._popper=qe(this._element,n,this._getPopperConfig(r)),n.classList.add(sn);const l=this._resolvePossibleFunction(this._config.customClass);l&&n.classList.add(...l.split(" ")),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>{j.on(t,"mouseover",d)}));const c=this.tip.classList.contains(nn);this._queueCallback((()=>{const t=this._hoverState;this._hoverState=null,j.trigger(this._element,this.constructor.Event.SHOWN),t===rn&&this._leave(null,this)}),this.tip,c)}hide(){if(!this._popper)return;const t=this.getTipElement();if(j.trigger(this._element,this.constructor.Event.HIDE).defaultPrevented)return;t.classList.remove(sn),"ontouchstart"in document.documentElement&&[].concat(...document.body.children).forEach((t=>j.off(t,"mouseover",d))),this._activeTrigger.click=!1,this._activeTrigger.focus=!1,this._activeTrigger.hover=!1;const e=this.tip.classList.contains(nn);this._queueCallback((()=>{this._isWithActiveTrigger()||(this._hoverState!==on&&t.remove(),this._cleanTipClass(),this._element.removeAttribute("aria-describedby"),j.trigger(this._element,this.constructor.Event.HIDDEN),this._disposePopper())}),this.tip,e),this._hoverState=""}update(){null!==this._popper&&this._popper.update()}isWithContent(){return Boolean(this.getTitle())}getTipElement(){if(this.tip)return this.tip;const t=document.createElement("div");t.innerHTML=this._config.template;const e=t.children[0];return this.setContent(e),e.classList.remove(nn,sn),this.tip=e,this.tip}setContent(t){this._sanitizeAndSetContent(t,this.getTitle(),an)}_sanitizeAndSetContent(t,e,i){const n=V.findOne(i,t);e||!n?this.setElementContent(n,e):n.remove()}setElementContent(t,e){if(null!==t)return o(e)?(e=r(e),void(this._config.html?e.parentNode!==t&&(t.innerHTML="",t.append(e)):t.textContent=e.textContent)):void(this._config.html?(this._config.sanitize&&(e=Yi(e,this._config.allowList,this._config.sanitizeFn)),t.innerHTML=e):t.textContent=e)}getTitle(){const t=this._element.getAttribute("data-bs-original-title")||this._config.title;return this._resolvePossibleFunction(t)}updateAttachment(t){return"right"===t?"end":"left"===t?"start":t}_initializeOnDelegatedTarget(t,e){return e||this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return"function"==typeof t?t.call(this._element):t}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"onChange",enabled:!0,phase:"afterWrite",fn:t=>this._handlePopperPlacementChange(t)}],onFirstUpdate:t=>{t.options.placement!==t.placement&&this._handlePopperPlacementChange(t)}};return{...e,..."function"==typeof this._config.popperConfig?this._config.popperConfig(e):this._config.popperConfig}}_addAttachmentClass(t){this.getTipElement().classList.add(`${this._getBasicClassPrefix()}-${this.updateAttachment(t)}`)}_getAttachment(t){return Ji[t.toUpperCase()]}_setListeners(){this._config.trigger.split(" ").forEach((t=>{if("click"===t)j.on(this._element,this.constructor.Event.CLICK,this._config.selector,(t=>this.toggle(t)));else if("manual"!==t){const e=t===hn?this.constructor.Event.MOUSEENTER:this.constructor.Event.FOCUSIN,i=t===hn?this.constructor.Event.MOUSELEAVE:this.constructor.Event.FOCUSOUT;j.on(this._element,e,this._config.selector,(t=>this._enter(t))),j.on(this._element,i,this._config.selector,(t=>this._leave(t)))}})),this._hideModalHandler=()=>{this._element&&this.hide()},j.on(this._element.closest(ln),cn,this._hideModalHandler),this._config.selector?this._config={...this._config,trigger:"manual",selector:""}:this._fixTitle()}_fixTitle(){const t=this._element.getAttribute("title"),e=typeof 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t in this._activeTrigger)if(this._activeTrigger[t])return!0;return!1}_getConfig(t){const e=U.getDataAttributes(this._element);return Object.keys(e).forEach((t=>{Gi.has(t)&&delete e[t]})),(t={...this.constructor.Default,...e,..."object"==typeof t&&t?t:{}}).container=!1===t.container?document.body:r(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),a(Qi,t,this.constructor.DefaultType),t.sanitize&&(t.template=Yi(t.template,t.allowList,t.sanitizeFn)),t}_getDelegateConfig(){const t={};for(const e in this._config)this.constructor.Default[e]!==this._config[e]&&(t[e]=this._config[e]);return t}_cleanTipClass(){const t=this.getTipElement(),e=new 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file mode 100644 index 0000000..d9fd98f --- /dev/null +++ b/Dissertation/sharktooth_functions_files/libs/quarto-html/quarto-syntax-highlighting.css @@ -0,0 +1,203 @@ +/* quarto syntax highlight colors */ +:root { + --quarto-hl-ot-color: #003B4F; + --quarto-hl-at-color: #657422; + --quarto-hl-ss-color: #20794D; + --quarto-hl-an-color: #5E5E5E; + --quarto-hl-fu-color: #4758AB; + --quarto-hl-st-color: #20794D; + --quarto-hl-cf-color: #003B4F; + --quarto-hl-op-color: #5E5E5E; + --quarto-hl-er-color: #AD0000; + --quarto-hl-bn-color: #AD0000; + --quarto-hl-al-color: #AD0000; + --quarto-hl-va-color: #111111; + --quarto-hl-bu-color: inherit; + --quarto-hl-ex-color: inherit; + --quarto-hl-pp-color: #AD0000; + --quarto-hl-in-color: #5E5E5E; + --quarto-hl-vs-color: #20794D; + --quarto-hl-wa-color: #5E5E5E; + --quarto-hl-do-color: #5E5E5E; + --quarto-hl-im-color: #00769E; + --quarto-hl-ch-color: #20794D; + --quarto-hl-dt-color: #AD0000; + --quarto-hl-fl-color: #AD0000; + --quarto-hl-co-color: 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+code span.cv { + color: #5E5E5E; + font-style: italic; +} + +code span.cn { + color: #8f5902; + font-style: inherit; +} + +code span.sc { + color: #5E5E5E; + font-style: inherit; +} + +code span.dv { + color: #AD0000; + font-style: inherit; +} + +code span.kw { + color: #003B4F; + font-style: inherit; +} + +.prevent-inlining { + content: " { + // Find any conflicting margin elements and add margins to the + // top to prevent overlap + const marginChildren = window.document.querySelectorAll( + ".column-margin.column-container > * " + ); + + let lastBottom = 0; + for (const marginChild of marginChildren) { + if (marginChild.offsetParent !== null) { + // clear the top margin so we recompute it + marginChild.style.marginTop = null; + const top = marginChild.getBoundingClientRect().top + window.scrollY; + console.log({ + childtop: marginChild.getBoundingClientRect().top, + scroll: window.scrollY, + top, + lastBottom, + }); + if (top < lastBottom) { + const margin = lastBottom - top; + marginChild.style.marginTop = `${margin}px`; + } + const styles = window.getComputedStyle(marginChild); + const marginTop = parseFloat(styles["marginTop"]); + + console.log({ + top, + height: marginChild.getBoundingClientRect().height, + marginTop, + total: top + marginChild.getBoundingClientRect().height + marginTop, + }); + lastBottom = top + marginChild.getBoundingClientRect().height + marginTop; + } + } +}; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Recompute the position of margin elements anytime the body size changes + if (window.ResizeObserver) { + const resizeObserver = new window.ResizeObserver( + throttle(layoutMarginEls, 50) + ); + resizeObserver.observe(window.document.body); + } + + const tocEl = window.document.querySelector('nav.toc-active[role="doc-toc"]'); + const sidebarEl = window.document.getElementById("quarto-sidebar"); + const leftTocEl = window.document.getElementById("quarto-sidebar-toc-left"); + const marginSidebarEl = window.document.getElementById( + "quarto-margin-sidebar" + ); + // function to determine whether the element has a previous sibling that is active + const prevSiblingIsActiveLink = (el) => { + const sibling = el.previousElementSibling; + if (sibling && sibling.tagName === "A") { + return sibling.classList.contains("active"); + } else { + return false; + } + }; + + // fire slideEnter for bootstrap tab activations (for htmlwidget resize behavior) + function fireSlideEnter(e) { + const event = window.document.createEvent("Event"); + event.initEvent("slideenter", true, true); + window.document.dispatchEvent(event); + } + const tabs = window.document.querySelectorAll('a[data-bs-toggle="tab"]'); + tabs.forEach((tab) => { + tab.addEventListener("shown.bs.tab", fireSlideEnter); + }); + + // fire slideEnter for tabby tab activations (for htmlwidget resize behavior) + document.addEventListener("tabby", fireSlideEnter, false); + + // Track scrolling and mark TOC links as active + // get table of contents and sidebar (bail if we don't have at least one) + const tocLinks = tocEl + ? [...tocEl.querySelectorAll("a[data-scroll-target]")] + : []; + const makeActive = (link) => tocLinks[link].classList.add("active"); + const removeActive = (link) => tocLinks[link].classList.remove("active"); + const removeAllActive = () => + [...Array(tocLinks.length).keys()].forEach((link) => removeActive(link)); + + // activate the anchor for a section associated with this TOC entry + tocLinks.forEach((link) => { + link.addEventListener("click", () => { + if (link.href.indexOf("#") !== -1) { + const anchor = link.href.split("#")[1]; + const heading = window.document.querySelector( + `[data-anchor-id=${anchor}]` + ); + if (heading) { + // Add the class + heading.classList.add("reveal-anchorjs-link"); + + // function to show the anchor + const handleMouseout = () => { + heading.classList.remove("reveal-anchorjs-link"); + heading.removeEventListener("mouseout", handleMouseout); + }; + + // add a function to clear the anchor when the user mouses out of it + heading.addEventListener("mouseout", handleMouseout); + } + } + }); + }); + + const sections = tocLinks.map((link) => { + const target = link.getAttribute("data-scroll-target"); + if (target.startsWith("#")) { + return window.document.getElementById(decodeURI(`${target.slice(1)}`)); + } else { + return window.document.querySelector(decodeURI(`${target}`)); + } + }); + + const sectionMargin = 200; + let currentActive = 0; + // track whether we've initialized state the first time + let init = false; + + const updateActiveLink = () => { + // The index from bottom to top (e.g. reversed list) + let sectionIndex = -1; + if ( + window.innerHeight + window.pageYOffset >= + window.document.body.offsetHeight + ) { + sectionIndex = 0; + } else { + sectionIndex = [...sections].reverse().findIndex((section) => { + if (section) { + return window.pageYOffset >= section.offsetTop - sectionMargin; + } else { + return false; + } + }); + } + if (sectionIndex > -1) { + const current = sections.length - sectionIndex - 1; + if (current !== currentActive) { + removeAllActive(); + currentActive = current; + makeActive(current); + if (init) { + window.dispatchEvent(sectionChanged); + } + init = true; + } + } + }; + + const inHiddenRegion = (top, bottom, hiddenRegions) => { + for (const region of hiddenRegions) { + if (top <= region.bottom && bottom >= region.top) { + return true; + } + } + return false; + }; + + const categorySelector = "header.quarto-title-block .quarto-category"; + const activateCategories = (href) => { + // Find any categories + // Surround them with a link pointing back to: + // #category=Authoring + try { + const categoryEls = window.document.querySelectorAll(categorySelector); + for (const categoryEl of categoryEls) { + const categoryText = categoryEl.textContent; + if (categoryText) { + const link = `${href}#category=${encodeURIComponent(categoryText)}`; + const linkEl = window.document.createElement("a"); + linkEl.setAttribute("href", link); + for (const child of categoryEl.childNodes) { + linkEl.append(child); + } + categoryEl.appendChild(linkEl); + } + } + } catch { + // Ignore errors + } + }; + function hasTitleCategories() { + return window.document.querySelector(categorySelector) !== null; + } + + function offsetRelativeUrl(url) { + const offset = getMeta("quarto:offset"); + return offset ? offset + url : url; + } + + function offsetAbsoluteUrl(url) { + const offset = getMeta("quarto:offset"); + const baseUrl = new URL(offset, window.location); + + const projRelativeUrl = url.replace(baseUrl, ""); + if (projRelativeUrl.startsWith("/")) { + return projRelativeUrl; + } else { + return "/" + projRelativeUrl; + } + } + + // read a meta tag value + function getMeta(metaName) { + const metas = window.document.getElementsByTagName("meta"); + for (let i = 0; i < metas.length; i++) { + if (metas[i].getAttribute("name") === metaName) { + return metas[i].getAttribute("content"); + } + } + return ""; + } + + async function findAndActivateCategories() { + const currentPagePath = offsetAbsoluteUrl(window.location.href); + const response = await fetch(offsetRelativeUrl("listings.json")); + if (response.status == 200) { + return response.json().then(function (listingPaths) { + const listingHrefs = []; + for (const listingPath of listingPaths) { + const pathWithoutLeadingSlash = listingPath.listing.substring(1); + for (const item of listingPath.items) { + if ( + item === currentPagePath || + item === currentPagePath + "index.html" + ) { + // Resolve this path against the offset to be sure + // we already are using the correct path to the listing + // (this adjusts the listing urls to be rooted against + // whatever root the page is actually running against) + const relative = offsetRelativeUrl(pathWithoutLeadingSlash); + const baseUrl = window.location; + const resolvedPath = new URL(relative, baseUrl); + listingHrefs.push(resolvedPath.pathname); + break; + } + } + } + + // Look up the tree for a nearby linting and use that if we find one + const nearestListing = findNearestParentListing( + offsetAbsoluteUrl(window.location.pathname), + listingHrefs + ); + if (nearestListing) { + activateCategories(nearestListing); + } else { + // See if the referrer is a listing page for this item + const referredRelativePath = offsetAbsoluteUrl(document.referrer); + const referrerListing = listingHrefs.find((listingHref) => { + const isListingReferrer = + listingHref === referredRelativePath || + listingHref === referredRelativePath + "index.html"; + return isListingReferrer; + }); + + if (referrerListing) { + // Try to use the referrer if possible + activateCategories(referrerListing); + } else if (listingHrefs.length > 0) { + // Otherwise, just fall back to the first listing + activateCategories(listingHrefs[0]); + } + } + }); + } + } + if (hasTitleCategories()) { + findAndActivateCategories(); + } + + const findNearestParentListing = (href, listingHrefs) => { + if (!href || !listingHrefs) { + return undefined; + } + // Look up the tree for a nearby linting and use that if we find one + const relativeParts = href.substring(1).split("/"); + while (relativeParts.length > 0) { + const path = relativeParts.join("/"); + for (const listingHref of listingHrefs) { + if (listingHref.startsWith(path)) { + return listingHref; + } + } + relativeParts.pop(); + } + + return undefined; + }; + + const manageSidebarVisiblity = (el, placeholderDescriptor) => { + let isVisible = true; + let elRect; + + return (hiddenRegions) => { + if (el === null) { + return; + } + + // Find the last element of the TOC + const lastChildEl = el.lastElementChild; + + if (lastChildEl) { + // Converts the sidebar to a menu + const convertToMenu = () => { + for (const child of el.children) { + child.style.opacity = 0; + child.style.overflow = "hidden"; + } + + nexttick(() => { + const toggleContainer = window.document.createElement("div"); + toggleContainer.style.width = "100%"; + toggleContainer.classList.add("zindex-over-content"); + toggleContainer.classList.add("quarto-sidebar-toggle"); + toggleContainer.classList.add("headroom-target"); // Marks this to be managed by headeroom + toggleContainer.id = placeholderDescriptor.id; + toggleContainer.style.position = "fixed"; + + const toggleIcon = window.document.createElement("i"); + toggleIcon.classList.add("quarto-sidebar-toggle-icon"); + toggleIcon.classList.add("bi"); + toggleIcon.classList.add("bi-caret-down-fill"); + + const toggleTitle = window.document.createElement("div"); + const titleEl = window.document.body.querySelector( + placeholderDescriptor.titleSelector + ); + if (titleEl) { + toggleTitle.append( + titleEl.textContent || titleEl.innerText, + toggleIcon + ); + } + toggleTitle.classList.add("zindex-over-content"); + toggleTitle.classList.add("quarto-sidebar-toggle-title"); + toggleContainer.append(toggleTitle); + + const toggleContents = window.document.createElement("div"); + toggleContents.classList = el.classList; + toggleContents.classList.add("zindex-over-content"); + toggleContents.classList.add("quarto-sidebar-toggle-contents"); + for (const child of el.children) { + if (child.id === "toc-title") { + continue; + } + + const clone = child.cloneNode(true); + clone.style.opacity = 1; + clone.style.display = null; + toggleContents.append(clone); + } + toggleContents.style.height = "0px"; + const positionToggle = () => { + // position the element (top left of parent, same width as parent) + if (!elRect) { + elRect = el.getBoundingClientRect(); + } + toggleContainer.style.left = `${elRect.left}px`; + toggleContainer.style.top = `${elRect.top}px`; + toggleContainer.style.width = `${elRect.width}px`; + }; + positionToggle(); + + toggleContainer.append(toggleContents); + el.parentElement.prepend(toggleContainer); + + // Process clicks + let tocShowing = false; + // Allow the caller to control whether this is dismissed + // when it is clicked (e.g. sidebar navigation supports + // opening and closing the nav tree, so don't dismiss on click) + const clickEl = placeholderDescriptor.dismissOnClick + ? toggleContainer + : toggleTitle; + + const closeToggle = () => { + if (tocShowing) { + toggleContainer.classList.remove("expanded"); + toggleContents.style.height = "0px"; + tocShowing = false; + } + }; + + // Get rid of any expanded toggle if the user scrolls + window.document.addEventListener( + "scroll", + throttle(() => { + closeToggle(); + }, 50) + ); + + // Handle positioning of the toggle + window.addEventListener( + "resize", + throttle(() => { + elRect = undefined; + positionToggle(); + }, 50) + ); + + window.addEventListener("quarto-hrChanged", () => { + elRect = undefined; + }); + + // Process the click + clickEl.onclick = () => { + if (!tocShowing) { + toggleContainer.classList.add("expanded"); + toggleContents.style.height = null; + tocShowing = true; + } else { + closeToggle(); + } + }; + }); + }; + + // Converts a sidebar from a menu back to a sidebar + const convertToSidebar = () => { + for (const child of el.children) { + child.style.opacity = 1; + child.style.overflow = null; + } + + const placeholderEl = window.document.getElementById( + placeholderDescriptor.id + ); + if (placeholderEl) { + placeholderEl.remove(); + } + + el.classList.remove("rollup"); + }; + + if (isReaderMode()) { + convertToMenu(); + isVisible = false; + } else { + // Find the top and bottom o the element that is being managed + const elTop = el.offsetTop; + const elBottom = + elTop + lastChildEl.offsetTop + lastChildEl.offsetHeight; + + if (!isVisible) { + // If the element is current not visible reveal if there are + // no conflicts with overlay regions + if (!inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToSidebar(); + isVisible = true; + } + } else { + // If the element is visible, hide it if it conflicts with overlay regions + // and insert a placeholder toggle (or if we're in reader mode) + if (inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToMenu(); + isVisible = false; + } + } + } + } + }; + }; + + const tabEls = document.querySelectorAll('a[data-bs-toggle="tab"]'); + for (const tabEl of tabEls) { + const id = tabEl.getAttribute("data-bs-target"); + if (id) { + const columnEl = document.querySelector( + `${id} .column-margin, .tabset-margin-content` + ); + if (columnEl) + tabEl.addEventListener("shown.bs.tab", function (event) { + const el = event.srcElement; + if (el) { + const visibleCls = `${el.id}-margin-content`; + // walk up until we find a parent tabset + let panelTabsetEl = el.parentElement; + while (panelTabsetEl) { + if (panelTabsetEl.classList.contains("panel-tabset")) { + break; + } + panelTabsetEl = panelTabsetEl.parentElement; + } + + if (panelTabsetEl) { + const prevSib = panelTabsetEl.previousElementSibling; + if ( + prevSib && + prevSib.classList.contains("tabset-margin-container") + ) { + const childNodes = prevSib.querySelectorAll( + ".tabset-margin-content" + ); + for (const childEl of childNodes) { + if (childEl.classList.contains(visibleCls)) { + childEl.classList.remove("collapse"); + } else { + childEl.classList.add("collapse"); + } + } + } + } + } + + layoutMarginEls(); + }); + } + } + + // Manage the visibility of the toc and the sidebar + const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, { + id: "quarto-toc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + const sidebarScrollVisiblity = manageSidebarVisiblity(sidebarEl, { + id: "quarto-sidebarnav-toggle", + titleSelector: ".title", + dismissOnClick: false, + }); + let tocLeftScrollVisibility; + if (leftTocEl) { + tocLeftScrollVisibility = manageSidebarVisiblity(leftTocEl, { + id: "quarto-lefttoc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + } + + // Find the first element that uses formatting in special columns + const conflictingEls = window.document.body.querySelectorAll( + '[class^="column-"], [class*=" column-"], aside, [class*="margin-caption"], [class*=" margin-caption"], [class*="margin-ref"], [class*=" margin-ref"]' + ); + + // Filter all the possibly conflicting elements into ones + // the do conflict on the left or ride side + const arrConflictingEls = Array.from(conflictingEls); + const leftSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return false; + } + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + className.startsWith("column-") && + !className.endsWith("right") && + !className.endsWith("container") && + className !== "column-margin" + ); + }); + }); + const rightSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return true; + } + + const hasMarginCaption = Array.from(el.classList).find((className) => { + return className == "margin-caption"; + }); + if (hasMarginCaption) { + return true; + } + + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + !className.endsWith("container") && + className.startsWith("column-") && + !className.endsWith("left") + ); + }); + }); + + const kOverlapPaddingSize = 10; + function toRegions(els) { + return els.map((el) => { + const boundRect = el.getBoundingClientRect(); + const top = + boundRect.top + + document.documentElement.scrollTop - + kOverlapPaddingSize; + return { + top, + bottom: top + el.scrollHeight + 2 * kOverlapPaddingSize, + }; + }); + } + + let hasObserved = false; + const visibleItemObserver = (els) => { + let visibleElements = [...els]; + const intersectionObserver = new IntersectionObserver( + (entries, _observer) => { + entries.forEach((entry) => { + if (entry.isIntersecting) { + if (visibleElements.indexOf(entry.target) === -1) { + visibleElements.push(entry.target); + } + } else { + visibleElements = visibleElements.filter((visibleEntry) => { + return visibleEntry !== entry; + }); + } + }); + + if (!hasObserved) { + hideOverlappedSidebars(); + } + hasObserved = true; + }, + {} + ); + els.forEach((el) => { + intersectionObserver.observe(el); + }); + + return { + getVisibleEntries: () => { + return visibleElements; + }, + }; + }; + + const rightElementObserver = visibleItemObserver(rightSideConflictEls); + const leftElementObserver = visibleItemObserver(leftSideConflictEls); + + const hideOverlappedSidebars = () => { + marginScrollVisibility(toRegions(rightElementObserver.getVisibleEntries())); + sidebarScrollVisiblity(toRegions(leftElementObserver.getVisibleEntries())); + if (tocLeftScrollVisibility) { + tocLeftScrollVisibility( + toRegions(leftElementObserver.getVisibleEntries()) + ); + } + }; + + window.quartoToggleReader = () => { + // Applies a slow class (or removes it) + // to update the transition speed + const slowTransition = (slow) => { + const manageTransition = (id, slow) => { + const el = document.getElementById(id); + if (el) { + if (slow) { + el.classList.add("slow"); + } else { + el.classList.remove("slow"); + } + } + }; + + manageTransition("TOC", slow); + manageTransition("quarto-sidebar", slow); + }; + const readerMode = !isReaderMode(); + setReaderModeValue(readerMode); + + // If we're entering reader mode, slow the transition + if (readerMode) { + slowTransition(readerMode); + } + highlightReaderToggle(readerMode); + hideOverlappedSidebars(); + + // If we're exiting reader mode, restore the non-slow transition + if (!readerMode) { + slowTransition(!readerMode); + } + }; + + const highlightReaderToggle = (readerMode) => { + const els = document.querySelectorAll(".quarto-reader-toggle"); + if (els) { + els.forEach((el) => { + if (readerMode) { + el.classList.add("reader"); + } else { + el.classList.remove("reader"); + } + }); + } + }; + + const setReaderModeValue = (val) => { + if (window.location.protocol !== "file:") { + window.localStorage.setItem("quarto-reader-mode", val); + } else { + localReaderMode = val; + } + }; + + const isReaderMode = () => { + if (window.location.protocol !== "file:") { + return window.localStorage.getItem("quarto-reader-mode") === "true"; + } else { + return localReaderMode; + } + }; + let localReaderMode = null; + + const tocOpenDepthStr = tocEl?.getAttribute("data-toc-expanded"); + const tocOpenDepth = tocOpenDepthStr ? Number(tocOpenDepthStr) : 1; + + // Walk the TOC and collapse/expand nodes + // Nodes are expanded if: + // - they are top level + // - they have children that are 'active' links + // - they are directly below an link that is 'active' + const walk = (el, depth) => { + // Tick depth when we enter a UL + if (el.tagName === "UL") { + depth = depth + 1; + } + + // It this is active link + let isActiveNode = false; + if (el.tagName === "A" && el.classList.contains("active")) { + isActiveNode = true; + } + + // See if there is an active child to this element + let hasActiveChild = false; + for (child of el.children) { + hasActiveChild = walk(child, depth) || hasActiveChild; + } + + // Process the collapse state if this is an UL + if (el.tagName === "UL") { + if (tocOpenDepth === -1 && depth > 1) { + el.classList.add("collapse"); + } else if ( + depth <= tocOpenDepth || + hasActiveChild || + prevSiblingIsActiveLink(el) + ) { + el.classList.remove("collapse"); + } else { + el.classList.add("collapse"); + } + + // untick depth when we leave a UL + depth = depth - 1; + } + return hasActiveChild || isActiveNode; + }; + + // walk the TOC and expand / collapse any items that should be shown + + if (tocEl) { + walk(tocEl, 0); + updateActiveLink(); + } + + // Throttle the scroll event and walk peridiocally + window.document.addEventListener( + "scroll", + throttle(() => { + if (tocEl) { + updateActiveLink(); + walk(tocEl, 0); + } + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 5) + ); + window.addEventListener( + "resize", + throttle(() => { + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 10) + ); + hideOverlappedSidebars(); + highlightReaderToggle(isReaderMode()); +}); + +// grouped tabsets +window.addEventListener("pageshow", (_event) => { + function getTabSettings() { + const data = localStorage.getItem("quarto-persistent-tabsets-data"); + if (!data) { + localStorage.setItem("quarto-persistent-tabsets-data", "{}"); + return {}; + } + if (data) { + return JSON.parse(data); + } + } + + function setTabSettings(data) { + localStorage.setItem( + "quarto-persistent-tabsets-data", + JSON.stringify(data) + ); + } + + function setTabState(groupName, groupValue) { + const data = getTabSettings(); + data[groupName] = groupValue; + setTabSettings(data); + } + + function toggleTab(tab, active) { + const tabPanelId = tab.getAttribute("aria-controls"); + const tabPanel = document.getElementById(tabPanelId); + if (active) { + tab.classList.add("active"); + tabPanel.classList.add("active"); + } else { + tab.classList.remove("active"); + tabPanel.classList.remove("active"); + } + } + + function toggleAll(selectedGroup, selectorsToSync) { + for (const [thisGroup, tabs] of Object.entries(selectorsToSync)) { + const active = selectedGroup === thisGroup; + for (const tab of tabs) { + toggleTab(tab, active); + } + } + } + + function findSelectorsToSyncByLanguage() { + const result = {}; + const tabs = Array.from( + document.querySelectorAll(`div[data-group] a[id^='tabset-']`) + ); + for (const item of tabs) { + const div = item.parentElement.parentElement.parentElement; + const group = div.getAttribute("data-group"); + if (!result[group]) { + result[group] = {}; + } + const selectorsToSync = result[group]; + const value = item.innerHTML; + if (!selectorsToSync[value]) { + selectorsToSync[value] = []; + } + selectorsToSync[value].push(item); + } + return result; + } + + function setupSelectorSync() { + const selectorsToSync = findSelectorsToSyncByLanguage(); + Object.entries(selectorsToSync).forEach(([group, tabSetsByValue]) => { + Object.entries(tabSetsByValue).forEach(([value, items]) => { + items.forEach((item) => { + item.addEventListener("click", (_event) => { + setTabState(group, value); + toggleAll(value, selectorsToSync[group]); + }); + }); + }); + }); + return selectorsToSync; + } + + const selectorsToSync = setupSelectorSync(); + for (const [group, selectedName] of Object.entries(getTabSettings())) { + const selectors = selectorsToSync[group]; + // it's possible that stale state gives us empty selections, so we explicitly check here. + if (selectors) { + toggleAll(selectedName, selectors); + } + } +}); + +function throttle(func, wait) { + let waiting = false; + return function () { + if (!waiting) { + func.apply(this, arguments); + waiting = true; + setTimeout(function () { + waiting = false; + }, wait); + } + }; +} + +function nexttick(func) { + return setTimeout(func, 0); +} diff --git a/Dissertation/sharktooth_functions_files/libs/quarto-html/tippy.css b/Dissertation/sharktooth_functions_files/libs/quarto-html/tippy.css new file mode 100644 index 0000000..e6ae635 --- /dev/null +++ b/Dissertation/sharktooth_functions_files/libs/quarto-html/tippy.css @@ -0,0 +1 @@ +.tippy-box[data-animation=fade][data-state=hidden]{opacity:0}[data-tippy-root]{max-width:calc(100vw - 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b/Dissertation/sharktooth_functions_multidimensional.ipynb new file mode 100644 index 0000000..be2234e --- /dev/null +++ b/Dissertation/sharktooth_functions_multidimensional.ipynb @@ -0,0 +1,260 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Sharktooth functions in 2-dimensions" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "def max_conv_operator(samples, f_samples, input, L):\n", + " return np.max(f_samples - L * np.linalg.norm(input - samples, ord=1))" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'f' is not defined", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "\u001b[1;32m/Users/shakilrafi/Library/Mobile Documents/com~apple~CloudDocs/Dissertation/sharktooth_functions_multidimensional.ipynb Cell 4\u001b[0m line \u001b[0;36m4\n\u001b[1;32m 2\u001b[0m y \u001b[39m=\u001b[39m np\u001b[39m.\u001b[39mlinspace(\u001b[39m0\u001b[39m,\u001b[39m1\u001b[39m,\u001b[39m100\u001b[39m)\n\u001b[1;32m 3\u001b[0m samples \u001b[39m=\u001b[39m np\u001b[39m.\u001b[39mrandom\u001b[39m.\u001b[39muniform([\u001b[39m0\u001b[39m,\u001b[39m0\u001b[39m], [\u001b[39m1\u001b[39m,\u001b[39m1\u001b[39m],size\u001b[39m=\u001b[39m(\u001b[39m100\u001b[39m,\u001b[39m2\u001b[39m))\n\u001b[0;32m----> 4\u001b[0m f_samples \u001b[39m=\u001b[39m f(samples[:,\u001b[39m0\u001b[39m],samples[:,\u001b[39m1\u001b[39m])\n\u001b[1;32m 5\u001b[0m points \u001b[39m=\u001b[39m []\n\u001b[1;32m 6\u001b[0m \u001b[39mfor\u001b[39;00m i \u001b[39min\u001b[39;00m \u001b[39mrange\u001b[39m(\u001b[39mlen\u001b[39m(x)):\n", + "\u001b[0;31mNameError\u001b[0m: name 'f' is not defined" + ] + } + ], + "source": [ + "x = np.linspace(0,1,100)\n", + "y = np.linspace(0,1,100)\n", + "samples = np.random.uniform([0,0], [1,1],size=(100,2))\n", + "f_samples = f(samples[:,0],samples[:,1])\n", + "points = []\n", + "for i in range(len(x)):\n", + " for j in range(len(y)):\n", + " points.append([x[i],y[j],max_conv_operator(samples, f_samples,[x[i],y[j]],L)])" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = plt.figure()\n", + "ax = fig.add_subplot(111,projection='3d') # Create a 3D axes object\n", + "fig = plt.figure()\n", + " # Create a 3D axes object\n", + "ax.scatter(points[0], points[1], points[2], c='b', marker='o')\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "samples = np.random.uniform([0,0], [1,1],size=(10,2))\n", + "f_samples = f(samples[:,0],samples[:,1])\n", + "x = np.linspace(0,1,100)\n", + "y = np.linspace(0,1,100)\n", + "X, Y = np.meshgrid(x,y)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-43.00870965517393" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "max_conv_operator(samples, f_samples, [4,5],1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 18, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "fig = plt.figure()\n", + "ax = fig.add_subplot(111, projection='3d')\n", + "ax.plot_surface(X,Y,X**2+Y**2, cmap = \"viridis\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def sharktooth_function(function, mins, maxes, number_of_sharkteeth, L):\n", + " samples = np.random.uniform(mins, maxes,size=(number_of_sharkteeth,len(mins)))\n", + " f_samples = f(samples[:,0],samples[:,1])\n", + " X_cords = np.linspace(mins[0],maxes[0],100)\n", + " Y_cords = np.linspace(mins[1],maxes[1],100)\n", + " approximate_z = []\n", + " error = []\n", + " for i in range(len(X_cords)):\n", + " for j in range(len(Y_cords)):\n", + " approximate_z = max_conv_operator(samples, f_samples, [X_cords[i],Y_cords[j]],L)\n", + " error.append(np.abs(f(X_cords[i],Y_cords[j]) - approximate_z))\n", + " return np.max(error)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Text(0.5, 1.0, 'Sup of deviance from f(x) as the number of teeth increase and as we get closer to L')" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "\n", + "L = [0.1,0.5,0.75,1]\n", + "plt.figure(figsize=(17,5))\n", + "for L in L:\n", + " errors = []\n", + " for i in range(1,20):\n", + " errors.append(sharktooth_function(f,[0,0],[1,0],i,L))\n", + " plt.plot(errors, label =L)\n", + "\n", + "plt.xlabel(\"Number of teeth\")\n", + "plt.ylabel(r\"$\\sup_{x\\in x_i} | f(x)- \\mathfrak{R}_{\\mathfrak{r}} ( \\mathsf{P})|$\")\n", + "plt.legend()\n", + "plt.title(\"Sup of deviance from f(x) as the number of teeth increase and as we get closer to L\")" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "base", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.4" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Dissertation/sharktooth_functions_multidimensional.pdf b/Dissertation/sharktooth_functions_multidimensional.pdf new file mode 100644 index 0000000..b6140b7 Binary files /dev/null and b/Dissertation/sharktooth_functions_multidimensional.pdf differ diff --git a/Dissertation/sharktooth_multidimensional.jl b/Dissertation/sharktooth_multidimensional.jl new file mode 100644 index 0000000..12e13be --- /dev/null +++ b/Dissertation/sharktooth_multidimensional.jl @@ -0,0 +1,12 @@ +using Random +using Distributions +using Plots +using LinearAlgebra +# =============================================================== + +function max_conv_operator(samples, f_samples, input, L) + return maximum(f_samples .- L .* norm(input .- samples, 1)) +end + +x = rand(Uniform(0,1),100) +y = rand(Uniform(0,1),100) \ No newline at end of file diff --git a/Dissertation/sharktooth_multidimensional.nb b/Dissertation/sharktooth_multidimensional.nb new file mode 100644 index 0000000..13da533 --- /dev/null +++ b/Dissertation/sharktooth_multidimensional.nb @@ -0,0 +1,44 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 13.3' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 158, 7] +NotebookDataLength[ 815, 34] +NotebookOptionsPosition[ 561, 21] +NotebookOutlinePosition[ 955, 37] +CellTagsIndexPosition[ 912, 34] +WindowFrame->Normal*) + +(* Beginning of Notebook Content *) +Notebook[{ +}, +WindowSize->{808, 796}, +WindowMargins->{{315, Automatic}, {21, Automatic}}, +FrontEndVersion->"13.3 for Mac OS X ARM (64-bit) (July 24, 2023)", +StyleDefinitions->"Default.nb", +ExpressionUUID->"3e63ad8f-9f2d-4c31-8e59-19c54a579337" +] 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We do this by first introducing viscosity solutions to Kolmogorov PDEs as given in Crandall \& Lions \cite{crandall_lions} and further extended, esp. in \cite{Beck_2021}. +%\subsection{The case without $f$} +%\subsection{Linear Algebra Preliminaries} +%\begin{lemma} +% For a matrix $A \in \R^{m\times n}$, $A[A]^*$ is symmetric. +%\end{lemma} +%\begin{proof} +% Note that $[A]_{i,j} = [A^*]_{j,i}$, and hence: +% \begin{align} +% [AA^*]_{i,j} = \sum^n_{k=1} [A]_{i,k}[A^*]_{k,j} = [AA^*]_{j,i} +% \end{align} +%\end{proof} +%\begin{lemma} +% A symmetric matrix $A\in \R^{m\times n}$: +% \begin{enumerate}[label = (\roman*)] +% \item Has only real eigenvalues. +% \item Is always diagonalizable. +% \end{enumerate} +%\end{lemma} +%\begin{proof} +% \textit{(i)} Assume $\lambda \in \mathbb{C}$ is an eigenvalue for the symmetric matrix A. This indicates that $Av = \lambda v$, where $v\neq 0$ for some $v$ in our vector space and further that: +% \begin{align} +% v^*Av = \lambda v^*v +% \end{align} +% where $v^* = \overline{v}^T$. +% Note that since $A$ is symmetric we have: +% \begin{align} +% \lambda v^* v = v^*Av = \lp v^*Av\rp^* = \overline{\lambda}v^*v +% \end{align} +% This indicates that $\lambda = \overline{\lambda}$ whence $\lambda$ is real. +% \medskip +% +% \textit{(ii)} Assume $A$ is not diagonalizable. Then there must exist some eigenvalue $\lambda_i$ of order 2 or more. This would indicate that there is some repeated eigenvalue $\lambda$, and $v\neq 0$ such that: +% \begin{align} +% (A-\lambda I)^2v = 0 & \text{ and }(A-\lambda I)v \neq 0 +% \end{align} +% +% But note that: +% \begin{align} +% 0 = v^*(A-\lambda I)^2v = v^*(A-\lambda I)(A - \lambda I) \neq 0 +% \end{align} +% Leading to a contradiction. Thus there are no generalized eigenvectors of order 2 or higher, and so $A$ must be diagonalizable. +%\end{proof} +\section{Some Preliminaries} + We take work previously pioneered by \cite{Ito1942a} and \cite{Ito1946}, and then seek to re-apply concepts first applied in \cite{Beck_2021} and \cite{BHJ21}. + \begin{lemma}\label{lemma:2.7} + Let $d,m \in \N$, $T \in (0,\infty)$. Let $\mu \in C^{1,2}([0,T] \times \R^d, \R^d)$ and $\sigma \in C^{1,2}([0,T] \times \R^d, \R^{d\times m})$ satisfying that they have non-empty compact supports and let $\mathfrak{S}= \supp(\mu)\cup \supp(\sigma) \subseteq [0,T] \times \R^d$. Let $( \Omega, \mathcal{F}, \mathbb{P}, ( \mathbb{F}_t )_{t \in [0,T]})$ be a filtered probability space satisfying usual conditions. Let $W:[0,T ]\times \Omega \rightarrow \R^m$ be a standard $(\mathbb{F}_t)_{t\in [0,T]}$ -Brownian motion, and let $\mathcal{X}:[0,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_t)_{t\in [0,T]}$-adapted stochastic process with continuous sample paths satisfying for all $t \in [0,T]$ with $\mathbb{P}$-a.s. that: + \begin{align} + \mathcal{X}_t = \mathcal{X}_0 + \int^t_0 \mu(s, \mathcal{X}_s) ds + \int^t_0 \sigma(s, \mathcal{X}_s)dW_s + \end{align} + It then holds that: + \begin{enumerate}[label = (\roman*)] + \item $\lb \lp \mathbb{P} \lp \mathcal{X}_0 \not \in \mathfrak{S} \rp = 1 \rp \implies \lp \mathbb{P} \lp \forall t \in [0,T]: \mathcal{X}_t = \mathcal{X}_0 \rp =1 \rp \rb $ + \item $\lb \lp \mathbb{P} \lp \mathcal{X}_0 \in \mathfrak{S} \rp = 1 \rp \implies \lp \mathbb{P} \lp \forall t \in [0,T]: \mathcal{X}_t \in \mathfrak{S} \rp = 1 \rp \rb$ + \end{enumerate} + \end{lemma} + \begin{proof} + Assume that $\mathbb{P}(\mathcal{X}_0 \not \in \mathfrak{S} ) =1$, meaning that the particle almost surely starts outside $\mathfrak{S}$. It is then the case that $\mathbb{P}( \forall t \in [0, T]: \|\mu(t,\mathcal{X}_0)\|_E + \|\sigma(t,\mathcal{X}_0)\|_F =0)=1$ as the $\mu$ and $\sigma$ are outside their supports, and we integrate over zero over time. + \medskip + + It is then the case that: + \begin{align} + \mathcal{Y}:= \lp \lb 0,T \rb \times \Omega \ni \lp t, \omega \rp \mapsto \mathcal{X}_0(\omega)\in \R^d \rp + \end{align} + is an $(\mathbb{F}_t)_{t \in [0,T]}$ adapted stochastic process with continuous sample paths satisfying that for all $t \in [0,T]$ with $\mathbb{P}$-almost surety that: + \begin{align} + \mathcal{Y}_t = \mathcal{X}_0 + \int^t_0 0 ds + \int^t_0 0 dW_s &= \mathcal{X}_0 + \int^t_0 \mu(s,\mathcal{X}_0)ds + \int^t_0 \sigma(s, \mathcal{X}_0) dW_s \nonumber\\ + &= \mathcal{X}_0 + \int^t_0 \mu(s,\mathcal{Y}_s) ds + \int^t_0 \sigma(s,\mathcal{Y}_s) dW_s + \end{align} + Note that since $\mu \in C^{1,2}([0, T] \times \R^d, \R^d)$ and $\sigma \in C^{1,2}([0, T] \times \R^d, \R^{d \times m})$, and since continuous functions are locally Lipschitz, and since this is especially true in the space variable for $\mu$ and $\sigma$, the fact that $\mathfrak{S}$ is compact and continuous functions over compact sets are Lipschitz and bounded, and \cite[Theorem~5.2.5]{karatzas1991brownian} allows us to conclude that strong uniqueness holds, that is to say: + \begin{align} + \mathbb{P}\lp \forall t \in [0,T]: \mathcal{X}_t = \mathcal{X}_0 \rp = \mathbb{P} \lp \forall t \in [0,T]: \mathcal{X}_t = \mathcal{Y}_t \rp=1 + \end{align} + establishing the case (i). + + Assume now that $\mathbb{P}(\mathcal{X}_0 \in \mathfrak{S})=1$ that is to say that the particle almost surely starts inside $\mathfrak{S}$. We define $\tau: \Omega \rightarrow [0,T]$ as $\tau= \inf\{t \in [0,T]: \mathcal{X}_t \not \in \overline{\mathfrak{S}}\}$. $\tau$ is an $(\mathbb{F}_t)_{t\in [0,T]}$-adapted stopping time. On top of $\tau$ we can define $\mathcal{Y}:[0,T] \times \Omega \rightarrow\R^d$, for all $t\in [0,T]$, $\omega \in \Omega$ as $\mathcal{Y}_t(\omega) =\mathcal{X}_{\min \{t,\tau\}}(\omega)$. $\mathcal{Y}$ is thus an $(\mathbb{F}_t)_{t \in [0,T]}$-adapted stochastic process with continuous sample paths. Note however that for $t > \tau$ it is the case $\|\mu(t, \mathcal{Y}_t)+\sigma(t, \mathcal{Y}_t) \|_E=0$ as we are outside their supports. For $t < \tau$ it is also the case that $\mathcal{Y}_t = \mathcal{X}_t$. This yields with $\mathbb{P}$-a.s. that: + \begin{align} + \mathcal{Y}_t = \mathcal{X}_{\min\{t,\tau \}} &= \mathcal{X}_0 + \int^{\min\{t,\tau\}}_0 \mu(s, \mathcal{X}_s) ds + \int^{\min\{t,\tau\}}_0 \sigma(s,\mathcal{X}_s)dW_s \nonumber \\ + &=\mathcal{X}_0 + \int^t_0 \mathbbm{1}_{\{0 < s \leqslant \tau \}} \mu (s, \mathcal{X}_s) ds+\int^t_0 \mathbbm{1}_{\{0 0$, there exists $\delta > 0$ such that: + \begin{align} + f(x) < f(x_0) + \ve \text{ for all } x \in B\lp x_0, \delta \rp \cap U + \end{align} +\end{definition} +\begin{definition}[Lower semi-continuity] + A function $f: U \rightarrow \R$ is lower semi-continuous at $x_0$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that: + \begin{align} + f(x) > f(x_0) - \varepsilon \text{ for all } x \in B\lp x_0, \delta \rp \cap U + \end{align} +\end{definition} +\begin{corollary}\label{sumofusc} + Given two upper semi-continuous functions $f,g: \R^d \rightarrow \R$, their sum $(f+g): \R^d \rightarrow \R$ is also upper semi-continuous. +\end{corollary} +\begin{proof} + From definitions, at any given $x_0 \in \R^d$, for any $\ve \in (0, \infty)$ there exist neighborhoods $U$ and $V$ around $x_0$ such that: + \begin{align} + \lp \forall x \in U \rp \lp f(x) \leqslant f(x_0) + \varepsilon \rp \\ + \lp \forall x \in V \rp \lp g(x) \leqslant g(x_0) + \varepsilon \rp + \end{align} + and hence: + \begin{align} + \lp \forall x \in U \cap V \rp \lp f(x) + g(x) \leqslant f(x_0)+g(x_0) + 2\varepsilon \rp + \end{align} +\end{proof} +\begin{corollary}\label{neglsc} + Given an upper semi-continuous function $f:\R^d \rightarrow \R$, it is the case that $(-f):\R^d \rightarrow \R$ is lower semi-continuous. +\end{corollary} + +\begin{proof} + Let $f:\R^d \rightarrow \R$ be upper semi-continuous. At any given $x_0 \in \R^d$, for any $\ve \in (0, \infty)$ there exists a neighborhood $U$ around $x_0$ such that: + \begin{align} + \lp \forall x \in U \rp \lp f(x) \leqslant f(x_0) + \ve \rp + \end{align} + This also means that: + \begin{align} + &\lp \forall x \in U \rp \lp -f(x) \geqslant -f(x_0) - \ve \rp \nonumber\\ + \end{align} + This completes the proof. +\end{proof} + +\begin{definition}[Degenerate Elliptic Functions] Let $d \in \N$, $T \in \lp 0, \infty \rp$, let $\mathcal{O} \subseteq \R^d$ be a non-empty open set, and let $\la \cdot, \cdot \ra: \R^d \times \R^d \rightarrow \R$ be the standard Euclidean inner product on $\R^d$. $G$ is degenerate elliptic on $\lp 0,T \rp \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ if and only if: +\begin{enumerate}[label = (\roman*)] + \item $G: \lp 0,T \rp \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ is a function, and + \item for all $t \in \lp 0 ,T \rp$, $x \in \mathcal{O}$, $r \in \R$, $p \in \R^d$, $A,B \in \mathbb{S}_d$, with $\forall y \in \R^d$: $\la Ay,y \ra \leqslant \la By, y \ra$ that $G(t,x,r,p,A) \leqslant G(t,x,r,p,B)$. +\end{enumerate} +\end{definition} +\begin{remark} + Let $t \in (0,T)$, $x \in \R^d$, $r\in \R$, $p\in \R^d$, $A \in \mathbb{S}_d$. Let $u \in C^{1,2} ([0,T] \times \R^d, \R)$, and let $\sigma: \R^d \rightarrow \R^{d\times d}$ and $\mu: \R^d \rightarrow \R^d$ be infinitely often differentiable. The function $G:(0,T) \times \R^d \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ given by: + \begin{align} + G(t,x,r,p,A) = \frac{1}{2} \Trace \lp \sigma(x) \lb \sigma(x)\rb^* \lp \Hess_x u \rp \lp t,x \rp \rp + \la \mu(t,x), \nabla_xu\lp t,x\rp \ra + \end{align} + where $\lp t,x,u(t,x),\mu(x), \sigma(x) \lb \sigma(x) \rb^* \rp \in \lp 0,T\rp \times \R^d \times \R \times \R^d \times \mathbb{S}_d$, is degenerate elliptic. +\end{remark} + +\begin{lemma}\label{negdegel} + Given a function $G: (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ that is degerate elliptic on $(0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ it is also the case that $H: (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ given by $H(t,x,r,p,A)=-G(t,x,-r, -p,-A)$ is degenerate elliptic on $(0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$. +\end{lemma} + +\begin{proof} + Note that $H$ is a function. Assume for $y\in \R^d$ it is the case that $\la Ay, y \ra \leqslant \la By,y \ra$ then it is also the case by (\ref{bigsum}) that $\la -Ay,y\ra \geqslant \la -By, y \ra$ for $y\in \R^d$. However since $G$ is monotoically increasing over the subset of $(0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ where $\la Ay,y \ra \leqslant \la By,y \ra$ then it is also the case that $H(t,x,r,p,A) =-G(t,x,-r,-p, -A) \geqslant -G(t,x,-r,-p,-B)=H(t,x,r,p,B)$. + + + +\end{proof} + +\begin{definition}[Viscosity subsolutions]\label{def:viscsubsolution} + Let $d \in \N$, $T \in \lp 0, \infty \rp$, let $\mathcal{O} \subseteq \R^d$ be a non-empty open set, and let $G: \lp 0, T \rp \times \mathcal{O} \times \R \times \R^d \times\mathbb{S}_d \rightarrow \R$ be degenrate elliptic. Then we say that $u$ is a viscosity solution of $\lp \frac{\partial}{\partial t} u \rp \lp t,x \rp + G \lp t,x,u(t,x), \lp \nabla_x u \rp \lp t,x \rp, \lp \Hess_x u \rp \lp t,x \rp \rp \geqslant 0$ for $\lp t,x, \rp \in \lp 0,T \rp \times \mathcal{O}$ if and only if there exists a set $A$ such that: + \begin{enumerate}[label = (\roman*)] + \item we have that $\lp 0,T \rp \times \mathcal{O} \subseteq A$. + \item we have that $u: A \rightarrow \R$ is an upper semi-continuous function from $A$ to $\R$, and + \item we have that for all $t\in \lp 0, T \rp$, $x \in \mathcal{O}$, $\phi \in C^{1,2} \lp \lp 0,T \rp \times \mathcal{O}, \R \rp$ with $\phi (t,x) = u (t,x)$ and $\phi \geqslant u$ that: + \begin{align} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + G \lp t,x, \phi(t,x), \lp \nabla_x \phi \rp \lp t,x \rp, \lp \Hess_x \phi \rp \lp t,x \rp \rp \geqslant 0 + \end{align} + \end{enumerate} +\end{definition} + +\begin{definition}[Viscosity supersolutions]\label{def:viscsupsolution} + Let $d \in \N$, $T \in \lp 0, \infty \rp$, let $\mathcal{O} \subseteq \R^d$ be a non-empty open set, and let $G: \lp 0, T \rp \times \mathcal{O} \times \R \times \R^d \times\mathbb{S}_d \rightarrow \R$ be degenrate elliptic. Then we say that $u$ is a viscosity solution of $\lp \frac{\partial}{\partial t} u \rp \lp t,x \rp + G \lp t,x,u(t,x), \lp \nabla_x u \rp \lp t,x \rp, \lp \Hess_x u \rp \lp t,x \rp \rp \leqslant 0$ for $\lp t,x, \rp \in \lp 0,T \rp \times \mathcal{O}$ if and only if there exists a set $A$ such that: + \begin{enumerate}[label = (\roman*)] + \item we have that $\lp 0,T \rp \times \mathcal{O} \subseteq A$. + \item we have that $u: A \rightarrow \R$ is an upper semi-continuous function from $A$ to $\R$, and + \item we have that for all $t\in \lp 0, T \rp$, $x \in \mathcal{O}$, $\phi \in C^{1,2} \lp \lp 0,T \rp \times \mathcal{O}, \R \rp$ with $\phi (t,x) = u (t,x)$ and $\phi \leqslant u$ that: + \begin{align} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + G \lp t,x, \phi(t,x), \lp \nabla_x \phi \rp \lp t,x \rp, \lp \Hess_x \phi \rp \lp t,x \rp \rp \leqslant 0 + \end{align} + \end{enumerate} +\end{definition} +\begin{definition}[Viscosity solution]\label{def:viscsolution} + Let $d \in \N$, $T \in \lp 0, \infty \rp$, $\mathcal{O} \subseteq \R^d$ be a non-empty open set and let $G: \lp 0, T \rp \times \mathcal{O} \times R \times \R^d \times \mathbb{S}_d \rightarrow \R$ be degenerate elliptic. Then we say that $u_d$ is a viscosity solution to $\lp \frac{\partial }{\partial t} u_d \rp (t,x) + G(t,x,u(t,x), \nabla_x(x,t), (\Hess_x u_d)(t,x))$ if and only if: + \begin{enumerate}[label = (\roman*)] + \item $u$ is a viscosity subsolution of $\lp \frac{\partial }{\partial t} u_d \rp (t,x) + G(t,x,u(t,x), \nabla_x(x,t), (\Hess_x u_d)(t,x))=0$ for $(t,x) \in (0,T) \times \mathcal{O}$ + \item $u$ is a viscosity supersolution of $\lp \frac{\partial }{\partial t} u_d \rp (t,x) + G(t,x,u(t,x), \nabla_x(x,t), (\Hess_x u_d)(t,x))=0$ for $(t,x) \in (0,T) \times \mathcal{O}$ + \end{enumerate} + +\end{definition} + +% +%\begin{lemma} +% Let $T\in (0, \infty)$, let $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, let $\sigma_d: \R^d \rightarrow \R^{d\times d}$, $d\in \N$, be infinitely often differentiable functions, let $u_d \in C^{1,2} \left( \left[ 0,T \right] \times \R^d, \R \right)$, $d\in \N$, satisfy for all $d\in \N$, $t \in \left[ 0,T \right]$, $x \in \R^d$ that: +% \begin{align} +% \left( \frac{\partial}{\partial t} u_d \right)\left( t,x \right) + \text{Trace} \left( \sigma(x) \left[ \sigma(x) \right]* \left( \text{Hess}_x u_d \right)(t,x) \right) =0 +% \end{align} +% +%Let $W^d: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: [t,T] \times \Omega \rightarrow \R^d$, $d \in \N$, $t \in [0,T]$, $x \in \R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in \N$, $t\in [0,T]$, $s \in [t,T]$, $x \in \R^d$, we have $\mathbb{P}$-a.s. that: +%\begin{align} +% \mathcal{X}^{d,t,x}_s = x + \int^t_s \sqrt{2} \sigma \left( \mathcal{X}^{d,t,x} \right) dW^d_r +%\end{align} +%Then for all $d\in \N$, $t \in [0,T]$, $x \in \R^d$ it holds that: +%\begin{align} +% u_d \left( t,x \right) = \E \left[ u_d \left( T, \mathcal{X}^{d,t,x}_T \right) \right] +%\end{align} +%\end{lemma} +% +%First we consider the case where $\sigma: \R^d \rightarrow \R^{d\times d}$ is the constant function: +%\begin{align*} +%\sigma: \begin{pmatrix} +% x_1 \\ +% x_2 \\ +% \vdots\\ +% x_d +%\end{pmatrix} \rightarrow \begin{pmatrix} +% c_{1,1} & c_{2,2} & \hdots & c_{1,d}\\ +% c_{2,1} & & & \vdots \\ +% \vdots & & \ddots\\ +% c_{d,1} & \hdots & & c_{d,d} +%\end{pmatrix} +%\end{align*} +% +%and hence $\sigma \sigma^*$ is a $d \times d$ matrix where $[\sigma \sigma^*]_{i,j} = \sum^d_{k=1}\left( c_{i,k} \times c_{j,k} \right)$. Observe also that: +%\begin{align*} +% \text{Hess}_x u_d &= \begin{pmatrix} +% \frac{\partial^2 u_d(t,x)}{\partial x_1 \partial x_1} & \cdots & \frac{\partial^2 u_d(t,x)}{\partial x_1 \partial x_d} \\ +% \vdots & & \vdots \\ +% \frac{\partial^2 u_d(t,x)}{\partial x_d \partial x_1} & \cdots & \frac{\partial^2 u_d(t,x )}{\partial x_d^2} +% \end{pmatrix} +%\end{align*} +% +%Their product is the matrix +%\begin{align} +%[(\sigma [\sigma^*])(\text{Hess}_x u)(t,x)]_{i,j} = \sum^d_{l=1}\left( \sum^d_{k=1} \left( c_{i,k} \times c_{l,k} \right) \left(\frac{\partial u_d(t,x)}{\partial x_l \partial x_j} \right) \right) +%\end{align} +%And hence: +%\begin{align} +% \text{Trace}[(\sigma [\sigma^*])(\text{Hess}_x u)(t,x)] +% &= \sum^d_{m=1} \left(\sum^d_{l=1}\left( \sum^d_{k=1} \left( c_{m,k} \times c_{l,k} \right) \left(\frac{\partial u_d(t,x)}{\partial x_l \partial x_m} \right) \right) \right) \nonumber\\ +% &= \sum^d_{m=1} \left(\sum^d_{l=1}\left( \sum^d_{k=1} \left( c_{m,k} \times c_{l,k} \right) \right)\left(\frac{\partial u_d(t,x)}{\partial x_l \partial x_m} \right) \right) \nonumber\\ +% &= \sum^d_{m=1} \sum^d_{l=1}\mathfrak{C}^{m,d,l} \frac{\partial u_d(t,x)}{\partial x_l \partial x_m} +%\end{align} +%\medskip +% +%Where $\mathfrak{C}^{m,d,l} = \sum^d_{k=1} \left( c_{m,k}\times c_{l,k} \right)$ +%\medskip +% +%Note that for each row it is the case that: +% +% +% +%This also renders (2.12) as: +%\begin{align} +%\left( \frac{\partial}{\partial t} u_d \right)\left( t,x \right) + \sum^d_{m=1} \sum^d_{l=1}\mathfrak{C}^{m,d,l} \frac{\partial u_d(t,x)}{\partial x_l \partial x_m} = 0 +%\end{align} +% +%This renders (2.13) as: +% +%\begin{align} +% \mathcal{X}^{d,t,x}_s &= \begin{pmatrix} +% x_1 \\ +% x_2 \\ +% \vdots\\ +% x_d +% \end{pmatrix} + \int^t_s \sqrt{2} \begin{pmatrix} +% c_{1,1} & c_{1,2} & \hdots & c_{1,d}\\ +% c_{2,1} & & & \vdots \\ +% \vdots & & \ddots\\ +% c_{d,1} & \hdots & & c_{d,d} +% \end{pmatrix} dW^d_r \nonumber\\ +% &= \begin{pmatrix} +% x_1 \\ +% x_2 \\ +% \vdots\\ +% x_d +% \end{pmatrix} + \sqrt{2} \begin{pmatrix} +% c_{1,1} & c_{1,2} & \hdots & c_{1,d}\\ +% c_{2,1} & & & \vdots \\ +% \vdots & & \ddots\\ +% c_{d,1} & \hdots & & c_{d,d} +% \end{pmatrix} W^d_{t-s} +%\end{align} +% +%\medskip +% +%Note that $W^d \in \R^d$, and specifically let $W^d_{t-s}$ be the $d$-dimensional vector: +%\begin{align} +% W^d_{t-s} = \begin{pmatrix} +% W_{1,t-s} \\ +% W_{2,t-s} \\ +% \vdots \\ +% W_{d,t-s} +% \end{pmatrix} +%\end{align} +%\medskip +% +%This then renders (2.16) as: +%\begin{align} +% \mathcal{X}^{d,t,x}_s &= +% \begin{pmatrix} +% x_1 \\ +% x_2 \\ +% \vdots\\ +% x_d +% \end{pmatrix} + \sqrt{2} \begin{pmatrix} +% c_{1,1} & c_{1,2} & \hdots & c_{1,d}\\ +% c_{2,1} & & & \vdots \\ +% \vdots & & \ddots\\ +% c_{d,1} & \hdots & & c_{d,d} +% \end{pmatrix} \begin{pmatrix} +% W_{1,t-s} \\ +% W_{2,t-s} \\ +% \vdots \\ +% W_{d,t-s} +% \end{pmatrix} \nonumber \\ +% &= \begin{pmatrix} +% x_1 \\ +% x_2 \\ +% \vdots\\ +% x_d +% \end{pmatrix} + \sqrt{2} \begin{pmatrix} +% \sum^d_{i=1}\left(c_{1,i} W_{i,t-s}\right) \\ +% \sum^d_{i=1}\left(c_{2,i} W_{i,t-s}\right) \\ +% \vdots \\ +% \sum^d_{i=1}\left(c_{d,i} W_{i,t-s}\right) \\ +% \end{pmatrix} \nonumber \\ +% &= \begin{pmatrix} +% x_1 + \sqrt{2}\sum^d_{i=1}\left(c_{1,i} W_{i,t-s}\right) \\ +% x_2 + \sqrt{2}\sum^d_{i=1}\left(c_{2,i} W_{i,t-s}\right) \\ +% \vdots \\ +% x_d + \sqrt{2}\sum^d_{i=1}\left(c_{d,i} W_{i,t-s}\right) \\ +% \end{pmatrix} +%\end{align} +% +%Let $\mathfrak{W}^{i,j,d}_{t-s} = \sum^d_{i=1} \left( c_{j, i} W_{i,t-s}\right)$, given that the product of a Brownian motion with a constant is a Brownian motion and the sum of Brownian motion is also a Brownian motion, we have that: $\mathfrak{W}^{i,j,d}_{t-s} = \sum^d_{i=1} \left( c_{j, i} W_{i,t-s}\right)$ is also a Brownian motion. +%\medskip +% +%For each row $j$ we therefore have $x_k + \sqrt{2} \mathfrak{W}^{i,j,d}_{t-s}$ +% +\begin{lemma} \label{maxviscosity} + Let $d\in \N$, $T \in \lp 0,\infty \rp$, $\mathfrak{t} \in \lp 0,T \rp$, let $\mathcal{O} \subseteq \R^d$ be an open set, let $\mathfrak{r} \in \mathcal{O}$, $\phi \in C^{1,2}\lp \lp 0,T \rp \times \mathcal{O},\R \rp$, let $G: \lp 0,T \rp \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ be degenerate elliptic and let $u_d (0,T) \times \mathcal{O} \rightarrow \R$ be a viscosity solution of $\lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + G \lp t,x,u(t,x), \lp \nabla_x u_D \rp \lp t,x \rp, \lp \Hess_x u_d \rp \lp t,x \rp \rp \geqslant 0$ for $(t,x) \in (0,T) \times \mathcal{O}$, and assume that $u-\phi$ has a local maximum at $(\mathfrak{t}, \mathfrak{r}) \in (0,T) \times \mathcal{O}$, then: + \begin{align} + \lp \frac{\partial}{\partial t} \phi \rp \lp \mathfrak{t},\mathfrak{r}\rp + G \lp \mathfrak{t}, \mathfrak{r}, u(\mathfrak{t}, \mathfrak{r}), \lp \nabla _x \phi \rp \lp \mathfrak{t}, \mathfrak{r} \rp, \lp \Hess_x \phi\rp\lp \mathfrak{t}, \mathfrak{r} \rp \rp \geqslant 0 + \end{align} +\end{lemma} +\begin{proof} + That $u$ is upper semi-continuous ensures that there exists as a neighborhood $U$ around $(\mathfrak{t}, \mathfrak{r})$ and $\psi \in C^{1,2} ((0,T) \times \mathcal{O},\R)$ where: + \begin{enumerate}[label = (\roman*)] + \item for all $(t,x) \in (0,T) \times \mathcal{O}$ that $u(\mathfrak{t}, \mathfrak{r}) - \psi(\mathfrak{t}, \mathfrak{r}) \geqslant u(t,x) - \psi(t,x)$ + \item for all $(t,x) \in U$ that $\phi(t,x) = \phi(t,x)$. + \end{enumerate} + We therefore obtain that: + \begin{align} + &\lp \frac{\partial}{\partial t} \phi \rp \lp \mathfrak{t}, \mathfrak{r} \rp + G \lp \mathfrak{t}, \mathfrak{r}, u(\mathfrak{t},\mathfrak{r}), (\nabla_x) (\mathfrak{t}, \mathfrak{r}), (\Hess_x \phi)(\mathfrak{t}, \mathfrak{r}) \rp \nonumber \\ + &= \lp \frac{\partial}{\partial t} \psi \rp \lp \mathfrak{t}, \mathfrak{r} \rp + G \lp \mathfrak{t}, \mathfrak{r}, u(\mathfrak{t},\mathfrak{r}), (\nabla_x) (\mathfrak{t}, \mathfrak{r}), (\Hess_x \psi)(\mathfrak{t}, \mathfrak{r}) \rp \geqslant 0 + \end{align} +\end{proof} +\begin{lemma}\label{ungeq0} + Let $d \in \N$, $T \in (0, \infty)$, let $\mathcal{O} \subseteq \R^d$ be a non-empty open set, let $u_n:(0,T) \times \mathcal{O} \rightarrow \R$, $n \in \N_0$ be functions, let $G_n: (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$, $n \in \N$ be degenerate elliptic, assume that $G_\infty$ is upper semi-continuous for all non-empty compact $\mathcal{K} \subseteq (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ that: + \begin{align} + \limsup_{n \rightarrow \infty} \lb\sup_{(t,x,r,p,A) \in \mathcal{K}} \lp \lv u_n(t,x) -u_0(t,x) \rv + \lv G_n(t,x,r,p,A) - G_0(t,x,r,p,A) \rv \rp \rb=0 \label{limitofun} + \end{align} + and assume for all $n\in \N$ that $u_n$ is a viscosity solution of: + \begin{align} + \lp \frac{\partial}{\partial t} u_n \rp \lp t,x \rp + G_n \lp t,x,u_n(t,x),(\nabla_x u_n)(t,x), (\Hess_x u_n)(t,x) \rp \geqslant 0 + \end{align} + then $u_0$ is a viscosity solution of: + \begin{align}\label{hessungeq0} + \lp \frac{\partial}{\partial t} u_0 \rp \lp t,x \rp + G_n \lp t,x,u_0(t,x),(\nabla_x u_0)(t,x), (\Hess_x u_0)(t,x) \rp \geqslant 0 + \end{align} +\end{lemma} +\begin{proof} + Let $(t_o, x_o) \in (0,T) \times \mathcal{O}$. Let $\phi_\epsilon \in C^{1,2}((0,T) \times \mathcal{O}, \R)$ satisfy for all $\epsilon \in (0, \infty)$, $s \in (0,T)$, $y \in \mathcal{O}$ that $\phi_0(t_0,x_0) = u_0(t_0,x_0)$, $\phi_0(t_0,x_0) \geqslant u_0(t_0,x_0)$, and: + \begin{align}\label{phieps} + \phi_\varepsilon(s,y) = \phi_o(s,y) + \varepsilon\lp \lv s - t_0 \rv + \| y - x_0 \|_E \rp + \end{align} + Let $\delta \in (0,\infty)$ be such that $\{(s,y) \in \R^d \times \R: \max \lp|s-t_0|^2, \|y-x_0\|_E^2 \rp\leqslant \delta \}$. Note that this and (\ref{limitofun}) + then imply for all $\varepsilon \in (0,\infty)$ there exists an $\nu_\varepsilon \in \N$ such that for all $n \geqslant \nu_\varepsilon$, and $\max \lp|s-t_0|, \|y-x_0\|_E \rp\leqslant \delta$, it is the case that: + \begin{align} + \sup \lp \lv u_n(s,y) - u_0(s,y) \rv \rp \leqslant \frac{ \varepsilon \delta}{2} + \end{align} + Note that this combined with (\ref{phieps}) tells us that for all $\varepsilon \in (0,\infty)$, $n \in \N \cap [\nu_\epsilon, \infty)$, $s\in (0,T)$, $y\in \mathcal{O}$, with $|s-t_0| < \delta$, $\|y-x_0\|_E \leqslant \delta$, $|s-t_0| + \|y-x_0\|_E > \delta$ that: + \begin{align} + u_n(t_0,x_0) - \phi_\varepsilon(t_0,x_0) &= u_n(t_0,x_0) - \phi_0(t_0,x_0) \\ + &= u_n(t_0,x_0) - u_0(t_0,x_0) \nonumber\\ + &\geqslant \frac{-\varepsilon \delta}{2} \nonumber \\ + &\geqslant u_n(s,y) - u_0(s,y)-\varepsilon \lp |s-t_0|+\|y-x_0\|_E\rp \nonumber \\ + &\geqslant u_n(s,y) - \phi_0(s,y)-\varepsilon \lp |s-t_0|+\|y-x_0\|_E\rp \nonumber \\ + &= u_n(s,y) - \phi_\varepsilon (s,y) + \end{align} + \medskip + + Note that Corollary \ref{sumofusc} implies that for all $\epsilon \in (0,\infty)$ and $n\in \N$ that $u_n - \phi_\varepsilon$ is upper semi-continuous. There therefore exists for all $\epsilon \in (0,\infty)$ and $n \in \N$, a $\tau^\varepsilon_n \in (t_0 - \delta, t_0+\delta)$ and a $\rho^\varepsilon_n$, where $\|\rho_n^\varepsilon - x_0 \| \leqslant \delta$ such that: + \begin{align}\label{maxphiu} + u_n(\tau^\varepsilon_n, \rho_n^\varepsilon) - \phi_\epsilon(\tau_n^\varepsilon, \rho^\ve_n) \geqslant u_n(s,y) - \phi_\varepsilon(s,y) + \end{align} + By Lemma \ref{maxviscosity}, it must be the case that for all $\varepsilon \in (0,\infty)$ and $n \in \N \cap [\nu_\varepsilon, \infty)$: + \begin{align} + \lp \frac{\partial}{\partial t} \phi_\varepsilon \rp \lp \tau^\varepsilon_n, \rho^\varepsilon_n \rp + G_n \lp \tau_n^\ve, \rho_n^\ve, u_n \lp \tau_n^\ve, \rho_n^\ve\rp, \lp \nabla_x \phi_\ve \rp \lp \tau^\ve_n, \rho^\ve_n \rp, \lp \Hess_x \phi_\ve \rp \lp \tau^\ve_n, \rho^\ve_n \rp \rp \geqslant 0 + \end{align} + Note however that (\ref{maxphiu}) along with (\ref{phieps}) and (\ref{limitofun}) yields that for all $\ve \in (0,\infty)$ that: + \begin{align} + &\limsup_{n\rightarrow \infty}\lb u_n(\tau_n^\varepsilon, \rho_n^\varepsilon) - \phi_\epsilon(\tau_n^\varepsilon, \rho^\ve_n) \rb \nonumber\\ + &\geqslant \limsup_{n\rightarrow \infty} \lb u_n(\tau^\ve_n,\rho^\ve_n) - \lp\phi_0(\tau^\ve_n,\rho^\ve_n) + \ve \lp |\tau^\ve_n-t_0| + \|\rho^\ve_n- x_0\|_E \rp \rp \rb \nonumber \\ + &\geqslant \limsup_{n\rightarrow \infty} \lb u_n(\tau^\ve_n,\rho^\ve_n) - u_0(\tau^\ve_n,\rho^\ve_n) - \ve \lp |\tau^\ve_n-t_0| + \|\rho^\ve_n- x_0\|_E \rp \rb \nonumber\\ + &= \limsup_{n\rightarrow \infty} \lb - \ve \lp |\tau^\ve_n-t_0| + \|\rho^\ve_n- x_0\|_E \rp \rb \leqslant 0 + \end{align} + + However note also that since $G_0$ is upper semi-continuous, further the fact that, $\phi_0 \in \lp \lp0,T\rp \times \mathcal{O}, \R \rp$, and then $(\ref{limitofun})$, and $(\ref{phieps})$, imply for all $\ve \in (0,\infty)$ we have that: + $\limsup_{n\rightarrow \infty} \lv \lp \frac{\partial}{\partial t}\phi_\ve \rp\lp \tau^\ve_n, \rho^\ve_n \rp - \\ + \lp\frac{\partial}{\partial t} \phi_0\rp \lp t_0,x_0 \rp\rv = 0$ and: + \begin{align} + & G_0 \lp t_0, x_0, \phi_0 \lp t_0, x_0 \rp, \lp \nabla_x \phi_0 \rp \lp t_0,x_0 \rp, \lp \Hess_x \phi_0 \rp \lp t_0,x_0 \rp+ \text{Id}_{\R^d} \rp \nonumber \\ + &= G_0 \lp t_0, x_0, u_0 \lp t_0, x_0 \rp, \lp \nabla_x \phi_\ve \rp \lp t_0,x_0 \rp, \lp \Hess_x \phi_\ve \rp \lp t_0,x_0\rp \rp \nonumber \\ + & \geqslant \limsup_{n \rightarrow \infty} \lb G_0 \lp \tau^\ve_n, \rho^\ve_n, u_n \lp \tau^\ve_n, \rho^\ve_n \rp, \lp \nabla_x \phi_\ve \rp \lp \tau^\ve_n,\rho^\ve_n \rp, \lp \Hess_x \phi_\ve \rp \lp \tau^\ve_n,\rho^\ve_n\rp \rp \rb \\ + &\geqslant \limsup_{n \rightarrow \infty} \lb G_n \lp \tau^\ve_n, \rho^\ve_n, u_n \lp \tau^\ve_n, \rho^\ve_n \rp, \lp \nabla_x \phi_\ve \rp \lp \tau^\ve_n,\rho^\ve_n \rp, \lp \Hess_x \phi_\ve \rp \lp \tau^\ve_n,\rho^\ve_n\rp \rp \rb + \end{align} + + This with (\ref{maxphiu}) assures for all $\epsilon \in (0,\infty)$ that: + \begin{align} + \lp \frac{\partial}{\partial t} \phi_0 \rp \lp t_0,x_0 \rp + G_0 \lp t_0, x_0, \phi_0 \lp t_0, x_0 \rp, \lp \nabla_x \phi_0 \rp \lp t_0, x_0 \rp, \lp \Hess_x \phi_0 \rp \lp t_0, x_0 \rp + \ve \text{Id}_{\R^d} \rp \geqslant 0 + \end{align} + That $G_0$ is upper semi-continuous then yields that: + + \begin{align} + \lp \frac{\partial}{\partial t} \phi_0 \rp \lp t_0,x_0 \rp + G_0 \lp t_0, x_0, \phi_0\lp t_0, x_0 \rp, \lp \nabla_x \phi_0 \rp \lp t_0, x_0 \rp, \lp \Hess_x \phi_0 \rp \lp t_0, x_0 \rp + \ve \text{Id}_{\R^d} \rp \geqslant 0 + \end{align} + This establishes $(\ref{hessungeq0})$ which establishes the lemma. + + + +\end{proof} + +\begin{corollary}\label{unleq0} + Let $d \in \N$, $T \in (0, \infty)$, let $\mathcal{O} \subseteq \R^d$ be a non-empty open set, let $u_n:(0,T) \times \mathcal{O} \rightarrow \R$, $n \in \N_0$ be functions, let $G_n: (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$, $n \in \N_0$ be degenerate elliptic, assume that $G_0$ is lower semi-continuous for all non-empty compact $\mathcal{K} \subseteq (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ that: + \begin{align} + \limsup_{n \rightarrow \infty} \lb\sup_{(t,x,r,p,A) \in \mathcal{K}} \lp \lv u_n(t,x) -u_0(t,x) \rv + \lv G_n(t,x,r,p,A) - G_0(t,x,r,p,A) \rv \rp \rb = 0 \label{limitofun} + \end{align} + and assume for all $n\in \N$ that $u_n$ is a viscosity solution of: + \begin{align}\label{viscsolutionvn} + \lp \frac{\partial}{\partial t} u_n \rp \lp t,x \rp + G_n \lp t,x,u_n(t,x),(\nabla_x u_n)(t,x), (\Hess_x u_n)(t,x) \rp \leqslant 0 + \end{align} + then $u_0$ is a viscosity solution of: + \begin{align}\label{hessungeq0} + \lp \frac{\partial}{\partial t} u_0 \rp \lp t,x \rp + G_n \lp t,x,u_0(t,x),(\nabla_x u_0)(t,x), (\Hess_x u_0)(t,x) \rp \leqslant 0 + \end{align} +\end{corollary} + +\begin{proof} + Let $v_n:(0,T) \times \mathcal{O} \rightarrow \R$, $n \in \N_0$ and $H_n:(0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ satisfy for all $n\in \N_0$, $t\in (0,T)$, $x \in \mathcal{O}$, $r\in \R$, $p \in \R^d$, $A \in \mathbb{S}_d$ that $v_n(t,x) = -u_n(t,x)$ and that $H_n(t,x) = -G_n(t,x,-r,-p,-A)$. + \medskip + + Note that Corollary \ref{neglsc} gives us that $H_0$ is upper semi-continuous. Note also that since it is the case that for all $n\in \N_0$, $G_n$ is degenerate elliptic then it is also the case by Lemma \ref{negdegel} that $H_n$ is degenerate elliptic for all $n\in \N_0$. These together with (\ref{viscsolutionvn}) ensure that for all $n\in \N$, $v_n$ is a viscosity solution of: + \begin{align}\label{hgeq0} + \lp \frac{\partial}{\partial t} v_n \rp \lp t,x \rp + H_n \lp t,x,v_n \lp t,x \rp, \lp \nabla_x v_n \rp \lp t,x \rp, \lp \Hess_x v_n \rp \lp t,x \rp \rp \geqslant 0 + \end{align} + This together with (\ref{limitofun}) establish that: + \begin{align} + \limsup_{n \rightarrow \infty} \lb\sup_{(t,x,r,p,A) \in \mathcal{K}} \lp \lv u_n(t,x) -u_0(t,x) \rv + \lv H_n(t,x,r,p,A) - H_0(t,x,r,p,A) \rv \rp \rb = 0 + \end{align} + + This (\ref{hgeq0}) and the fact that $H_0$ is upper semi-continuous then establish that: + \begin{align} + \lp \frac{\partial}{\partial t} v_0 \rp \lp t,x \rp + H_0 \lp t,x,v_0(t,x),(\nabla_x v_0)(t,x), (\Hess_x v_0)(t,x) \rp \geqslant 0 + \end{align} + for $(t,x) \in (0,T) \times \mathcal{O}$. Hence $v_0$ is a viscosity solution of: + \begin{align} + \lp \frac{\partial}{\partial t} u_0 \rp \lp t,x \rp + H_0 \lp t,x,u_0(t,x),(\nabla_x u_0)(t,x), (\Hess_x u_0)(t,x) \rp \leqslant 0 + \end{align} + This completes the proof. +\end{proof} + +\begin{corollary}\label{unneq0} + Let $d \in \N$, $T \in (0,\infty)$, let $\mathcal{O} \subseteq \R^d$ be a non-empty set, let $u_n: (0,T) \times \mathcal{O} \rightarrow \R$, $n \in \N_0$, be functions, let $G_n: (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$, $n \in \N_0$ be degenerate elliptic, assume also that $G_0: (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ be consinuous and assume for all non-empty compact $\mathcal{K} \subseteq (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ it is the case that: + \begin{align} + \limsup_{n\rightarrow \infty} \lb \sup_{(t,x,r,p,A) \in \mathcal{K}} \lp \lv G_n \lp t,x,r,p,A \rp - G_0 \lp t,x,r,p,A \rp \rv + \lv u_n \lp t,x \rp - u_0 \lp t,x \rp \rv \rp \rb = 0 + \end{align} + and further assume for all $n \in \N$, that $u_n$ is a viscosity solution of: + \begin{align} + \lp \frac{\partial}{\partial t} u_n \rp \lp t,x \rp + G_n \lp t,x ,u_n \lp t,x \rp, \lp \nabla_x u_n \rp \lp t,x \rp, \lp \Hess_x u_n \rp \lp t,x \rp \rp = 0 + \end{align} + for $(t,x) \in (0,T) \times \mathcal{O}$, then we have that $u_0$ is a viscosity solution of: + \begin{align} + \lp\frac{\partial}{\partial t} u_0 \rp \lp t,x \rp + G_0 \lp t,x,u_0 \lp t,x \rp, \lp \nabla_x u_0 \rp \lp t,x \rp, \lp \Hess_x u_0 \rp \lp t,x \rp \rp= 0 + \end{align} +\end{corollary} +\begin{proof} + Note that Lemma \ref{ungeq0} gives us that $u_0$ is a viscosity solution of: + \begin{align} + \lp \frac{\partial}{\partial t} u_0 \rp \lp t,x \rp + G_n \lp t,x,u_0(t,x),(\nabla_x u_0)(t,x), (\Hess_x u_0)(t,x) \rp \geqslant 0 + \end{align} + +for $(t,x) \in (0,T) \times \mathcal{O}$. Also note that Corollary \ref{unleq0} ensures that $u_0$ is a viscosity solution of: + \begin{align} + \lp \frac{\partial}{\partial t} u_0 \rp \lp t,x \rp + G_n \lp t,x,u_0(t,x),(\nabla_x u_0)(t,x), (\Hess_x u_0)(t,x) \rp \leqslant 0 + \end{align} +Taken together these prove the corollary. +\end{proof} +\begin{lemma}\label{absq} + For all $a,b \in \R$ it is the case that $(a+b)^2 \leqslant 2a^2+2b^2$. +\end{lemma} +\begin{proof} + Since for all $a,b\in\R$ it is the case that $(a-b)^2 \geqslant 0$ we then have that: + \begin{align} + (a+b)^2 &\leqslant (a+b)^2 + (a-b)^2 \nonumber \\ + &\leqslant a^2 + 2ab + b^2 + a^2 -2ab+b^2 \nonumber\\ + &=2a^2+2b^2 \nonumber + \end{align} + This completes the proof. +\end{proof} + + +\begin{lemma}\label{ugoesto0} + Let $d,m \in \N$, $T \in (0,\infty)$. Let $\mathcal{O} \subseteq \R^d$ be a non-empty compact set, and for all $n\in \N_0$, $\mu_n \in C([0,T] \times \mathcal{O}, \R)$, $\sigma_n \in C([0,T] \times \mathcal{O}, \R^{d \times m})$ +% \begin{align} +% \sup_{t\in [0,T]} \sup_{x\in \R^d} \sup_{y\in \R^d \setminus\{x\}} \lb \frac{\|\mu_n(t,x)-\mu_n(t,y)\|_E+\|\sigma_n(t,x) - \sigma_n(t,y)\|_F}{\|x-y\|_E} \rb < \infty +% \end{align} + assume also: + \begin{align}\label{limsupis0} + \limsup_{n \rightarrow \infty} \lb \sup_{t\in[0,T]} \sup_{x\in \mathcal{O}} \lp \right\|\mu_n(t,x) - \mu_0(t,x)\left\|_E + \left\|\sigma_n(t,x)-\sigma_0(t,x)\right\|_F \rp \rb = 0 + \end{align} + Let $\lp \Omega, \mathcal{F}, \mathbb{R} \rp$ be a stochastic basis and let $W: [0,T] \times \Omega \rightarrow \R^m$ be a standard $(\mathbb{F}_t)_{t\in [0,T]}$-Brownian motion for every $t\in [0,T]$, $x \in \mathcal{O}$, let $\mathcal{X}^{t,x} = (\mathcal{X}^{t,x}_s)_{s\in [t,T]}: [t,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_s)_{s\in [t,T]}$ adapted stochastic process with continuous sample paths, satisfying for all $s \in [t, T]$ we have $\mathbb{P}$-a.s. + \begin{align}\label{xnasintuvxn} + \mathcal{X}^{n,t,x}_s = x + \int^s_t \mu_n(r,\mathcal{X}^{n,t,x}_s) dr + \int^s_t \sigma_n(r,\mathcal{X}^{n,t,x}_r) dW_r + \end{align} + then it is the case that: + \begin{align} + \limsup_{n\rightarrow \infty} \lb \sup_{t\in[0,T]}\sup_{s \in[t,T]}\sup_{x \in \mathcal{O}} \lp \E \lb \left\|\mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s\right\|_E^2\rb \rp \rb = 0 + \end{align} + for $(t,x) \in (0,T) \times \R^d$. +\end{lemma} + +\begin{proof} + Since $\mathcal{O}$ is compact, let $L \in \R$ be such that for all $t \in [0,T]$, $x,y\in\mathcal{O}$ it is the case that: + \begin{align}\label{lipformun} + \|\mu_0(t,x) - \mu_0(t,y)\|_E - \|\sigma_0(t,x) + \sigma_0(t,y) \|_F \leqslant L\|x-y\|_E + \end{align} + + Furthermore \cite[Theorem~5.2.9]{karatzas1991brownian} tells us that: + \begin{align} \label{expofxisbounded} + \sup_{s\in[t,T]}\E \lb \|\mathcal{X}^{n,t,x}_s\|_E\rb < \infty + \end{align} + Note now that (\ref{xnasintuvxn}) tells us that: + \begin{align}\label{mathcalxn-mathclx0} + \mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s = \int^s_t \mu_n(r,\mathcal{X}^{n,t,x}_s) - \mu_0(r,\mathcal{X}^{0,t,x}_s) dr + \int^s_t \sigma_n(r,\mathcal{X}^{n,t,x}_r) - \sigma_0(r,\mathcal{X}^{0,t,x}_r) dW_r + \end{align} + Minkowski's Inequality applied to (\ref{mathcalxn-mathclx0}) then tells us for all $n\in \N$, $t\in [0,T]$, $s \in [t,T]$, and $x\in \mathcal{O}$ that: + \begin{align} + \lp \E \lb \left\|\mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s\right\|_E\rb \rp^{\frac{1}{2}} &\leqslant \int^s_t\lp \E \lb \left\|\mu_n(r, \mathcal{X}^{n,t,x}_r) - \mu_0(r,\mathcal{X}^{0,t,x}_r) \right\|_E^2 \rb\rp^{\frac{1}{2}} dr + \nonumber\\ + &\lp \E \lb \left\|\int^s_t (\sigma_n(r,\mathcal{X}^{n,t,x}_r)-\sigma_0(r,\mathcal{X}^{0,t,x}_r)) dW_r \right\|_E^2 \rb \rp^{\frac{1}{2}} + \end{align} + It\^o's isometry applied to the second summand yields: + \begin{align} + \lp \E \lb \left\|\mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E\rb \rp^{\frac{1}{2}} &\leqslant \int^s_t\lp \E \lb \left\|\mu_n(r, \mathcal{X}^{n,t,x}_r) - \mu_0(r,\mathcal{X}^{0,t,x}_r)\right\|_E^2 \rb\rp^{\frac{1}{2}} dr + \nonumber\\ + &\lp \int^s_t \E \lb \left\|\sigma_n(r,\mathcal{X}^{n,t,x}_r) - \sigma_0(r,\mathcal{X}^{0,t,x})\right\|_F^2 \rb dr \rp^\frac{1}{2} + \end{align} + Applying Lemma \ref{absq} followed by the Cauchy-Schwarz Inequality then gives us for all $n \in \N$, $t\in[0,T]$, $s \in [t,T]$, and $x\in \mathcal{O}$ that: + \begin{align} + \E \lb \|\mathcal{X}^{n,t,x}_s-\mathcal{X}^{n,t,x}_s \|^2_E\rb &\leqslant 2 \lb \int^s_t \lp \E \lb \left\| \mu_n(r,\mathcal{X}^{n,t,x}_r) - \mu_0(r,\mathcal{X}^{0,t,x}_r)\right\|^2_E \rb \rp^{\frac{1}{2}} dr \rb^2 \nonumber\\ + &+2\int^s_t \E \lb \left\|\sigma_n(r,\mathcal{X}^{nt,x}_r) - \sigma_0(r,\mathcal{X}^{0,t,x}_r)\right\|_F^2 \rb dr \nonumber\\ + &\leqslant 2T \int^s_t \E \lb \left\|\mu_n(r,\mathcal{X}^{n,t,x}_r) -\mu_0(r,\mathcal{X}^{0,t,x}_r) \right\|_E^2\rb dr \nonumber\\ + &+ 2 \int^s_t \E \lb \left\|\sigma_n(r,\mathcal{X}^{n,t,x}_r) - \sigma_0(r,\mathcal{X}^{0,t,x}_r) \right\|_F^2\rb dr + \end{align} + Applying Lemma \ref{absq} again to each summand then yields for all $n\in \N$, $t\in [0,T]$ $s\in [t,T]$, and $x\in \mathcal{O}$ it is the case that: + \begin{align} + &\E \lb \left\|\mathcal{X}^{n,t,x}_s-\mathcal{X}^{0,t,x}_s \right\|^2 \rb \nonumber \\ + &\leqslant 2T \int^s_t \lp 2\E \lb \left\|\mu_n(r,\mathcal{X}^{n,t,x}_r)-\mu_0(r,\mathcal{X}^{n,t,x}_r)\right\|^2_E\rb + 2\E \lb \left\|\mu_0(r,\mathcal{X}^{n,t,x}_r)-\mu_0(r,\mathcal{X}^{0,t,x}_r)\right\|^2_E\rb \rp dr \nonumber \\ + &+2\int^2_t \lp 2\E \lb \left\| \sigma_n(r,\mathcal{X}^{n,t,x}_r) - \sigma_0(r,\mathcal{X}^{n,t,x}_r) \right\|_F^2 \rb + 2\E \lb\left\|\sigma_0(r,\mathcal{X}^{n,t,x}_r)-\sigma_0(r,\mathcal{X}^{0,t,x}_r)\right\|_F\rb \rp dr + \end{align} + However assumption (\ref{lipformun}) then gives us that for all $n\in \N$, $t \in [0,T]$, $s \in [t,T]$, and $x \in \mathcal{O}$ that: + \begin{align} + \E \lb \left\|\mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb &\leqslant 4L^2(T+1) \int^s_t\E\lb \left\|\mathcal{X}^{n,t,x}_r-\mathcal{X}^{0,t,x}_r \right\|_E^2 \rb dr \nonumber \\ + &+4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y) - \mu_0(r,y) \right\|_E^2 + \left\| \sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb \nonumber + \end{align} + + Finally Gronwall's Inequality with assumption (\ref{expofxisbounded}) gives us for all $n\in \N$, $t\in [0,T]$, $s\in [t,T]$, $x \in \mathcal{O}$ that: + \begin{align} + &\E \lb \left\| \mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb \nonumber \\ + &\leqslant 4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \| \mu_n(r,y)-\mu_0(r,y) \|_E^2 + \|\sigma_n(r,y) - \sigma_)(r,y) \|_F^2 \rp \rb e^{4L^2T(T+1)} + \end{align} + Applying $\limsup_{n\rightarrow \infty}$ to both sides and applying (\ref{limsupis0}) gives us for all $n \in \N$, $t \in [0,T]$, $s\in [t,T]$, $x \in \mathcal{O}$ that: + \begin{align} + &\limsup_{n\rightarrow \infty} \E \lb \left\| \mathcal{X}^{n,t,x}_s - \mathcal{X}^{0,t,x}_s \right\|_E^2 \rb \nonumber\\ + &\leqslant \limsup_{n\rightarrow \infty} \lb 4T(T+1) \lb \sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y)-\mu_0(r,y) \right\|_E^2 + \left\|\sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb e^{4L^2T(T+1)} \rb \nonumber \\ + &\leqslant 4T(T+1) \lb \limsup_{n\rightarrow \infty} \lb\sup_{r\in [0,T]}\sup_{y\in \R^d} \lp \left\| \mu_n(r,y)-\mu_0(r,y) \right\|_E^2 + \left\|\sigma_n(r,y) - \sigma_0(r,y) \right\|_F^2 \rp \rb\rb e^{4L^2T(T+1)} \nonumber \\ + &\leqslant 0 \nonumber + \end{align} + This completes the proof. +\end{proof} + +\begin{lemma}\label{2.19} + Let $d,m \in \N$, $T \in (0,\infty)$, let $\mathcal{O} \subseteq [0,T] \times \R^d$, let $\mu \in C([0,T] \times \mathcal{O},\R^d)$ and $\sigma \in C([0,T] \times \mathcal{O}, \R^{d\times m})$ have compact supports such that $\supp(\mu) \cup \supp(\sigma) \subseteq [0,T] \times \mathcal{O}$ + let $g\in C(\R^d,\R)$. Let $\lp \Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t\in [0,T]} \rp$ be a stochastic basis, let $W:[0,T] \times \Omega \rightarrow \R^m$ be a standard $(\mathbb{F}_t)_{t\in [0,T]}$ Brownian motion, for every $t\in [0,T]$, $x\in \R^d$, let $\mathcal{X}^{t,x} = (\mathcal{X}^{t,x}_s)_{s\in [t,T]}: [t,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_s)_{s\in [t,T]}$ adapted stochastic process with continuous sample paths satisfying for all $s\in [t,T]$ with $\mathbb{F}$-a.s. that: + \begin{align}\label{2.59} + \mathcal{X}^{t,x}_s = x + \int^s_t \mu \lp r, \mathcal{X}^{t,x}_r \rp dr + \int^s_t \sigma \lp r, \mathcal{X}^{t,x}_r \rp dW_r + \end{align} + and further let $u:[0,T] \times \R^d \rightarrow \R$ satisfy for all $t\in [0,T]$, $x\in \R^d$ that: + \begin{align}\label{2.60} + u(t,x) = \E \lb g \lp \mathcal{X}^{t,x}_T \rp \rb + \end{align} + Then $u$ is a viscosity solution of: + \begin{align} + \lp \frac{\partial}{\partial t} u \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \sigma(t,x)\lb \sigma(t,x) \rb^* \lp \Hess_xu \rp \lp t,x \rp \rp + \la \mu(t,x), \lp \nabla_x u \rp \lp t,x \rp \ra = 0 + \end{align} + and where $u(T,x) = g(x)$ for $(t,x) \in (0,T) \times \mathcal{O}$. +\end{lemma} + +\begin{proof} + Let $\mathcal{S} = \supp(\mu) \cup \supp(\sigma) \subseteq [0,T] \times \mathcal{O}$ be bounded in space by $\rho \in (0,\infty)$, as $\mathcal{S} \subseteq [0,T] \times (-\rho, \rho)^d$. This exists as the supports are compact and thus by Hiene-B\"orel is closed and bounded. Let $\mathfrak{s}_n, \mathfrak{m}_n \in C^\infty([0,T] \times \R^d, \R^{d\times n})$ where $\bigcup_{n\in \N} \lb \supp(\mathfrak{s}_n) \cup \supp(\mathfrak{m}_n) \rb \subseteq [0,T] \times (-\rho, \rho)^d$ satisfy for $n\in \N$ that: + \begin{align}\label{2.62} + \limsup_{n\rightarrow \infty} \lb \sup_{t\in [0,T]} \sup_{x\in \R} \lp \left\| \mathfrak{m}_n(t,x) - \mu(t,x) \right\|_E + \left\|\mathfrak{s}_n - \sigma(t,x)\right\|_F \rp \rb = 0 + \end{align} + We construct a set of degenerate elliptic functions, $G^{n}:(0,T) \times \R^d \times \R \times \R^d \times \mathbb{S}_d \rightarrow \R$ with $n \in \N_0$ such that: + \begin{align} + &G^{0}(t,x,r,p,A) = \frac{1}{2} \Trace\lp \sigma (t,x)[\sigma(t,x)]^*A \rp + \la \mu(t,x),p \ra \label{2.63}\\ + &\text{and} \nonumber\\ + &G^{n}(t,x,r,p,A) = \frac{1}{2} \Trace \lp \mathfrak{s}_n (t,x)[\mathfrak{s}_n(t,x)]^*A \rp + \la \mu(t,x),p \label{2.64}\ra + \end{align} + Also let $\mathfrak{g}_n \in C^\infty(\R^d,\R)$ for $n\in \N$ satisfy for all $n\in \N$ that: + \begin{align}\label{2.65} + \limsup_{n\rightarrow \infty} \sup_{t\in [0,T]} \sup_{x\in \R^d} \lp \| \mathfrak{g}_n(x) - g(x) \|_E \rp = 0 + \end{align} + + Further let $\mathfrak{X}^{n,t,x} = (\mathfrak{X}^{n,t,x}_s)_{s\in [t,T]}:[t,T] \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_s)_{s\in [t,T]}$-adapted stochastic process with continuous sample paths that satisfy: + \begin{align}\label{2.66} + \mathcal{X}^{n,t,x}_s = x + \int^s_t \mathfrak{m}_n(r, \mathcal{X}^{n,t,x}_r)dr + \int^s_t \mathfrak{s}_n(r,\mathcal{X}^{n,t,x}_r) dW_r + \end{align} + % TODO Need to talk about karatzas and how the proof really isnt there the one it's referencing +% Note that then \cite{karatzas1991brownian} tells us that + Finally let $\mathfrak{u}^n: [0,T] \times \R^d \rightarrow \R$ for $n\in \N$ be: + \begin{align}\label{ungn} + \mathfrak{u}^n = \E \lb \mathfrak{g}_n \lp \mathfrak{X}^{n,t,x}_T \rp \rb + \end{align} + and: + \begin{align}\label{u0gn} + \mathfrak{u}^0 = \E \lb \mathfrak{g}_n \lp \mathcal{X}^{t,x}_T \rp \rb + \end{align} + + Note that \cite[Lemma~2.2]{BHJ21} with $g \curvearrowleft \mathfrak{g}_k$, $\mu \curvearrowleft \mathfrak{m}_n$, $\sigma \curvearrowleft \mathfrak{s}_n$, $\mathcal{X}^{t,x} \curvearrowleft \mathcal{X}^{n,t,x}$ gives us $\mathfrak{u}^n \in C^{1,2}([0,T] \times \R^d, \R)$, and $\mathfrak{u}^n(t,x) = \mathfrak{g}_k(x)$ where: + \begin{align} + \lp \frac{\partial}{\partial t} \mathfrak{u}^n \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \mathfrak{s}_n(t,x)\lb \mathfrak{s}_n(t,x) \rb^* \lp \Hess_x \mathfrak{u}^n \rp \lp t,x \rp \rp + \la \mathfrak{m}_n(t,x), \lp \nabla_x \mathfrak{u}^n \rp \lp t,x \rp \ra = 0 + \end{align} + And by Definitions \ref{def:viscsubsolution}, \ref{def:viscsupsolution}, and \ref{def:viscsolution} we have that $\mathfrak{u}^n$ is a viscosity solution of + \begin{align}\label{2.70} + \lp \frac{\partial}{\partial t} \mathfrak{u}^n \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \mathfrak{s}_n(t,x)\lb \mathfrak{s}_n(t,x) \rb^* \lp \Hess_x \mathfrak{u}^n \rp \lp t,x \rp \rp + \la \mathfrak{m}_n(t,x), \lp \nabla_x \mathfrak{u}^n \rp \lp t,x \rp \ra = 0 + \end{align} + for $(t,x) \in (0,T) \times \R^d$. +\medskip + +Since for all $n\in \N$, it is the case that $\mathcal{S} = \lp \supp(\mathfrak{m}_n)\cup \supp(\mathfrak{s}_n) \cup \supp(\mu) \cup \supp(\sigma) \rp \subseteq [0,T] \times (-\rho, \rho)^d$ and because of (\ref{2.59}) of (\ref{2.66}) we have that \cite[Lemma~3.2, Item~(ii)]{Beck_2021} which yields that for all $n\in \N$, $t\in [0,T]$, $x \in \R^d\setminus(-\rho, \rho)^d$ that $\mathbb{P}(\forall s \in [t,T]: \mathfrak{X}^{n,t,x}_s =x=\mathcal{X}^{t,x}_s)=1$. This in turn shows that for all $n\in \N$, $x \in \R^d \setminus(-\rho,\rho)^d$ that $\mathfrak{u}^n(t,x) = \mathfrak{u}^0(t,x)$ which along with (\ref{ungn}) and (\ref{u0gn}) yields that: + \begin{align}\label{unu0} + \sup_{t\in[0,T]}\sup_{x \in \R^d} \lb \lv \mathfrak{u}^n (t,x) - \mathfrak{u}^0(t,x) \rv \rb &= \sup_{t\in [0,T]} \sup_{x \in (-\rho,\rho)^d} \lb \lv \mathfrak{u}^n(t,x) - \mathfrak{u}^0(t,x) \rv \rb \nonumber\\ + &\leqslant \sup_{t\in [0,T]}\sup_{x \in (-\rho,\rho)^d} \lp \E \lb \lv \mathfrak{g}_k \lp \mathfrak{X}^{n,t,x}_T \rp - \mathfrak{g} \lp \mathcal{X}^{t,x}_T \rp \rv \rb \rp + \end{align} + Note that Lemma \ref{ugoesto0} allows us to conclude that: + + \begin{align} + \limsup_{n\rightarrow \infty} \lb \sup_{t\in[0,T]} \sup_{x\in (-\rho, \rho)^d} \lp \E \lb \|\mathfrak{X}^{n,t,x}_T-\mathcal{X}^{t,x}_s \| \rb \rp \rb = 0 + \end{align} + But then we have that (\ref{unu0}) which yields that: + \begin{align}\label{2.73} + \limsup_{n\rightarrow 0} \lb \sup_{t\in [0,T]} \sup_{x \in \R^d} \lp \lv \mathfrak{u}^n(t,x) - \mathfrak{u}^0(t,x) \rv\rp \rb =0 + \end{align} + However now note that (\ref{2.64}) and (\ref{2.70}) thus yield that for $n \in \N_0$, $\mathfrak{u}^n$ is a viscosity solution to: + \begin{align}\label{2.74} + \lp \frac{\partial}{\partial t} \mathfrak{u}^n \rp \lp t,x \rp + G^n \lp t,x,\mathfrak{u}^n \lp t,x \rp, \lp \nabla_x \mathfrak{u}^n \rp \lp t,x \rp, \lp \Hess_x \mathfrak{u}^n \rp \lp t,x \rp \rp =0 + \end{align} + But since we've established (\ref{2.62}) we have that for a non-empty compact set $\mathcal{C} \subseteq (0,T) \times \mathcal{O} \times \R \times \R^d \times \mathbb{S}_d$ that: + \begin{align} + &\limsup_{n \rightarrow \infty} \lb \sup_{(t,x,r,p,A) \in \mathcal{C}} \lv G^n \lp t,x,r,p,A \rp - G^0 \lp t,x,r,p,A \rp \rv \rb \nonumber \\ + &\leqslant \limsup_{n\rightarrow \infty} \lb \sup_{(t,x,r,p,A) \in \mathcal{C}} \left\|\mu(t,x) - \mathfrak{m}_n(t,x) \|_E \right\| p \|_E \rb \nonumber \\ + &+ \limsup_{n\rightarrow \infty} \lb \sup_{(t,x,r,p,A) \in \mathcal{C}} \left\| \sigma(t,x)\lb \sigma(t,x) \rb^* - \mathfrak{s}_n(t,x) \lb \mathfrak{s}_n(t,x) \rb^* \right\|_F \left\|A\right\|_F \rb = 0 + \end{align} + + This, together with (\ref{2.73}), (\ref{2.74}) and Corollary \ref{unneq0} yields that $\mathfrak{u}^0$ is also a viscosity solution to: + \begin{align}\label{2.76} + \lp \frac{\partial}{\partial t} \mathfrak{u}^0 \rp \lp t,x \rp + G^0 \lp t,x,\mathfrak{u}^0(t,x), \lp \nabla_x \mathfrak{u}^0\rp \lp t,x \rp, \lp \Hess_x \rp \lp t,x \rp \rp =0 + \end{align} + Finally note that (\ref{2.62}), (\ref{2.66}), (\ref{u0gn}), and (\ref{2.76}) yield that $u$ is a viscosity solution of:: + \begin{align} + \lp \frac{\partial}{\partial t} u \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \sigma(t,x) \lb \sigma(t,x) \rb^* \lp \Hess_x u \rp \lp t,x \rp \rp + \la \mu(t,x), \lp \nabla_x \rp \lp t,x \rp \ra = 0 + \end{align} + for $(t,x) \in [0,T] \times \R^d$. Finally (\ref{2.59}) and (\ref{2.60}) allows us to conclude that for all $x \in \R^d$ it is the case that $u(T,x) = g(x)$. This concludes the proof. +\end{proof} + +\begin{lemma} + Let $d,m \in \N$, $T \in (0,\infty)$, further let $\mathcal{O} \subseteq \R^d$ be a non, empty compact set. Let every $r \in (0,\infty)$ satisfy the condition that $O_r \subseteq \mathcal{O}$, where $O_r = \{x \in \mathcal{O}: \lp \|x\|_E \leqslant r \text{ and }\{y \in \R^d: \| y-x\|_E < \frac{1}{r} \} \subseteq \mathcal{O} \rp \}$ let $g \in C(\mathcal{O},\R)$, $\mu \in C([0,T] \times \mathcal{O},\R)$, $V \in C^{1,2}([0,T] \times \mathcal{O},(0,\infty))$, assume that for all $t \in [0,T]$, $x \in \mathcal{O}$ that: + \begin{align} + \sup \lp \left\{ \frac{\| \mu(t,x)-\mu(t,y)\|_E+\|\sigma(t,x)-\sigma(t,y)\|_F}{\|x-y\|_E}:t\in [0,T], x,y\in O_r, x \neq y \right\} \cup \{0\}\rp < \infty + \end{align} + \begin{align} + \lp \frac{\partial}{\partial t} V \rp \lp t,x \rp + \frac{1}{2}\Trace \lp \sigma(t,x)\lb \sigma(t,x) \rb^* \lp \Hess_x V\rp \lp t,x \rp \rp + \la \mu(t,x), \lp \nabla_x V \rp (t,x) \ra \leqslant 0 + \end{align} + assume that $\sup_{r \in (0,\infty)} \lb \inf_{x \in \mathcal{O}\setminus O_r} V(t,x)\rb = \infty$ and $\inf_{r \in (0, \infty)} \lb \sup_{t \in [0,T]} \sup _{x \in \mathcal{O} \setminus O_r} \lp \frac{g(x)}{V(T,x)} \rp \rb = 0$. + Let $\lp \Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t\in [0,T]} \rp$ be a stochastic basis and let $W: [0,T] \times \Omega \rightarrow \R^m$ be a standard $(\mathbb{F}_t)_{t \in [0,T]}$-Brownian motion, for every $t \in [0,T]$, $x \in \mathcal{O}$ let $\mathcal{X}^{t,x} = (\mathcal{X}^{t,x}_s)_{s\in [t,T]}: [t,T] \times \Omega \rightarrow \mathcal{O}$ be an $(\mathbb{F}_s)_{s\in [t,T]}$-adapted stochastic process with continuous sample paths satisfying that for all $s\in [t, T]$, we have $\mathbb{P}$-a.s. that: + \begin{align}\label{2.79} + \mathcal{X}^{t,x}_s = x+\int^s_t \mu(r,\mathcal{X}^{t,x}_r) dr + \int^s_t \sigma(r, \mathcal{X}^{t,x}_n) dW_r + \end{align} + also let $u:[0,T] \times \R^d \rightarrow \R$ satisfy for all $t \in [0,T]$, $x \in \R^d$ that: + \begin{align} + u(t,x) = \E \lb u(T,\mathcal{X}^{t,x}_T) \rb + \end{align} + It is then the case that $u$ is a viscosity solution to: + \begin{align}\label{2.81} + \lp \frac{\partial}{\partial t} u \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \sigma(t,x) \lb \sigma(t,x) \rb^* \lp \Hess_x u \rp \lp t,x \rp \rp + \la \mu(t,x), \lp \nabla_x \rp \lp t,x \rp \ra = 0 + \end{align} + for $(t,x) \in (0,T) \times \mathcal{O}$ with $u(T,x) = g(x)$. + \end{lemma} +\begin{proof} + Let it be the case, that throughout the proof, for $n\in \N$, we have that $\mathfrak{g}_n \in C(\R^d, \R)$, compactly supported and that $\lb \bigcup_{n\in\N} \supp(\mathfrak{g}_m) \rb \subseteq [0, T] \times \mathcal{O}$ and further that: + \begin{align}\label{2.82} + \limsup_{n\rightarrow \infty} \lb \sup_{t\in [0,T]}\sup_{x\in \mathcal{O}} \lp \frac{|\mathfrak{g}_n(x)-g(x)|}{V(T,x)} \rp \rb = 0 + \end{align} + Let is also be the case that for $n\in \N$, $\mathfrak{m}_n \in C([0,T] \times \R^d, \R^d)$ and $\mathfrak{s}_n \in C([0,T] \times \R^d, \R^{d \times m})$ satisfy: + \begin{enumerate}[label = (\roman*)] + \item for all $n\in \N$: + \begin{align} + \sup_{t \in [0,T]} \sup_{x,y \in \R^d,x\neq y} \lb \frac{\|\mathfrak{m}_n(t,y)-\mathfrak{m}_n(t,y)\|_E + \|\mathfrak{s}_n(t,x)-\mathfrak{s}_n(t,y)\|_E}{\| x-y\|_E} \rb = 0 + \end{align} + \item for all all $n\in \N$, $t \in [0,T]$, $x \in \mathcal{O}$: + \begin{align}\label{2.84} + \mathbbm{1}_{\{V \leqslant n \}}(t,x)\lb \| \mathfrak{m}_n(t,x) - \mu(t,x)\|_E + \| \mathfrak{s}_n(t,x) - \sigma(t,x) \|_F \rb = 0 + \end{align} + and + \item for all $n\in \N$, $t \in [0,T]$, $x \in \R^d \setminus \{V \leqslant n+1\}$ that: + \begin{align} + \| \mathfrak{m}_n(t,x)\|_E + \| \mathfrak{s}_n (t,x) \|_F = 0 + \end{align} + \end{enumerate} + Next for every $n\in \N$, $t\in [0,T]$ and $x\in \R^d$ let it be the case that $\mathfrak{X}^{n,t,x}_s = (\mathfrak{X}^{n,t,x}_s)_{s\in [t,T]}: [t,t] \times \Omega. \rightarrow \R^d$ be a stochastic process with continuous sample paths satisfying: + \begin{align}\label{2.86} + \mathfrak{X}^{n,t,x}_s = x + \int^s_t \mathfrak{m}_n(r, \mathfrak{X}^{n,t,x}_s) dr + \int^s_t \mathfrak{s}_n(r, \mathfrak{X}^{n,t,x}_s) dW_r + \end{align} + Let $\mathfrak{u}^n: [0,T] \times \R^d \rightarrow \R$, $k \in \N$, $n \in \N_0$, satisfy for all $n\in \N$, $t \in [0,T]$, $x \in \R^d$ that: + \begin{align} + \mathfrak{u}^{n,k}(t,x) = \E \lb \mathfrak{g}_k (\mathfrak{X}^{n,t,x}_T) \rb + \end{align} + and + \begin{align}\label{2.88} + \mathfrak{u}^{0,k}(t,x) = \E \lb \mathfrak{g}_k \lp \mathcal{X}^{t,x}_T \rp \rb + \end{align} + and finally let, for every $n\in \N$, $t \in [0,T]$, $x \in \mathcal{O}$, there be $\mathfrak{t}^{t,x}_n: \Omega \rightarrow [t,T]$ which satisfy $\mathfrak{t}^{t,x}_n = \inf \lp \{ s \in [t,T], \max \{V(s,\mathfrak{X}^{t,x}_s),V(s,\mathcal{X}^{t,x}_s)\} \geqslant n \} \cup \{T\} \rp$. We may apply Lemma \ref{2.19} with $\mu \curvearrowleft \mathfrak{m}_n$, $\sigma \curvearrowleft \mathfrak{s}_n$, $g \curvearrowleft \mathfrak{g}_k$ to show that for all $n,k \in \N$ we have that $\mathfrak{u}^{n,k}$ is a viscosity solution to: + \begin{align}\label{2.89} + \lp \frac{\partial}{\partial t} \mathfrak{u}^{n,k} \rp (t,x) + \frac{1}{2} \Trace \lp \mathfrak{s}_n(t,x) \lb \mathfrak{s}_n(t,x) \rb^* \lp \Hess_x \mathfrak{u}^{n,k} \rp (t,x) \rp + \la \mathfrak{m}_n(t,x), \lp \nabla_x(\mathfrak{u}^{n,k} \rp(t,x) \ra = 0 + \end{align} + for $(t,x) \in (0,T) \times \R^d$. But note that items (i)-(iii) and \ref{2.86} give us that, in line with \cite[Lemma~3.5]{Beck_2021}: + \begin{align} + \mathbb{P}\lp \forall s \in [t,T]: \mathbbm{1}_{\{s \leqslant \mathfrak{t}^{t,x}_n\}} \mathfrak{X}^{n,t,x}_s = \mathbbm{1}_{\{s\leqslant \mathfrak{t}^{t,x}_n\}} \mathcal{X}^{t,x}_s \rp =1 + \end{align} + Further this implies that for all $n,k \in \N$, $t \in [0,T]$, $x \in \mathcal{O}$ that: + \begin{align} + \E \lb \lv \mathfrak{g}_k \lp \mathfrak{X}^{n,t,x}_T) - \mathfrak{g}_k(\mathcal{X}^{t,x}_T \rp \rv \rb &= \E \lb \mathbbm{1}_{\{\mathfrak{t}^{t,x}_n < T\}} \lv \mathfrak{g}_k(\mathfrak{X}^{n,t,x}_T) - \mathfrak{g}_k(\mathcal{X}^{t,x}_T) \rv \rb \nonumber\\ + &\leqslant 2 \lb \sup_{y \in \mathcal{O}} \lv \mathfrak{g}_k(y) \rv \rb \mathbb{P} \lp \mathfrak{t}^{t,x}_n < T \rp \nonumber + \end{align} + Note that this combined with \cite[Lemma~3.1]{Beck_2021} implies for all $t \in [0,T]$, $x \in \mathcal{O}$, $n \in \N$ we have that $\E \lb V \lp \mathfrak{t}^{t,x}_n, \mathcal{X}^{t,x}_{\mathfrak{t}^{t,x}_n}\rp \rb \leqslant V(t,x)$, which then further proves that: + \begin{align} + \lv \mathfrak{u}^{n,k}(t,x) - \mathfrak{u}^{0,k}(t,x) \rv &\leqslant 2 \lb \sup_{y\in \mathcal{O}} \lv \mathfrak{g}_k(y) \rv \rb \mathbb{P}\lp \mathfrak{t}^{t,x}_n < T \rp \nonumber\\ + &\leqslant 2 \lb\sup _{y \in \mathcal{O}} \lv \mathfrak{g}_k(y) \rv \rb \mathbb{P} \lp V \lp \mathfrak{t}^{t,x}_n, \mathcal{X}^{t,x}_{\mathfrak{t}^{t,x}_n}\rp \geqslant n\rp \nonumber \\ + %TODO: Ask Dr. Padgett how this came about + &\leqslant \frac{2}{n} \lb \sup_{y\in \mathcal{O}} \lv \mathfrak{g}_k(y) \rv \rb \E \lb V \lp \mathfrak{t}^{t,x}_n, \mathcal{X}^{t,x}_{\mathfrak{t}^{t,x}_n} \rp \rb \nonumber \\ + &\leqslant \frac{2}{n} \lb \sup_{y \in \mathcal{O}} \lv \mathfrak{g}_k(y) \rv \rb V \lp t,x, \rp \nonumber + \end{align} + Together these imply that for all $k\in\N$ and compact $\mathcal{K} \subseteq [0,T] \times \mathcal{O}$: + \begin{align}\label{2.90} + \limsup_{k \rightarrow \infty}\lb \sup_{(t,x) \in \mathcal{K}} \lp \lv \mathfrak{u}^{n,k}(t,x) - \mathfrak{u}^{0,k}(t,x) \rv \rp \rb = 0 + \end{align} + But again note that since have that $\sup_{r\in (0,\infty)}\lb \inf_{t \in [0,T], x \in \R^d \setminus O_r} V(t,x) \rb = \infty$ and (\ref{2.84}) tell us that for all compact $\mathcal{K} \subseteq [0,T] \times \mathcal{O}$ we have that: + \begin{align} + \limsup_{n \rightarrow \infty} \lb \sup_{(t,x) \in \mathcal{K}} \lp \|\mathfrak{m}_n(t,x) - \mu(t,x) \|_E + \| \mathfrak{s}_n(t,x)-\sigma(t,x) \|_F \rp \rb = 0 + \end{align} + Note that (\ref{2.89}), (\ref{2.90}) and Corollary \ref{unneq0} tell us that for all $k \in \N$ we have that $\mathfrak{u}^{0,k}$ is a viscosity solution to: + \begin{align}\label{2.93} + \lp \frac{\partial}{\partial t} \mathfrak{u}^{0,k} \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \sigma(t,x) \lb \sigma(t,x) \rb^* \lp \Hess_x \mathfrak{u}^{0,k} \rp (t,x) \rp + \la \mu(t,x), \lp \nabla_x \mathfrak{u}^{0,k} \rp (t,x) \ra = 0 + \end{align} + for $(t,x) \in (0,T) \times \mathcal{O}$. However note that (\ref{2.79}),(\ref{2.82}), (\ref{2.88}) prove that for all compact $\mathcal{K} \subseteq [0,T] \times \mathcal{O}$ we have: + \begin{align} + \limsup_{k \rightarrow \infty} \lb \sup_{(t,x) \in \mathcal{K}} \lv \mathfrak{u}^{0,k}(t,x) - u(t,x) \rv \rb = 0 + \end{align} + This together with (\ref{2.93}), (\ref{2.82}), Corollary \ref{unneq0} shows that $u_0$ is a viscosity solution to: + \begin{align}\label{2.95} + \lp \frac{\partial}{\partial t} u \rp \lp t,x \rp + \frac{1}{2} \Trace \lp \sigma(t,x) \lb \sigma(t,x) \rb^*\lp \Hess_x u \rp (t,x) \rp + \la \mu(t,x), \lp \nabla_x u \rp \ra = 0 + \end{align} + for $(t,x) \in (0,T) \times \mathcal{O}$. By (\ref{2.81}) we are ensured that for all $x\in \R^d$ we have that $u(T,x) = g(x)$ which together with proves the proposition. + +\end{proof} + + +\section{Solutions, Characterization, and Computational Bounds to the Kolmogorov Backward Equations} +% \begin{proof} +% From Feynman-Kac, especially from \cite[(1.5)]{hutzenthaler_strong_2021} and setting $f=0$ in the notation of \cite[(1.5)]{hutzenthaler_strong_2021} we have that: +% \begin{align} +% u_d(t,x) = \E \left[ u_d \left( 0,x+ \sqrt{2}W^{d}_t \right) \right] +% \end{align} + +% Substituting (2.2) and applying the time reversal property of Brownian motions then gives us: +% \begin{align} +% u_d(t,x) &= \E \left[ u_d \left( 0,x+ \sqrt{2}W^{d}_t \right) \right] \nonumber \\ +% &= \E \left[ u_d \left( 0,x+ \sqrt{2}W^{d}_{-t} \right) \right] \nonumber \\ +% &= \E \left[ u_d \left( 0,x+ \sqrt{2}W^{d}_{0-t} \right) \right] \nonumber \\ +% &= \E \left[ u_d \left( 0, \mathcal{X}^{d,t,x}_{0} \right) \right] +% \end{align} +% Looking closely at (2.3), where according to the notation of (2.2), the time reversal property of Brownian motions, (2.5), and the shift-invariance of Brownian motions we have that: + +% \begin{align} +% u_d(t,x) = \E \left[ u_d \left( T,\mathcal{X}_T^{d,t,x} \right) \right] &= \E \left[ u_d\left( T,x+\sqrt{2}W^{d,\theta}_{T-t}\right) \right] \nonumber \\ +% &= \E \left[ u_d\left( T,x+\sqrt{2}W^{d,\theta}_{t-0}\right) \right] \nonumber \\ +% &= \E \left[ u_d\left( T,x+\sqrt{2}W^{d,\theta}_{0-t}\right) \right] \nonumber \\ +% &= \E \left[ u_d\left(T, \mathcal{X}^{d,t,x}_0\right) \right] +% \end{align} +% The independence of the Brownian motions then indicates that (2.5) and (2.6) are equal. + +% This completes the proof of Lemma 2.1. +% \end{proof} + + + +\begin{theorem}[Existence and characterization of $u_d$]\label{thm:3.21} +Let $T \in (0,\infty)$. Let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a probability space. Let $\sigma_d \in C \lp \R^d,\R^{d\times d} \rp $ and $\mu_d\in C \lp \R^d, \R^d \rp$ for $d \in \N$, let $u_d \in C^{1,2} \lp \lb 0,T \rb \times \R^d, \R \rp$ satisfy for all $d \in \N$, $t \in \lb 0,T \rb$ , $x \in \R^d$ that: + \begin{align}\label{3.3.1} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + \frac{1}{2}\Trace\lp \sigma_d(x) \lb \sigma_d(x) \rb^* \lp \Hess_x u_d \rp (t,x) \rp + \la \mu_d(x), \lp \nabla_x u_d \rp(t,x) \ra = 0 + \end{align} + let $\mathcal{W}^d:[0,T] \times \Omega \rightarrow \R^d$, $d \in \N$ be a standard Brownian motions and let $\mathcal{X}^{d,t,x}: \lb t, T\rb \times \Omega \rightarrow \R^d$, $d \in \N$, $ t\in \lb 0,T \rb$, be a stochastic process with continuous sample paths satisfying for all $d \in \N$, $t \in \lb 0,T \rb$, $s \in \lb t,T \rb$, $x \in \R^d$, we have $\mathbb{P}$-a.s. that: + \begin{align}\label{3.102} + \mathcal{X}^{d,t,x} = x + \int^t_s \mu_d \lp \mathcal{X}^{d,t,x}_r \rp dr + \int^t_s \sigma \lp \mathcal{X}^{d,t,x}_r \rp d\mathcal{W}^d_r + \end{align} + Then for all $d \in N$, $t \in \lb 0,T \rb$, $x \in \R$, it holds that: + \begin{align} + u_d(t,x) = \E \lb u_d \lp T,\mathcal{X}_t^{d,t,x} \rp \rb + \end{align} + Furthermore, $u_d$ is a viscosity solution to (\ref{3.3.1}). +\end{theorem} + + +\begin{proof} + This is a consequence of Lemma \ref{lem:3.4} and \ref{2.19}. +\end{proof} +\newpage +\begin{corollary}\label{lem:3.19} Let $T \in (0,\infty)$, let $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, let $u_d \in C^{1,2} \left( \left[ 0,T \right] \times \R^d, \R \right)$, $d \in \N$ satisfy for all $d \in \N$, $t \in [0,T]$, $x \in \R^d$ that: +\begin{align} + \left( \frac{\partial}{\partial t} u_d \right) \left(t,x\right) + \frac{1}{2}\left(\nabla^2_x u_d\right) \left(t,x\right) = 0 +\end{align} +Let $\mathcal{W}^d: [0,T] \times \Omega \rightarrow \R^d$, $d \in \N$ be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: [t,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $t \in [0,T]$, $x \in \R^d$, be a stochastic process with continuous sample paths satisfying that for all $d \in N$, $t \in [0,T]$, $s \in [t,T]$, $x \in \R^d$ we have $\mathbb{P}$-a.s. that: +\begin{align} + \mathcal{X}^{d,t,x}_s = x + \int^s_t d\mathcal{W}^d_r = x + \mathcal{W}^d_{t-s} +\end{align} +Then for all $d\in \N$, $t \in [0,T]$, $x \in \R^d$ it holds that: +\begin{align} + u_d(t,x) = \E \left[u_d \left(T,\mathcal{X}^{d,T,x}_t\right)\right] +\end{align} +\end{corollary} +\begin{proof} + This is a special case of Theorem \ref{thm:3.21}. It is the case where $\sigma_d(x) = \mathbb{I}_d$, the uniform identity function where $\mathbb{I}_d$ is the identity matrix in dimension $d$ for $d \in \N$, and $\mu_d(x) = \mymathbb{0}_{d,1}$ where $\mymathbb{0}_d$ is the zero vector in dimension $d$ for $d \in \N$. +\end{proof} + + + +\begin{lemma}\label{3.3.2} + Let $T \in \lp 0,\infty \rp$, let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$, be a probability space, let $\alpha_d\in C^2_b \lp \R^d,\R \rp$, and $\alpha \in \mathcal{O}\lp x^2 \rp $ for $d \in N$, be infinitey often differentiable function, let $u_d \in C^{1,2} \lp \lb 0,T \rb \times \R^d,\R \rp$, $d \in \N$, satisfy for all $d \in \N$, $t \in \lb 0,T \rb$, $x \in \R^d$, that: + \begin{align}\label{3.3.7} + \lp \frac{\partial}{\partial t} u_d\rp \lp t,x \rp + \frac{1}{2}\lp \nabla^2_x u_d \rp \lp t,x \rp +\alpha_d \lp x \rp u_d \lp t,x \rp = 0 + \end{align} + Let $\mathcal{W}^d: \lb 0,T \rb \times \Omega \rightarrow \R^d$ be standard Brownian motions and let $\mathcal{X}^{d,t,x}: \lb t,T \rb \times \Omega \rightarrow \R^d$, $d \in \N$, $t \in \lb 0,T \rb$, $x \in \R^d$ be a stochastic process with continuous sample paths satisfying that for all $d \in \N$, $ t\in \lb 0,T \rb$, $s \in \lp t,T \rb$, $x \in \R^d$, we have $\mathbb{P}$-a.s. that: + \begin{align} + \mathcal{X}^{d,t,x}_s = x + \int^t_s \frac{1}{2} d \mathcal{W}^d_r =\frac{1}{2} \mathcal{W}^d_{t-r} + \end{align} + Then for all $d \in \N$, $t \in \lb 0,T \rb$, $x \in \R^d$ it holds that: + \begin{align} + u_d \lp t,x \rp = \E \lb \exp \lp \int^T_t \alpha_d \lp \mathcal{X}^{d,t,x}_r \rp dr \rp u_d \lp T, \mathcal{X}^{d,t,x}_T \rp \rb + \end{align} +\end{lemma} +\begin{proof} Let $v_d: \R^d \rightarrow \R$ be continuous. Throughout the proof let $u_d \lp t,x\rp = e^{-t\alpha_d(x)}v_d(t,x)$ for all $d \in \N$, $t \in [0,T]$, $x \in \R^d$. For notational simplicity, we will drop the $d,t,x$ wherever it is obvious. Therefore the derivatives become: +%\begin{align} +% \lp \frac{\partial}{\partial t}u_d \rp \lp t,x \rp &= -\alpha_d(x)e^{-t\alpha_d(x)}v_d(t,x)+e^{-t\alpha_d(x)} v_d^{(1,0)}(t,x) \nonumber\\ +% \lp \nabla_x^2 u_d \rp \lp t,x \rp &= e^{-t\alpha_d(x)} \lb -2t\alpha_d'(x) (\nabla_x v_d) (t,x) + tv(t,x) \lb t\lp \alpha_d'(x) \rp^2 -\alpha_d''(x)\rb + \lp \nabla_x^2v_d \rp (t,x)\rb +%\end{align} +% Substituting this into (\ref{3.3.7}) renders it as: +% \begin{align} +% 0 &= - \cancel{\alpha_d(x)e^{-t\alpha_d(x)}v_d(t,x)}+e^{-t\alpha_d(x)} v_d^{(1,0)} (t,x) \nonumber\\ +% &+ e^{-t\alpha_d(x)} \lb -2t\alpha_d'(x) (\nabla_x v_d) (t,x) + tv(t,x) \lb t\lp \alpha_d'(x) \rp^2 -\alpha_d''(x)\rb + v_d^{(0,2)}(t,x)\rb \nonumber\\ +% &+ \cancel{\alpha_d(x)e^{-t\alpha_d(x)}v_d(t,x)} \nonumber \\ +% &= e^{-t\alpha_d(x)} v_d^{(1,0)}(t,x) \nonumber\\ +% &+ e^{-t\alpha_d(x)} \lb -2t\alpha_d'(x) v_d^{(0,1)} (t,x) + tv(t,x) \lb t\lp \alpha_d'(x) \rp^2 -\alpha_d''(x)\rb + v_d^{(0,2)}(t,x)\rb \nonumber +% \end{align} +% Multiplying both sides by $e^{-t\alpha_d(x)}$ then gives us: +% \begin{align} +% v_d^{(1,0)}(t,x) -2t\alpha'_d(x)v_d^{(0,1)}(t,x)+tv(t,x)\lb t \lp \alpha_d'(x) \rp^2 - \alpha''_d(x) \rb + v_d^{(0,2)}(t,x) &= 0 \nonumber\\ +% v_d^{(1,0)}(t,x) = 2t\alpha'_d(x)v_d^{(0,1)}(t,x)-v(t,x)\lb t^2 \lp \alpha_d'(x) \rp^2 - t\alpha''_d(x) \rb - v_d^{(0,2)}(t,x) \nonumber\\ +% \end{align} +% From the variable substitution observe that: +% \begin{align} +% v_d(t,x) = e^{t \alpha_d(x)}u_d(t,x) \nonumber +% \end{align} +% And hence: +% \begin{align} +% v_d(0,x) = e^{0}u_d(0,x) \nonumber\\ +% v_d(0,x) = u_d(0,x) \nonumber +% \end{align} +\begin{align} + u_t &= -\alpha e^{-t\alpha}v+e^{-t\alpha} v_t \\ + \frac{1}{2}\nabla^2_x u &= \frac{1}{2} \lb e^{-t\alpha} \nabla^2_xv+2\la \nabla_x v, \nabla_x e^{-t\alpha} \ra + v\nabla^2_x e^{-t\alpha} \rb +% u_{xx} &= \frac{1}{2}e^{-t\alpha} \lb -2t\alpha_x v_x + tv \lb t(\alpha_x)^2-\alpha_{xx}\rb + v_{xx}\rb +\end{align} +This then renders (\ref{3.3.7}) as: +\begin{align} + \cancel{-\alpha e^{-t\alpha}v} + e^{-t \alpha}v_t + \frac{1}{2} \lb e^{-t\alpha }\nabla^2_xv + 2\la \nabla_xv,\nabla_xe^{-t\alpha}\ra +v\nabla_x^2e^{-t\alpha} \rb + \cancel{\alpha e^{-t\alpha}v} &= 0 \nonumber\\ + e^{-t \alpha}v_t + \frac{1}{2} \lb e^{-t\alpha }\nabla^2_xv - 2te^{-t\alpha}\la \nabla_x v, \nabla_x \alpha \ra +v\nabla_x^2e^{-t\alpha} \rb &= 0 \nonumber\\ + e^{-t \alpha}v_t + \frac{1}{2} \lb e^{-t\alpha }\nabla^2_xv - 2te^{-t\alpha}\la \nabla_x v, \nabla_x \alpha \ra -tve^{-t\alpha}\nabla^2_x\alpha \rb &= 0 \nonumber\\ + v_t + \frac{1}{2} \lb \nabla_x^2v-2t\la \nabla_xv,\nabla_x\alpha \ra -tv\nabla^2_x\alpha \rb &= 0 \nonumber \\ + v_t + \frac{1}{2} \lb \nabla_x^2v-2t\la \nabla_x \alpha,\nabla_x v \ra -tv\nabla^2_x\alpha \rb &= 0 \nonumber \\ + v_t + \frac{1}{2}\nabla_x^2v+\la -t\nabla_x \alpha,\nabla_x v \ra -\frac{1}{2}tv\nabla^2_x\alpha &= 0 \label{3.3.12} +\end{align} +%\begin{align} +% \cancel{-\alpha e^{-t\alpha}v}+e^{-t\alpha} v_t + \frac{1}{2}e^{-t\alpha} \lb -2t\alpha_xv_x + tv \lb t(\alpha_x)^2-\alpha_{xx}\rb + v_{xx}\rb + \cancel{\alpha e^{-t\alpha}v} &= 0 \nonumber\\ +% e^{-t\alpha}v_t + \frac{1}{2} e^{-t\alpha} \lb -2t\alpha_x v_x + tv \lb t(\alpha_x)^2- \alpha_{xx} \rb + v_{xx} \rb &= 0 \nonumber \\ +% v_t -t\alpha_xv_x+\frac{1}{2}tv\lb t(\alpha_x )^2-\alpha_{xx} \rb + v_{xx} &= 0 \nonumber \\ +% v_t = 2t\alpha_xv_x - tv \lb t(\alpha_x)^2-\alpha_{xx} \rb &-v_{xx} +%\end{align} +Let $\sigma(t,x) = \mathbb{I}_d$, i.e. the uniform identity function. Let $\mu(t,x) = -t\nabla_x \alpha$ for $t \in [0,T], x\in \R^d$, and for fixed $\alpha$. Let $f(t,x,v) = -\frac{1}{2}tv\nabla_x^2 \alpha$ for $t \in [0,T], x \in \R^d$. + +\begin{claim} + It is the case that for for all $x \in \R^d$ and $t \in [0,T]$ that $\la x, \mu(t,x) \ra \leqslant L \lp 1 + \|x\|_E \rp$ for some constant $L \in (0,\infty)$. +\end{claim} +\begin{proof} +Since $\alpha$ has bounded first and second derivatives let: +\begin{align} + \mathfrak{B} = \max \left\{\sup_{x \in \R^d} \| \nabla_x \alpha \|_E, \sup_{x\in \R^d} \lv \nabla_x^2\alpha \rv \right\} \label{3.3.13} + \end{align} + Note that we then have the Cauchy-Schwarz inequality: +\begin{align} + \la x, \mu(t,x) \ra \leqslant \| \la x, -t\nabla_x \alpha \ra \|_E &\leqslant \| x\|_E\|t\nabla_x\alpha\|_E \nonumber\\ + &\leqslant T \lp \| x \|_E \mathfrak{B} \rp \nonumber \\ + &\leqslant T \lp\mathfrak{B} +d \rp \|x\|_E \nonumber\\ + &= L\| x\|_E \leqslant L\lp 1 + \|x \|_E^2 \rp +\end{align} +It also follows that $\|\sigma(t,x) \|_F =\sqrt{d}\leqslant L \leqslant L(1+\|x\|_E)$. +\end{proof} +\begin{claim} + It is the case that for all $x,y \in \R^d$, and $t \in [0,T]$ that: $\|\mu(t,x) - \mu(t,y) \|_E + \| \sigma(t,x) - \sigma(t,y)\|_E \leqslant \mathfrak{C} \lp \|x\|_E+\|y\|_E \rp \lp \|x-y\|_E\rp$ for some constant $\mathfrak{C} \in (0,\infty)$. +\end{claim} +\begin{proof} +The fact that for all $x,y \in \R^d$ and $t \in [0,T]$ it is the case that $\|\sigma(t,x)-\sigma(t,y)\|_F=0$, the fact that for all $x,y \in \R^d$ it is the case that $(\|x\|_E+\|y\|_E)(\|x-y\|_E) \geqslant 0$ and (\ref{3.3.13}) tells us that: +\begin{align} + \| \mu(t,x) - \mu(t,y) \|_E+\|\sigma(t,x) - \sigma(t,y) \|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0 \nonumber\\ + &= \|t\nabla_x\alpha(x) - t\nabla_x \alpha(y) \|_E \nonumber\\ + &\leqslant T\|\nabla_x\alpha(x)-\nabla_x\alpha(y)\|_E \nonumber\\ + &\leqslant 2T\mathfrak{B} \label{3.3.15} +\end{align} +Now consider a function $\mathfrak{f} \in C \lp [0,T] \times \R^d,\R^d \rp$, where for all $x,y \in \R^d$ it is the case that $\mathfrak{f}(x) - \mathfrak{f}(y) \leqslant \mathscr{C}\lp \|x\|_E+\|y\|_E \rp \lp \|x+y\|_e \rp $. Note then that setting $y=x+h$ gives us: +\begin{align} + \left|\frac{\mathfrak{f}(x+h)-\mathfrak{f}(x)}{h} \right| &\leqslant \mathscr{C} \lp \|x\|_E + \|x+h\|_E \rp \nonumber \\ + \lim_{h \rightarrow 0} \left|\frac{\mathfrak{f}(x+h)-\mathfrak{f}(x)}{h} \right| &\leqslant \lim_{h\rightarrow 0}\mathscr{C} \lp \|x\|_E + \|x+h\|_E \rp \nonumber \\ + \left| \nabla_x\mathfrak{f}\lp x \rp \right| &\leqslant 2\mathscr{C}\|x\|_E = \mathscr{K}\|x\|_E +\end{align} +This suggests that $\nabla_x\mathfrak{f} \in O \lp x \rp$ and in particular that $\mathfrak{f} \in O \lp x^2 \rp$. However with $\mathfrak{f} \curvearrowleft \mu$ we first notice that because $\mu \leqslant 2T\mathfrak{B}$ in (\ref{3.3.15}) it must also be that case that $\mu \in O(1)$ by Corollary \ref{1.1.20.1}. However since $O(c) \subseteq O(x) \subseteq O \lp x^2 \rp$ by Corollary \ref{1.1.20.2} it is also the case that $\mu \in O \lp x^2 \rp$, and hence there exists a $\mathfrak{C}$ satisfying the claim. This proves the claim. +\end{proof} +\begin{claim}\label{3.3.5} + It is the case that $\lv f(t,x,v) - f(t,x,w) \rv \leqslant L \lv v-w \rv$ +\end{claim} +\begin{proof} +Note that by the absolute homogeneity property of norms, we have: +\begin{align} + \left|f(t,x,v) -f(t,x,w) \right| &= \left|\frac{1}{2}tv\nabla_x^2 \alpha - \frac{1}{2} tw \nabla_x^2 \alpha \right| \nonumber\\ + &= \left| \frac{1}{2}t\nabla_x^2\alpha \right| \left|v-w \right| \nonumber\\ + &\leqslant \frac{1}{2}T \left| \nabla_x^2 \alpha \right|\left| v-w \right| \nonumber\\ + &\leqslant \frac{1}{2}T \mathfrak{B} \lv v-w \rv \nonumber\\ + &\leqslant T(\mathfrak{B}+d)\lv v-w \rv \nonumber\\ + &=L \lv v-w \rv +\end{align} +\end{proof} + +Note that we may rewrite (\ref{3.3.12}) as: +\begin{align} + \lp \frac{\partial}{\partial t} v\rp \lp t,x \rp + \frac{1}{2}\Trace \lp \sigma \lp t,x \rp\lb \sigma \lp t,x \rp \rb^* \lp \Hess_x v\rp \lp t,x \rp \rp + \la \mu \lp t,x \rp , \lp \nabla_x v \rp \lp t,x \rp \ra \nonumber\\ + + f \lp t,x, v \lp t,x \rp \rp = 0 \nonumber +\end{align} + + + + + +We realize that (\ref{3.3.12}) is a case of \cite[Corollary~3.9]{bhj20} where it is the case that: $u(t,x) \curvearrowleft v(t,x)$, where $\sigma_d(x) = \mathbb{I}_d$ for all $x \in \R^d$, $d \in \N$, where $\mu(t,x) = -t\nabla_x\alpha$ for fixed $\alpha$ and for all $t \in [0,T]$, $x \in \R^d$, and where $f \lp t,x,u \lp t,x \rp \rp = -\frac{1}{2}tu\nabla_x^2\alpha$ for fixed $\alpha$ and for all $t\in [0,T]$, $x \in \R^d$. + +We thus have that there exists a unique, at most polynomially growing viscosity solution $v \in C\lp \lb 0,T \rb \times \R^d, \R \rp$ given as: + +\begin{align} + v(t,x) &= \E \lb v \lp T, \mathcal{Y}^{t,x}_T \rp+ \int^T_t f \lp s,\mathcal{Y}^{t,x}_s, v \lp s,\mathcal{Y}^{t,x}_s \rp \rp ds \rb \label{3.2.21} +\end{align} +Let $\mathcal{V}: \lb 0,T \rb \times \Omega \rightarrow \R^m$ be a standard $\lp \mathbb{F}_t\rp _{t \in \lb 0,T \rb }$-Brownian motion. Note that this also implies that the $\mathcal{Y}$ in (\ref{3.2.21}) is characterized as: +\begin{align} + \mathcal{Y}^{t,x}_s = x + \int^s_t \mu \lp r,\mathcal{Y}^{t,x}_r \rp dr + \int^s_t \sigma\lp s, \mathcal{X}^{t,x}_r \rp d\mathcal{V}_r +\end{align} +With substitution, this is then: +\begin{align} + \mathcal{Y}^{t,x}_s &= x + \int^s_t -r\nabla_x\alpha \lp \mathcal{Y}^{t,x}_r \rp dr + \int^s_t \mathbb{I} d\mathcal{V}_r \nonumber\\ + \mathcal{Y}^{t,x}_s &=x - \int^s_t r\nabla_x\alpha \lp \mathcal{Y}^{t,x}_s \rp dr + \mathcal{V}_{s-t} \nonumber +\end{align} +Note that our initial substitution tells us: $v(t,x) = e^{t\alpha(x)}u(t,x)$. And so we have that: +\begin{align} + v(t,x) &= \E \lb v\lp T, \mathcal{X}_T^{t,x} \rp + \int ^T_t f \lp s, \mathcal{X}^{t,x}_s, v \lp s, \mathcal{X}^{t,x}_s \rp \rp ds\rb \label{3.3.20}\\ + v(t,x)&=\E \lb v\lp T, \mathcal{X}^{t,x}_T \rp -\frac{1}{2} \int^T_ttv\lp s,\mathcal{X}^{t,x}_s \rp \nabla_x^2 \alpha \lp \mathcal{X}^{t,x}_s \rp ds\rb \nonumber\\ + e^{t\alpha(x)} u(t,x)&= \E \lb \exp \lb T\alpha \lp \mathcal{X}^{t,x}_T \rp \rb u\lp T, \mathcal{X}^{t,x}_T \rp -\frac{1}{2} \int^T_t t \exp \lb t\alpha \lp \mathcal{X}^{t,x}_s \rp \rb u\lp t, \mathcal{X}^{t,x}_s \rp \nabla_x^2\alpha \lp \mathcal{X}^{t,x}_s \rp ds\rb \nonumber \\ + u(t,x) &= \E \lb \exp \lb T \alpha\lp \mathcal{X}^{t,x}_T \rp -t\alpha (x)\rb u \lp T, \mathcal{X}^{t,x}_T \rp \rb \nonumber\\ + &- \E \lb\frac{1}{2e^{t\alpha(x)}} \int^T_t t \exp \lb t\alpha \lp \mathcal{X}^{t,x}_s \rp \rb u\lp t,\mathcal{X}^{t,x}_s \rp \nabla_x^2 \alpha \lp \mathcal{X}^{t,x}_s \rp ds \rb \nonumber +\end{align} +\end{proof} diff --git 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mode 100644 index 0000000..3342fe3 --- /dev/null +++ b/Homework/Iserles/Chapter 1/solutions.tex @@ -0,0 +1,150 @@ +\documentclass{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage{cancel} + +\setlength\parindent{0pt} + +\title{Numerical Analysis of DiffEq HW 1} +\author{Shakil Rafi} + +\begin{document} + +\maketitle + +\textbf{1.1} We first need to establish a lemma: +\newtheorem{lem}{Lemma} +\begin{lem} +$hf(t_{{n}+\frac{1}{2}},\frac{1}{2}(y(t_n)+y(t_{n+1})) = hf(t_{n+\frac{1}{2}},y(t_{n+\frac{1}{2}})) +\eta$ where $\eta$ is $\mathcal{O}(h^3)$. +\end{lem} +\begin{align*} + ||\eta|| = h||f(t_{n+\frac{1}{2}},\frac{1}{2}(y(t_n)+y(t_{n+1})) - f(t_{n+\frac{1}{2}},y(t_{n+\frac{1}{2}})|| \\ + \leq h\lambda||\frac{1}{2}(y(t_n)+y(t_{n+1})) = y(t_{n+\frac{1}{2}}) \\ + \leq h\frac{\lambda}{2}|y(t_n)+y(t_n+1) - 2y(t_{n+\frac{1}{2}})|| +\end{align*} +Then Taylor expansion around $y(t_{n})$ gives us: +\begin{align*} + &||\eta|| \leq \frac{h\lambda}{2}||y+(y+hy'-2y(y+\frac{1}{2}hy')+\mathcal{O}(h^2)||\\ + &= \frac{h\lambda}{2}||y+y+hy^2-2y^2-hyy'+\mathcal{O}(h^2)|| \\ + &=\mathcal{O}(h^3) +\end{align*} +\textbf{Proving the convergence of the implicit midpoint rule}: + +The implicit midpoint rule is: +\begin{align} +y_{n+1}=y_n+hf(t_n+\frac{h}{2},\frac{1}{2}(y_n+y_{n+1})). +\end{align} +Substituting the exact value then gives us: +\begin{align} +y(t_{n+1}) = y(t_n)+hf(t_n+\frac{h}{2}, \frac{1}{2}(y(t_n)+y(t_{n+1}))+\mathcal{O}(h^2) +\end{align} +Following closely the proof of $(1.9)$ and $(1.4)$ we subtract $(2)$ from $(1)$, and applying the lemma to get: +\begin{align*} + e_{n+1} &= e_n +h(f(t_{n+\frac{1}{2}},\frac{1}{2}(y_n+y_{n+1})) -f(t_{n+\frac{1}{2}},\frac{1}{2}(y(t_n)+y(t_{n+\frac{1}{2}})) +\mathcal{O}(h^3) \\ + ||e_{n+1}|| &\leq ||e_{n}|| +\frac{\lambda h}{2}(||e_{n+1}||+||e_n||)+\mathcal{O}(h^3)\\ +\end{align*} +We then reproduce the steps in Iserles: +\begin{align*} + ||e_{n+1}||-\frac{\lambda h}{2}||e_{n-1}|| &\leq ||e_n|| + \frac{\lambda h}{2}||e_n|| + \mathcal{O}(h^3) \\ + ||e_{n+1}|| &\leq \bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)||e_n|| + \bigg(\frac{c}{1+\frac{\lambda}{2}}\bigg) + \mathcal{O}(h^3) \\ +\end{align*} +And similar to Iserles we claim that: +\begin{align*} + ||e_n|| \leq \frac{c}{\lambda}\bigg[ \bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)^n -1 \bigg]h^2 +\end{align*} +We will use induction to prove this step. Clearly it holds for $n=0$. Assume that the above inequality holds upto and including $n\in \mathbb{N}$, then for $n+1$ we have: +\begin{align*} + ||e_{n+1}|| &\leq \bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)\frac{c}{\lambda}\bigg[ \bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)^n -1 \bigg]h^2+ \bigg(\frac{c}{1+\frac{\lambda}{2}}\bigg) + \mathcal{O}(h^3) \\ + &= \frac{c}{\lambda} \bigg[\bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)^{n+1}\bigg]h^2 - \frac{c}{\lambda}\bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)h^2 +\mathcal{O}(h^3) \\ + &= ||e_n|| \leq \frac{c}{\lambda}\bigg[ \bigg(\frac{1+\frac{\lambda h}{2}}{1-\frac{\lambda h}{2}}\bigg)^{n+1} -1 \bigg]h^2 +\end{align*} + +\textbf{To show that the $\theta$ method is convergent}, we define the theta method as: +\begin{align*} + y_{n+1} &= y_{n}+h[\theta f(t_n,y_n)+(1-\theta)f(t_{n+1},y_{n+1})] +\end{align*} +Whereas substituting exact solutions gives us: + +\begin{align*} + y(t_{n+1}) y(t_n) +h[\theta f(t_n,y(t_n))+(1-\theta)f(t_{n+1},y(t_{n+1}))]+\mathcal{O}(h^3) +\end{align*} + +Once again subtracting $(3)$ from $(4)$ gives us: +\begin{align*} + e_{n+1} &= e_n +h[\theta f(t_n,y_n)-\theta f(t_n,y(t_n))+(1-\theta)f(t_{n+1},y(t_{n+1}))+(1-\theta)f(t_{n+1},y(t_{n+1}))] +\mathcal{O}(h^3)\\ + ||e_{n+1}|| &\leq ||e_n||+h[\theta \lambda ||e_n||+(1-\theta)\lambda ||e_{n+1}||]+\mathcal{O}(h^3) \\ + ||e_{n+1}|| &\leq \bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg)||e_n||+\frac{c}{1-h(1-\theta)\lambda} +ch^3 \quad \text{for some }c +\end{align*} +Similar to the trapezoid method we will argue: + +\begin{align*} + ||e_n|| \leq \frac{c}{\lambda} \bigg[ \bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg) ^n -1 \bigg] h^2 +\end{align*} +We argue via induction. Clearly we have that it is true for $n=0$ as at that point the exact and approximate solutions are the same. Assume now that it is true upto and including $n$. We need to prove for $n+1$: + +\begin{align*} + ||e_{n+1}|| &\leq \bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg)||e_n||+\frac{c}{1-h(1-\theta)\lambda} +ch^3\\ + ||e_{n+1}|| &\leq \bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg)\bigg(\frac{c}{\lambda} \bigg[ \bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg) ^n -1 \bigg] h^2\bigg)+\frac{c}{1-h(1-\theta)\lambda} +ch^3 +\end{align*} + +Now observe that $\theta$ varies between $1$ and $0$. Thus $\bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg)$ varies between $1+h\lambda$ and $\frac{1}{1-h\lambda}$, both of which are bigger than one. As such we can continue and say: +\begin{align*} + ||e_n|| \leq \frac{c}{\lambda} \bigg[ \bigg(\frac{1+h\theta \lambda}{1-h(1-\theta)\lambda} \bigg) ^{n+1} -1 \bigg] h^2 +\end{align*} + +\textbf{1.2a} Let $y'=Ay$, and let $e_n=y_n-y(nh)$ +We want to prove using induction: +\begin{align*} + ||e_n||_2 \leq ||y_0||_2 \max_{\lambda \in \sigma(A)}|(1-h\lambda)^n-e^{nh\lambda}| +\end{align*} +But before that we make an observation: +\begin{align*} + e_n = y_n-y(nh) +\end{align*} +And since we are using Euler method, we can say: +\begin{align*} + y(t_{n+1}) &= y(t_n) + hy'(t) + \mathcal{O}(h^2) \\ + y_{n+1} - y(t_{n-1}) &= y_n - y(nh) +h[(f(t_n,y(t_n))-f(t_n,y_n))] +\end{align*} +Substitution $A$ gives us: +\begin{align*} + e_{n+1} &= e_n+h[Ay_n - Ay(nh)]+\mathcal{O}(h^2) \\ + ||e_{n+1}|| &\leq ||e_n||_2 +h\lambda ||e_n||_2+\mathcal{O}(h^2)\\ + ||e_{n+1}|| &\leq ||e_n||_2(1+h\lambda) +\mathcal{O}(h^2) +\end{align*} + +For the induction part we observe that the statement clearly holds true for $n=0$ since then we get: +\begin{align*} + ||e_0||_2 &\leq ||y_0||_2 \max_{\lambda \in \sigma(A)}|(1+h\lambda)^0 - e^0| \\ + 0 &\leq 0 +\end{align*} +Using it as our base case, assume the inequality holds for upto and including $n$ the for $||e_{n+1}||$ we have: +\begin{align*} + ||e_{n+1}||_2 &\leq ||e_n||_2(1+h\lambda)+\mathcal{O}(h^2) \\ + ||e_{n+1}||_2 &\leq ||y_0||_2 \max_{\lambda \in \sigma(A)}|(1+h\lambda)^n - e^{nh\lambda}|(1+h\lambda)+\mathcal{O}(h^2x)\\ + &\leq ||y_0||_2\max_{\lambda \in \sigma(A)}|(1+h\lambda)^{n+1}-[e^{nh\lambda}+h\lambda e^{nh\lambda}]| \\ + &\leq ||y_0||_2\max_{\lambda \in \sigma(A)}|(1+h\lambda)^{n+1}-e^{(n+1)h\lambda}| +\end{align*} + +\textbf{1.2b} From the hint, we first seek to prove $1+x \leq e^x$. Let $f(x) = e^x-x-1$, then $f'(x) = e^x-1$ and $f''(x) = e^x$. There is a global minimum of $0$ and this function is concave up, and so $f(x) > 0$ over all $x$ and hence $e^x \geq 1+x$. +\newline +Following the hint again we seek to prove that $1+x+\frac{x^2}{2} \geq e^x$. Observe the series expansion of $e^x$ is $e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}\mathcal{O}(x^3)$. Thus: +\begin{align*} + 1+x+\frac{x^2}{2} +\frac{x^3}{3!}+\mathcal{O}(h^4) - 1-x-\frac{x^2}{2} \\ + \cancel{1}+\cancel{x}+\cancel{\frac{x^2}{2}} +\frac{x^3}{3!}+\mathcal{O}(h^4) - \cancel{1}-\cancel{x}-\cancel{\frac{x^2}{2}} \\ + \frac{x^3}{3!} + \mathcal{O}(h^4) < 0 \text{ as } x\in[-1,0] +\end{align*} +We use a similar logic for the last part of the hint. Observe that: $(a-b)^n = \Sigma^n_{i=0}{n \choose i}a^{n-i}b^i = a^n -na^{n-1}b + ...$. So: +\begin{align*} + \Sigma^n_{i=1}{n \choose i}a^{n-i}b^i &- a^n+na^{n-1}b \\ + \Sigma^n_{i=2}{n \choose i}a^{n-i}b^i &\geq 0 \text{ as a is close to being 1 and b is small} +\end{align*} +For the actual proof, let $a=e^x$ and $b=\frac{1}{2}x^2$. We then get: +\begin{align*} + e^{nx} - \frac{1}{2}nx^2e^{(n-1)x} &\leq (e^x-\frac{x^2}{2})^n \leq (1+x)^n \leq e^{nx} +\end{align*} + +\textbf{1.4} +\end{document} diff --git a/MLP Ideas/MLP_ideas.aux b/MLP Ideas/MLP_ideas.aux new file mode 100644 index 0000000..050cb49 --- /dev/null +++ b/MLP Ideas/MLP_ideas.aux @@ -0,0 +1,64 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} 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+\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} + +\numberwithin{equation}{section} + +%%% + +\input{commands.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +% +% +% + +\title{MLP starting ideas} + +% +% +% + +\author{ +%%%Joshua Lee Padgett$^{1,2}$ +%%%\bigskip +%%%\\ +%%%\small{$^1$ Department of Mathematical Sciences, University of Arkansas,} +%%%\vspace{-0.1cm}\\ +%%%\small{Arkansas, USA, e-mail: \texttt{padgett@uark.edu}} +%%%\smallskip +%%%\\ +%%%\small{$^2$ Center for Astrophysics, Space Physics, and Engineering Research,} +%%%\vspace{-0.1cm}\\ +%%%\small{Baylor University, Texas, USA, e-mail: \texttt{padgett@uark.edu}} +} + +% +% +% + +\date{\today} + +% +% +% + +\maketitle + +\begin{abstract} +% +Abstract goes here\dots +% +\end{abstract} + +% +% +% + +\tableofcontents + +% +% +% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Introduction} +\label{sec:intro} + +% +% +% + +Add an appropriate introduction\dots +% +% + +% +% +% + +\section{Multilevel Picard approximations for the heat equation} +\label{sec:mlp} + +% +% +% + +\begin{athm}{theorem}{th:1} +% +% +Let $T,\kappa, \delta \in (0,\infty)$, +$\Theta = \bigcup_{n\in\N}\! \Z^n$, +% +let $\smallU_d \in C^{1,2}([0,T]\times \R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t \in [0,T]$, $x=(x_1,\allowbreak x_2,\allowbreak \dots, \allowbreak x_d)\in\R^d$ that +% +\begin{equation}\label{eq:1} +\abs{ \smallU_d(t,x)} \le \kappa d^\kappa \pr[\big]{ 1 + \textstyle\sum_{k=1}^d \abs{ x_k } }^\kappa +\qquad +\text{and} +\qquad +\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) = \pr[]{\Delta_x \smallU_d}(t,x) \dc +\end{equation} +% +let $(\Omega, \cF ,\P)$ be a probability space, +% +let $W^{d,\theta} \colon [0,T] \times \Omega\to \R^d$, $d\in\N$, $\theta\in\Theta$, be independent standard Brownian motions, +% +let $\mlp{\littleM}{d}{\theta} \colon [0,T] \times \R^d \times \Omega \to \R$, $d,\littleM \in \Z$, $\theta \in \Theta$, satisfy for all $d,\littleM \in \N$, $\theta \in \Theta$, $t \in [0,T]$, $x \in \R^d$ that +% +\begin{align} +& \mlp{\littleM}{d}{\theta}(t,x) += +\frac{1}{\littleM} \biggl[ \SmallSum{k=1}{\littleM} \smallU_d \pr[\big]{ 0,x + \sqrt{2}\,W_{t}^{d,(\theta,0,-k)} } \biggr], \nonumber +\end{align} +% +and for every $d,n,\littleM \in \N$ let $\cost{n}{\littleM}{d} \in \N$ be the number of function evaluations of $\smallU_d(0,\cdot)$ and the number of realizations of scalar random variables which are used to compute one realization of $\mlp{\littleM}{d}{0}(T,0) \colon \Omega \to \R$. +% +% +Then there exist $c\in\R$ and $\fR \colon \N \times (0,1] \to \N$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that +% +\begin{equation} +\textstyle +\pr[\Big]{\E\br[\big]{\abs{\smallU_d(T,0) - \mlp{\fR(d,\varepsilon)}{d}{0}(T,0)}^2}}^{\!\!\nicefrac{1}{2}} \le \varepsilon +\qquad +\text{and} +\qquad +\cost{\fR(d,\varepsilon)}{\fR(d,\varepsilon)}{d} \le c d^c \varepsilon^{-(2+\delta)} +\dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\section{Stochastic solutions to parabolic partial differential equations} +\label{sec:sfp} + +% +% +% + +\begin{athm}{lemma}{lem:feynman-kac_1} +% +% +Let $T \in (0,\infty)$, +% +let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, +% +let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that +% +\begin{equation}\label{eq:pde_1} +\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) ++ \pr[]{\Delta_x \smallU_d}(t,x) += 0 \dc +\end{equation} +% +let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions, +% +and +let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s \in [t,T]$, $x\in\R^d$ we have $\mathbb{P}$-a.s.\ that +% +\begin{equation}\label{eq:stochastic_process_1} +\cX_{s}^{d,t,x} += x + \int_t^s \sqrt{2} \dx \fwpr_r^d += x + \sqrt{2} \, \fwpr_{t-s}^d +\dpp +\end{equation} +% +Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that +% +\begin{equation}\label{eq:feynman-kac_sol_1} +\smallU_d(t,x) += +\E\br[\bpig]{ \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } } \dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +% +\end{aproof} + +% +% +% + +\newcommand{\Hess}{\operatorname{Hess}} +\newcommand{\Trace}{\operatorname{Trace}} + +\begin{athm}{lemma}{lem:feynman-kac_2} +% +% +Let $T \in (0,\infty)$, +% +let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, +% +let $\sigma_d \colon \R^d \to \R^{d\times d}$, $d\in\N$, be infinitely often differentiable functions, +% +let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that +% +\begin{equation}\label{eq:pde_2} +\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) ++ \Trace \pr[\pig]{ \sigma(x) \br[]{ \sigma(x) }^* \pr[]{\Hess_x \smallU_d}(t,x) } += 0 \dc +\end{equation} +% +let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions, +% +and +let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s\in[t,T]$, $x\in\R^d$ we have $\P$-a.s.\ that +% +\begin{equation}\label{eq:stochastic_process_2} +\cX_{s}^{d,t,x} += x + \int_s^t \sqrt{2} \, \sigma\pr[]{ \cX_r^{d,t,x} } \dx \fwpr_r^d +\dpp +\end{equation} +% +Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that +% +\begin{equation}\label{eq:feynman-kac_sol_2} +\smallU_d(t,x) += +\E\bpigl[ \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } \bpigr] \dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +% +\end{aproof} + +% +% +% + +\begin{athm}{lemma}{lem:feynman-kac_3} +% +% +Let $T \in (0,\infty)$, +% +%%%let $\vt{\cdot,\cdot} \colon \bigcup_{d\in\N} \pr[]{ \R^d \times \R^d} \to \bigcup_{d\in\N}\!\R^d$ satisfy for all $d\in\N$, $v = (v_1,\dots,v_d), w = (w_1,\dots,w_d) \in \R^d$ that $\vt{v,w} = \sum_{k=1}^d v_k w_k$, +% +let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, +% +let $\mu_d \in \R^d \to \R^d$, $d\in\N$, be infinitely often differentiable functions, +% +let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that +% +\begin{equation}\label{eq:pde_3} +\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) ++ \pr[]{\Delta_x \smallU_d}(t,x) ++ \br[]{ \mu_d(x) }^* \pr[]{ \nabla_x \smallU_d }(t,x) += 0 \dc +\end{equation} +% +let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions, +% +and +let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s\in[t,T]$, $x\in\R^d$ we have $\P$-a.s.\ that +% +\begin{equation}\label{eq:stochastic_process_3} +\cX_{s}^{d,t,x} += x + \int_s^t \mu_d\pr[]{ \cX_r^{d,t,x} } \dx r + \int_s^t \sqrt{2} \dx \fwpr_r^d +\dpp +\end{equation} +% +Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that +% +\begin{equation}\label{eq:feynman-kac_sol_3} +\smallU_d(t,x) += +\E\bpigl[ \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } \bpigr] \dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +% +\end{aproof} + +% +% +% + +\begin{athm}{lemma}{lem:feynman-kac_4} +% +% +Let $T \in (0,\infty)$, +% +% +let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, +% +let $\alpha_d \in \R^d \to \R$, $d\in\N$, be infinitely often differentiable functions, +% +let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that +% +\begin{equation}\label{eq:pde_4} +\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) ++ \pr[]{\Delta_x \smallU_d}(t,x) ++ \alpha_d(x) \smallU_d(t,x) += 0 \dc +\end{equation} +% +let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions, +% +and +let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s\in[t,T]$, $x\in\R^d$ we have $\P$-a.s.\ that +% +\begin{equation}\label{eq:stochastic_process_4} +\cX_{s}^{d,t,x} += x + \int_s^t \sqrt{2} \dx \fwpr_r^d +\dpp +\end{equation} +% +Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that +% +\begin{equation}\label{eq:feynman-kac_sol_4} +\smallU_d(t,x) += +\E\br[\bpig]{ \exp\pr[\big]{ \textstyle\int_t^T \alpha_d( \cX_r^{d,t,x} ) \dx r } \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } } \dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +% +\end{aproof} + +% +% +% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%%\bibliographystyle{acm} +%%%\bibliography{bibfile} + +\end{document} \ No newline at end of file diff --git a/MLP Ideas/MLP_ideas.toc b/MLP Ideas/MLP_ideas.toc new file mode 100644 index 0000000..b43b20e --- /dev/null +++ b/MLP Ideas/MLP_ideas.toc @@ -0,0 +1,3 @@ +\contentsline {section}{\numberline {1}Introduction}{1}{section.1}% +\contentsline {section}{\numberline {2}Multilevel Picard approximations for the heat equation}{1}{section.2}% +\contentsline {section}{\numberline {3}Stochastic solutions to parabolic partial differential equations}{2}{section.3}% diff --git a/MLP Ideas/bibfile.bib b/MLP Ideas/bibfile.bib new file mode 100644 index 0000000..671f37a --- /dev/null +++ b/MLP Ideas/bibfile.bib @@ -0,0 +1,21 @@ +% Encoding: UTF-8 + +@article{grohs2018proof, + title={A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of {B}lack-{S}choles partial differential equations}, + author={Grohs, Philipp and Hornung, Fabian and Jentzen, Arnulf and Von Wurstemberger, Philippe}, + journal={arXiv:1809.02362}, + note={\emph{To appear in Mem.\ Amer.\ Math.\ Soc.}}, + year={2018}, + pages={124 pages}, +} + +@article{rio2009moment, + title={Moment inequalities for sums of dependent random variables under projective conditions}, + author={Rio, Emmanuel}, + journal={J.\ Theor.\ Probab.}, + volume={22}, + number={1}, + pages={146--163}, + year={2009}, + publisher={Springer} +} \ No newline at end of file diff --git a/MLP Ideas/commands.tex b/MLP Ideas/commands.tex new file mode 100644 index 0000000..cbf6bc0 --- /dev/null +++ b/MLP Ideas/commands.tex @@ -0,0 +1,700 @@ +\DeclareFontEncoding{LS1}{}{} +\DeclareFontSubstitution{LS1}{stix}{m}{n} +\DeclareMathAlphabet{\mathscr}{LS1}{stixscr}{m}{n} + +\newcommand{\cA}{\mathcal{A}} +\newcommand{\cB}{\mathcal{B}} +\newcommand{\cC}{\mathcal{C}} 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+\newcommand{\fI}{\mathfrak{I}} +\newcommand{\fJ}{{\bf\mathfrak{J}}} +\newcommand{\fK}{\mathfrak{K}} +\newcommand{\fL}{\mathfrak{L}} +\newcommand{\fM}{\mathfrak{M} \cfadd{multi}} +\newcommand{\fN}{\mathfrak{N}} +\newcommand{\fO}{\mathfrak{O}} +\newcommand{\fP}{\mathfrak{P}} +\newcommand{\fQ}{\mathfrak{Q}} +\newcommand{\fR}{\mathfrak{R}} +\newcommand{\fS}{\mathfrak{S}} +\newcommand{\fT}{\mathfrak{T}} +\newcommand{\fU}{\mathfrak{U}} +\newcommand{\fV}{\mathfrak{V}} +\newcommand{\fW}{\mathfrak{W}} +\newcommand{\fX}{\mathfrak{X}} +\newcommand{\fY}{\mathfrak{Y}} +\newcommand{\fZ}{\mathfrak{Z}} + +\newcommand{\fa}{\mathfrak{a}} +\newcommand{\fb}{\mathfrak{b}} +\newcommand{\fc}{\mathfrak{c}} +\newcommand{\fd}{\mathfrak{d}} +\newcommand{\fe}{\mathfrak{e}} +\newcommand{\ff}{\mathfrak{f}} +\newcommand{\fg}{\mathfrak{g}} +\newcommand{\fh}{\mathfrak{h}} +%\newcommand{\fi}{\mathfrak{i}} +%\newcommand{\fj}{\mathfrak{j}} +\newcommand{\fk}{\mathfrak{k}} +\newcommand{\fl}{\mathfrak{l}} +\newcommand{\fm}{\mathfrak{m}} +\newcommand{\fn}{\mathfrak{n}} +\newcommand{\fo}{\mathfrak{o}} +\newcommand{\fp}{\mathfrak{p}} +\newcommand{\fq}{\mathfrak{q}} +\newcommand{\fr}{\mathfrak{r} \cfadd{rectifier}} +\newcommand{\fs}{\mathfrak{s}} +\newcommand{\ft}{\mathfrak{t}} +\newcommand{\fu}{\mathfrak{u}} +\newcommand{\fv}{\mathfrak{v}} +\newcommand{\fw}{\mathfrak{w}} +\newcommand{\fx}{\mathfrak{x}} +\newcommand{\fy}{\mathfrak{y}} +\newcommand{\fz}{\mathfrak{z}} + +\newcommand{\bfa}{\mathbf{a}} +\newcommand{\bfb}{\mathbf{b}} +\newcommand{\bfc}{\mathbf{c}} +\newcommand{\bfd}{\mathbf{d}} +\newcommand{\bfe}{\mathbf{e}} +\newcommand{\bff}{\mathbf{f}} +\newcommand{\bfg}{\mathbf{g}} +\newcommand{\bfh}{\mathbf{h}} +\newcommand{\bfi}{\mathbf{i}} +\newcommand{\bfj}{\mathbf{j}} +\newcommand{\bfk}{\mathbf{k}} +\newcommand{\bfl}{\mathbf{l}} +\newcommand{\bfm}{\mathbf{m}} +\newcommand{\bfn}{\mathbf{n}} +\newcommand{\bfo}{\mathbf{o}} +\newcommand{\bfp}{\mathbf{p}} +\newcommand{\bfq}{\mathbf{q}} +\newcommand{\bfr}{\mathbf{r}} +\newcommand{\bfs}{\mathbf{s}} +\newcommand{\bft}{\mathbf{t}} +\newcommand{\bfu}{\mathbf{u}} +\newcommand{\bfv}{\mathbf{v}} +\newcommand{\bfw}{\mathbf{w}} +\newcommand{\bfx}{\mathbf{x}} +\newcommand{\bfy}{\mathbf{y}} +\newcommand{\bfz}{\mathbf{z}} + +\newcommand{\bfA}{\mathbf{A}} +\newcommand{\bfB}{\mathbf{B}} +\newcommand{\bfC}{\mathbf{C}} +\newcommand{\bfD}{\mathbf{D}} +\newcommand{\bfE}{\mathbf{E}} +\newcommand{\bfF}{\mathbf{F}} +\newcommand{\bfG}{\mathbf{G}} +\newcommand{\bfH}{\mathbf{H}} +\newcommand{\bfI}{\mathbf{I}} +\newcommand{\bfJ}{\mathbf{J}} +\newcommand{\bfK}{\mathbf{K}} +\newcommand{\bfL}{\mathbf{L}} +\newcommand{\bfM}{\mathbf{M}} +\newcommand{\bfN}{\mathbf{N}} +\newcommand{\bfO}{\mathbf{O}} +\newcommand{\bfP}{\mathbf{P}} +\newcommand{\bfQ}{\mathbf{Q}} +\newcommand{\bfR}{\mathbf{R}} +\newcommand{\bfS}{\mathbf{S}} +\newcommand{\bfT}{\mathbf{T}} +\newcommand{\bfU}{\mathbf{U}} +\newcommand{\bfV}{\mathbf{V}} +\newcommand{\bfW}{\mathbf{W}} +\newcommand{\bfX}{\mathbf{X}} +\newcommand{\bfY}{\mathbf{Y}} +\newcommand{\bfZ}{\mathbf{Z}} + +\newcommand{\scrA}{\mathscr{A}} +\newcommand{\scrB}{\mathscr{B}} +\newcommand{\scrC}{\mathscr{C}} +\newcommand{\scrD}{\mathscr{D}} +\newcommand{\scrE}{\mathscr{E}} +\newcommand{\scrF}{\mathscr{F}} +\newcommand{\scrG}{\mathscr{G}} +\newcommand{\scrH}{\mathscr{H}} +\newcommand{\scrI}{\mathscr{I}} +\newcommand{\scrJ}{\mathscr{J}} +\newcommand{\scrK}{\mathscr{K}} +\newcommand{\scrL}{\mathscr{L} \cfadd{def:lin_interp}} +\newcommand{\scrM}{\mathscr{M}} +\newcommand{\scrN}{\mathscr{N}} +\newcommand{\scrO}{\mathscr{O}} +\newcommand{\scrP}{\mathscr{P}} +\newcommand{\scrQ}{\mathscr{Q}} +\newcommand{\scrR}{\mathscr{R}} +\newcommand{\scrS}{\mathscr{S}} +\newcommand{\scrT}{\mathscr{T}} +\newcommand{\scrU}{\mathscr{U}} +\newcommand{\scrV}{\mathscr{V}} +\newcommand{\scrW}{\mathscr{W}} +\newcommand{\scrX}{\mathscr{X}} +\newcommand{\scrY}{\mathscr{Y}} +\newcommand{\scrZ}{\mathscr{Z}} + +\newcommand{\LL}{\mathbb{L}} + +\newcommand{\induct}{\dashrightarrow} +\newcommand{\with}{\curvearrowleft} +\newcommand{\lrSpace}{\ensuremath{\mkern-1.5mu}} + +\newcommand{\eps}{\varepsilon} +\newcommand{\eg}{\unskip, e.g.,\ } +\newcommand{\Eg}{\unskip e.g.,\ } + +\newcommand{\dpp}{\text{.}} +\newcommand{\dc}{\text{,}} +\newcommand{\dx}{\, {\rm d}} +\newcommand{\dxx}{{\rm d}} + +\newcommand{\normmm}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 + \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert} + \cfadd{DNN_norm}} + +\DeclarePairedDelimiter{\pr}{(}{)} +\DeclarePairedDelimiter{\br}{[}{]} +\DeclarePairedDelimiter{\cu}{\{}{\}} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert\cfadd{DNN_norm}} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} +\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} +\DeclarePairedDelimiter{\vt}{\langle}{\rangle} + +\newcommand{\N}{\ensuremath{\mathbb N}} +\newcommand{\Z}{\ensuremath{\mathbb Z}} +\newcommand{\Q}{\ensuremath{\mathbb Q}} +\newcommand{\R}{\ensuremath{\mathbb R}} +\newcommand{\C}{\ensuremath{\mathbb C}} +\newcommand{\E}{\ensuremath{\mathbb E}} +\renewcommand{\P}{\ensuremath{\mathbb P}} +\newcommand{\parallelization}{\operatorname{P} \cfadd{def:generalParallelization}} +\newcommand{\parallelizationSpecial}{\mathbf{P} \cfadd{def:simpleParallelization}} +\newcommand{\pa}[1]{\left({#1}\right)} +\newcommand{\Rr}{\mathcal{R}_\mathfrak{r} \cfadd{def:ANNrealization} \cfadd{rectifier}} +\newcommand{\Ra}{\mathcal{R}_a \cfadd{def:ANNrealization}} +\newcommand{\1}{\mathbbm{1}} +\newcommand{\ii}{\mathfrak{i} \cfadd{padding}} +\newcommand{\pu}{\mathfrak{u}} +\newcommand{\fj}{\mathfrak{x}} +\newcommand{\network}{neural network \cfadd{def:ANN2}} +\newcommand{\interpolatingDNN}{\mathbf{F}} +\newcommand{\AffineANN}{\cfadd{linear}\mathbf{A}} +\newcommand{\bbigANNsum}{\cfadd{def:ANNsum:same}\mathop{\oplus}\limits} +\newcommand{\scalarMultANN}[2]{\cfadd{def:ANNscalar}#1\circledast#2} +\newcommand{\interpol}[2]{\cfadd{def:lin_interp}\mathscr{L}_{#1}^{#2}} +\newcommand{\infnorm}[1]{\cfadd{def:p-norm}\norm{#1}_{\infty}} +\newcommand{\MappingStructuralToVectorized}{\cfadd{def:TranslateStructuredIntoVectorizedDescription}\mathcal{T}} +\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}} + +\newcommand{\fwpr}{W} +\newcommand{\fwprr}{W} +\newcommand{\smallU}{u} +\newcommand{\smallF}{f} +\newcommand{\funcF}{F} +\newcommand{\funcG}{g} +\newcommand{\boundFG}{\mathfrak{L}} +\newcommand{\LipConstF}{L} +\newcommand{\mlp}[3]{U_{#1}^{#2,#3}} +\newcommand{\Msymb}{\mathcal{M}} +\newcommand{\littleM}{m} +\newcommand{\littleMM}{m} +\newcommand{\funcM}{\phi} +\newcommand{\funcMrep}[1]{\max\{ k \in \N \colon k \le \exp( \abs{\ln(#1)}^{1/2} ) \}} +\newcommand{\cost}[3]{{\mathfrak{C}}_{#3,#1,#2}} + +\newcommand{\firstConstant}[1]{\mathfrak{K}_{#1}} +\renewcommand{\fR}{\mathscr{n}} +\newcommand{\secondConstant}[1]{\firstConstant{#1}\sqrt{#1 -1}} +\newcommand{\maxfn}{\varphi} + +\renewcommand{\AA}{\mathbb{A}} +\newcommand{\PP}{\mathbf{P}} +\newcommand{\BB}{\mathbb{B}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\A}{\mathbf{A} \cfadd{linear}} +\newcommand{\U}{\mathbf{U}} +\newcommand{\V}{\mathbf{V}} +\newcommand{\F}{\mathbf{F}} +\newcommand{\G}{\mathbf{G}} +\newcommand{\I}{\mathbf{I} \cfadd{def:id_net}} +\renewcommand{\H}{\mathbf{H}} +\newcommand{\NN}{\mathbf{N}} +\newcommand{\lbd}{u} +\newcommand{\ubd}{v} +\newcommand{\len}{n} +\newcommand{\weight}{W \cfadd{linear}} +\newcommand{\bias}{B \cfadd{linear}} + +\newcommand{\ind}[1]{\mathbbm{1}_{\smash{#1}}} +\newcommand{\iind}[1]{\mathlarger{\mathbbm{1}}_{#1}} +\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} +\newcommand{\indicator}[1]{\mathbbm{1}_{\smash{#1}}} + +% and in equations +\newcommand{\qandqShort}{\quad\text{and}\quad} +\newcommand{\andShort}{\text{ and }} +\newcommand{\forallDist}{\forall\,} +\newcommand{\identity}[1]{\mathrm{Id}_{#1}} +\newcommand{\smallsum}{\textstyle\sum} +\newcommand{\SmallSum}[2]{ {\textstyle\sum\limits_{#1}^{#2}}} +\newcommand{\idRelu}{\mathfrak I} +\newcommand{\Exists}{\exists\,} +\newcommand{\Forall}{\forall\,} +\newcommand{\mc}[1]{\mathcal{#1}} +\newcommand{\mf}[1]{\mathfrak{#1}} +\newcommand{\affineMap}{\mathbb{A}} +\newcommand{\idMatrix}{\operatorname{I} \cfadd{def:identityMatrix}} +\newcommand{\ANNs}{\mathbf{N} \cfadd{def:ANN}} +\newcommand{\MatrixANN}[1]{\mf N_{#1}} +\newcommand{\activation}{a} +\newcommand{\activationDim}[1]{\mathfrak{M}_{\activation,#1} \cfadd{multi}} +\newcommand{\functionANN}[1]{\mathcal{R}_{#1} \cfadd{def:ANNrealization}} +\newcommand{\paramANN}{\mathcal{P} \cfadd{def:ANN}} +\newcommand{\paramNotZeroANN}{\mathscr{P}} +\newcommand{\lengthANN}{\mathcal{L} \cfadd{def:ANN}} +\newcommand{\inDimANN}{\mathcal{I} \cfadd{def:ANN}} +\newcommand{\compANN}[2]{{#1 \bullet \allowbreak #2} \cfadd{def:ANNcomposition}} +\newcommand{\concANN}[2]{{#1 \odot #2}} +\newcommand{\concPsiANN}[2]{{#1 \odot_{\Psi} #2}} +\newcommand{\paraANN}[1]{\mathbf{P}_{#1}} +\newcommand{\paraLANN}[2]{\mathbf{P}_{#1,#2}} +\newcommand{\outDimANN}{\mathcal{O} \cfadd{def:ANN}} +\newcommand{\longerANN}[1]{\mathcal{E}_{#1} \cfadd{def:ANNenlargement}} +\newcommand{\idANN}[2]{\mathbb{I}_{#1,#2}} +\newcommand{\idANNshort}[1]{\mathbb{I}_{#1}} +\newcommand{\dims}{\mathcal{D} \cfadd{def:ANN}} +\newcommand{\hiddenLength}{\mathcal{H} \cfadd{def:ANN}} +\newcommand{\hiddenDimId}{\mathfrak{i}} +\newcommand{\qandq}{\qquad\text{and}\qquad} +\newcommand{\andq}{\text{and}\qquad} +\newcommand{\rect}{\mathfrak r \cfadd{rectifier}} +\newcommand{\Rect}{\mathfrak R} +\newcommand{\multdim}{\mathfrak M} +\DeclareMathOperator{\id}{id} +\newcommand{\matrixANN}[1]{\mathfrak{W}_{#1}} +\newcommand{\vectorANN}{\mathfrak{B}} +\newcommand{\sumANN}{\mathfrak{S} \cfadd{def:ANN:sum}} +\newcommand{\extensionANN}{\mathfrak{T} \cfadd{def:ANN:extension}} +\newcommand{\dimANNlevel}{\mathbb{D} \cfadd{def:ANN}} +\newcommand{\power}[2]{#1^{\bullet #2} \cfadd{def:iteratedANNcomposition}} +\newcommand{\scalar}[2]{ #1 \circledast #2 \cfadd{def:ANNscalar}} +\newcommand{\transpose}{* \cfadd{def:Transpose}} +\newcommand{\oSum}{\oplus \cfadd{def:ANNsum:same}} +\newcommand{\OSum}[2]{{\mathop\oplus\limits_{#1}^{#2} } \cfadd{def:ANNsum:same} } +\newcommand{\bSum}{{\mathop\boxplus} \cfadd{def:ANN:sum_diff}} +\newcommand{\BSum}[3]{{\mathop\boxplus\limits_{#1,#2}^{#3}} \cfadd{def:ANN:sum_diff}} +\newcommand{\modCont}{w \cfadd{mod_cont_def}} + +\allowdisplaybreaks + +\usepackage{scalerel,stackengine} +\newcommand\pig[1]{\scalerel*[5pt]{\big#1}{% + \ensurestackMath{\addstackgap[1.5pt]{\big#1}}}} +\newcommand\pigl[1]{\mathopen{\pig{#1}}} +\newcommand\pigr[1]{\mathclose{\pig{#1}}} + +\newcommand\bpig[1]{\scalerel*[5pt]{\big#1}{% + \ensurestackMath{\addstackgap[1.5pt]{\big#1}}}} +\newcommand\bpigl[1]{\mathopen{\bpig{#1}}} +\newcommand\bpigr[1]{\mathclose{\bpig{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Enum +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentCommand{\enum}{ O{;} m o } + { + \my_enum:nnn { #1 } { #2 } { #3 } + } + +\seq_new:N \l__my_enum_seq +\tl_new:N \l__my_enum_item_tl +\prop_new:N \l__verbs +\prop_put:Nnn \l__verbs {show} {shows} +\prop_put:Nnn \l__verbs {imply} {implies} +\prop_put:Nnn \l__verbs {demonstrate} {demonstrates} +\prop_put:Nnn \l__verbs {prove} {proves} +\prop_put:Nnn \l__verbs {establish} {establishes} +\prop_put:Nnn \l__verbs {ensure} {ensures} +\prop_put:Nnn \l__verbs {assure} {assures} +\prop_put:Nnn \l__verbs {yield} {yields} + +\int_new:N \l__number_of_args + +\cs_new_protected:Nn \my_enum:nnn + { + %\seq_clear:N \l__my_enum_seq + %\clist_map_variable:nNn { #1 } \l__my_enum_item_tl + % { + % \__my_enum_add:V \l__my_enum_item_tl + % } + \seq_set_split:Nnn \l__my_enum_seq { #1 } { #2 } + \seq_remove_all:Nn \l__my_enum_seq {} + \int_set_eq:NN \l__number_of_args { \seq_count:N \l__my_enum_seq } + %triangle + %\int_compare:nF { \seq_count:N \l__my_enum_seq < 2 } { s } + %\nobreakspace + \seq_use:Nnnn \l__my_enum_seq { ~and~ } { ,~ } { ,~and~ } + \IfNoValueTF{#3}{}{ + \space + \int_compare:nNnTF{ \l__number_of_args } < {2}{ \prop_item:Nn \l__verbs {#3} }{ #3 } + } + } + +\ExplSyntaxOff + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic cf-lists + +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\seq_new:N \g_cflist_loaded + +\seq_new:N \g_cflist_pending + +\NewDocumentCommand{\cfadd}{ m } + +{ + + \seq_if_in:NnF \g_cflist_loaded { #1 } { + + \seq_if_in:NnF \g_cflist_pending { #1 } { + + \seq_gput_right:Nn \g_cflist_pending { #1 } + + } + + } + +} + +\NewDocumentCommand{\cfload}{ o } + +{ + + \seq_if_empty:NTF \g_cflist_pending {\unskip} { + + (cf.\ \cref{\seq_use:Nn \g_cflist_pending {,}})\IfValueTF{#1}{#1~}{\unskip} + + \seq_gconcat:NNN \g_cflist_loaded \g_cflist_loaded \g_cflist_pending + + \seq_gclear:N \g_cflist_pending + + } + +} + +\NewDocumentCommand{\cfclear} {} { + + \seq_gclear:N \g_cflist_loaded + + \seq_gclear:N \g_cflist_pending + +} + +\NewDocumentCommand{\cfout}{ o } + +{ + + \seq_if_empty:NTF \g_cflist_pending {\unskip} { + + (cf.\ \cref{\seq_use:Nn \g_cflist_pending {,}})\IfValueTF{#1}{#1~}{\unskip} + + \seq_gclear:N \g_cflist_pending + + } + +} + +\NewDocumentCommand{\ifnocf} { m } { + + \seq_if_empty:NT \g_cflist_pending { #1 } + +} + +\NewDocumentCommand{\cfconsiderloaded}{ m }{ + + \seq_gput_right:Nn \g_cflist_loaded {#1} + +} + +\ExplSyntaxOff + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic proof headers and endings, local labels +%%%%%%%%%%%%%%%%%%%%%%%% + +\NewDocumentEnvironment {athm} {m m} {% +%\newenvironment{athm}[2]{% +\begin{#1}\label{#2}\global\def\loc{#2}% +}{% +\end{#1}% +} + +\NewDocumentEnvironment{aproof} {} {% +\begin{proof}[Proof~of~\cref{\loc}]% +}{% +\finishproofthus +\end{proof}% +} + +\ExplSyntaxOff + +\newcommand{\eqqref}[1]{\cref{eq:\loc.#1}} +\newcommand{\eqlabel}[1]{\label{eq:\loc.#1}} + +\newcommand{\finishproofthus}{The proof of \cref{\loc} is thus complete.} +\newcommand{\finishproofthis}{This completes the proof of \cref{\loc}.} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Flexible math displays +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentEnvironment{flexmath}{ m o }{ + \str_if_eq:noTF {a} {#1} { + \begin{equation} + \IfValueT{#2}{\label{eq:\loc.#2}} + \begin{aligned} + } { + \catcode`&=9 + \renewcommand{\\}{} + \str_if_eq:noTF {d} {#1} { + \begin{equation} + \IfValueT{#2}{\label{eq:\loc.#2}} + } { + \begin{math} + } + } +}{ + \str_if_eq:noTF {i} {#1} { + \end{math} + \catcode`&=4 + } { + \str_if_eq:noTF {d} {#1} { + \end{equation} + } { + \end{aligned} + \end{equation} + } + } +} + +\ExplSyntaxOff + +\ExplSyntaxOn + +\bool_new:N \g_noteobserve + +\NewDocumentCommand{\setnote}{}{ + \bool_gset_true:N \g_noteobserve +} + +\NewDocumentCommand{\setobserve}{}{ + \bool_gset_false:N \g_noteobserve +} + +\NewDocumentCommand{\nobs}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \str_if_eq:noTF {Note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \bool_gset_false:N \g_noteobserve + } + } + } + \bool_if:nTF { \g_noteobserve } { + \bool_gset_false:N \g_noteobserve + note + } { + \bool_gset_true:N \g_noteobserve + observe + } + \IfValueF{#1}{~} +} + +\NewDocumentCommand{\Nobs}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \str_if_eq:noTF {Note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \bool_gset_false:N \g_noteobserve + } + } + } + \bool_if:nTF { \g_noteobserve } { + \bool_gset_false:N \g_noteobserve + Note + } { + \bool_gset_true:N \g_noteobserve + Observe + } + \IfValueF{#1}{~} +} + + +\bool_new:N \g_hencetherefore + +\NewDocumentCommand{\hence}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \str_if_eq:noTF {Hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \bool_gset_false:N \g_hencetherefore + } + } + } + \bool_if:nTF { \g_hencetherefore } { + \bool_gset_false:N \g_hencetherefore + hence + } { + \bool_gset_true:N \g_hencetherefore + therefore + } + \IfValueF{#1}{~} +} + +\NewDocumentCommand{\Hence}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \str_if_eq:noTF {Hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \bool_gset_false:N \g_hencetherefore + } + } + } + \bool_if:nTF { \g_hencetherefore } { + \bool_gset_false:N \g_hencetherefore + Hence + } { + \bool_gset_true:N \g_hencetherefore + Therefore + } + \IfValueF{#1}{~} +} + +\int_new:N \g_furthermore + +\NewDocumentCommand{\Moreover}{ o o }{ + \IfValueT{#1}{ + \str_case:nn {#1} { + {Next} {\int_gset:Nn {\g_furthermore} {0}} + {Furthermore} {\int_gset:Nn {\g_furthermore} {1}} + {Moreover} {\int_gset:Nn {\g_furthermore} {2}} + {In~addition} {\int_gset:Nn {\g_furthermore} {3}} + {note} {\bool_gset_true:N \g_noteobserve} + {observe} {\bool_gset_false:N \g_noteobserve} + } + \IfValueT{#2}{ + \str_case:nn {#2} { + {Next} {\int_gset:Nn {\g_furthermore} {0}} + {Furthermore} {\int_gset:Nn {\g_furthermore} {1}} + {Moreover} {\int_gset:Nn {\g_furthermore} {2}} + {In~addition} {\int_gset:Nn {\g_furthermore} {3}} + {note} {\bool_gset_true:N \g_noteobserve} + {observe} {\bool_gset_false:N \g_noteobserve} + } + } + } + \int_case:nn { \int_mod:nn {\g_furthermore} {4} } { + { 0 } { Next,~\nobs that} + { 1 } { Furthermore,~\nobs that} + { 2 } { Moreover,~\nobs that} + { 3 } { In~addition,~\nobs that} + } + \int_gincr:N \g_furthermore + \IfValueF{#1}{~} +} + +\ExplSyntaxOff + + + +\ExplSyntaxOn + +\int_new:N \g_enumit + +\NewDocumentCommand{\Enum}{ m o }{ + \IfValueT{#2}{ + \str_case:nn {#2} { + {ensure} {\int_gset:Nn {\g_enumit} {0}} + {ensures} {\int_gset:Nn {\g_enumit} {0}} + {assure} {\int_gset:Nn {\g_enumit} {1}} + {assures} {\int_gset:Nn {\g_enumit} {1}} + {prove} {\int_gset:Nn {\g_enumit} {2}} + {proves} {\int_gset:Nn {\g_enumit} {2}} + {show} {\int_gset:Nn {\g_enumit} {3}} + {shows} {\int_gset:Nn {\g_enumit} {3}} + {demonstrate} {\int_gset:Nn {\g_enumit} {4}} + {demonstrates} {\int_gset:Nn {\g_enumit} {4}} + {yield} {\int_gset:Nn {\g_enumit} {5}} + {yields} {\int_gset:Nn {\g_enumit} {5}} + {establish} {\int_gset:Nn {\g_enumit} {6}} + {establishes} {\int_gset:Nn {\g_enumit} {6}} + {imply} {\int_gset:Nn {\g_enumit} {7}} + {implies} {\int_gset:Nn {\g_enumit} {7}} + } + } + \int_case:nn { \int_mod:nn {\g_enumit} {8} } { + { 0 } { \enum{ #1 }[ensure]{ } } + { 1 } { \enum{ #1 }[assure] } + { 2 } { \enum{ #1 }[prove] } + { 3 } { \enum{ #1 }[show] } + { 4 } { \enum{ #1 }[demonstrate] } + { 5 } { \enum{ #1 }[yield] } + { 6 } { \enum{ #1 }[establish] } + { 7 } { \enum{ #1 }[imply] } + } + \int_gincr:N \g_enumit +} + +\ExplSyntaxOff \ No newline at end of file diff --git a/MLP and DNN Material/.DS_Store b/MLP and DNN Material/.DS_Store new file mode 100644 index 0000000..2666aff Binary files /dev/null and b/MLP and DNN Material/.DS_Store differ diff --git a/MLP and DNN Material/1-s2.0-S0885064X23000158-main.pdf b/MLP and DNN 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UU8UmXTSj]۱~힧?Q>WV^>&;1E.ɠT)R˙\eM~z{*ʰV^4mfrrϽmo8k;\ᴝzLs?":wci w[G,Mãа=2^m{ ,wɯʭ57MeS;kq S-==š}nshܜfoR7k7KC%ֶ)/fC#plK2@H{>uNSn`ecڎ;P=k$xx 1#{ҒI$?QGnw%&`{߫naRo7oTR@_Rݸ`,9%Iq?oYgwookWȱ4nS4http://ns.adobe.com/xap/1.0/ + + + + 2009-04-30T15:07:01-04:00 + 2009-04-30T15:07:01-04:00 + 2009-04-30T15:07:01-04:00 + Adobe Photoshop CS2 Macintosh + + + image/jpeg + + + untitled + + + + + Acrobat Distiller 8.1.0 (Macintosh) + + + uuid:E6B4002F372A11DE9709AD1BA492BA6A + uuid:E6B40030372A11DE9709AD1BA492BA6A + + uuid:5E3CDB4290BBDC11AEE3DA41F003DE35 + uuid:9920242c-2415-b846-8651-acbb855acc23 + + + + 1 + 3000000/10000 + 3000000/10000 + 2 + 256,257,258,259,262,274,277,284,530,531,282,283,296,301,318,319,529,532,306,270,271,272,305,315,33432;22E51E81479ACDC7538AD14F6A387570 + 1029 + 1624 + + + 8 + 8 + 8 + + + 5 + 2 + 3 + 1 + + + 1029 + 1624 + 1 + 36864,40960,40961,37121,37122,40962,40963,37510,40964,36867,36868,33434,33437,34850,34852,34855,34856,37377,37378,37379,37380,37381,37382,37383,37384,37385,37386,37396,41483,41484,41486,41487,41488,41492,41493,41495,41728,41729,41730,41985,41986,41987,41988,41989,41990,41991,41992,41993,41994,41995,41996,42016,0,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,22,23,24,25,26,27,28,30;49CFD7203C144815D485A71D9F4B3CED + + + 3 + sRGB IEC61966-2.1 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +CCjC  + 3!X 1"#Q$2Aa +; !1"AQTVaq#2$B ?=r)JR)JR)JR)JR)JR)JR)JV~؞fiϓx@pUq(=v>:N2~e7}w~K_S-@\ 39wm>Kmm0Aq`$VW5ǣU{F, lF8P3:yɳ!};l[~ X6>bySxogǔ q5aS]ſj2m rZ>#t 6Rtew/+pdTerx qR-4qkrRb|g W3 +-Ӎg>|}5cM'+9U,4T ES P#mkarѬFRv։y/AXkw?wcy+$ٵ•)Ym5*+d`jFVCq[,.M[Q =j^VY^t+~Ӄǿǜs֙ޛ%c4tCi 6<#r|X3AUC .U/z=,r;N뎞kyz~"H!;{5bR)JR͈OT:DZ5 ɨqrqvշ@!D7S. wbےB,%m{>6N{`s{T31w{,s~ú`E3",fkZfl-xIۮc64_ruZ)4)+hoLqNpcܑ_uQOvZn-MQyeu_dfdi0ڞc\3V%;ۼUN{p\6+=3 ^&A%!ۡkpU@ۂg"MDwŖOPخ=݁xw|#MGge N%D{R Nu{.U%kXyC,`#n[>^*ɖ#!aVidHwLI*2"SQ'T%LmOgyrR9%A9@㏀H@]?i[r]G<^lrjQb&mBn)}0|)2 )KS)c>/3 fBB4diWvZ2$ +tH2\1Yo{[,j ZlufҒ60omhvel{'rv@/YIK#z^*%Ӧ>%")NC>d)9p2bHJv7Ouk4Ŵ7(F*Wp F{*V/JRN4.C7(19Jctێ9& ~ ~&#R,.P Ԓ~^:_.4<Lx)BYyR҃qOf#vڋ(VkSzS4dN;1v yۿogbK꾱=7uPMs;*<$``W&؉Vr#)% +:іw9ܹǰߢҳiϧ`{:-$vJktSWAGOÜ+t`?'Z|VKɕ \\LpLHYbP[+4{=)!ݑ`nؙ<mo:*ސ&(lr]խ˪0`H$9A]KZM%C+ȗ2ZDe۴Z6;yNTxML虎7gf0FqjF` c!#&&)FZh[8M8~ r \w$^C.&ZmAt )rosu=Â$4rnRdze[^SyTי~T8I WśtȫM2sPKCns0$vFܟ}}f[:=iրBPNqb'*TVNآw)nm:!q"/9e?THΪLi0d IS SH;wzXwP ֢;iE_Q X?km0dZF8e숒1vQ1tGa@5=KSV%m"$FrWǒ~i[Dۣnl9+2><`r[| Pb(KH6w|%:B^)tt7!A BJ`ءԵfQJ.1>.ǸԓHӮ7]%n ,w-|G1XX9;'d U(*'29XSEM^qL77 p,$dcn '(XLA!2_$2C>9'd7E2Fl[|[¶cEⒾ 4yzMO~ o'o1t-HtkB0N̑x{k=w]wIXi.agv&! O:> u'0 Fgw-$ziiScc TI*ث *=TR)JUB*eP8P{rBF*T#= H^G5Z&;镓٤I=S*DT-HA%ԍ?0r/yO!7$2zcr0e:K3IHFM "(*Qgx]#'<]tީ]5&!XJ X@JsTVJR)JR)JR)JR)JR*LE&Dz +@ɏ(oۿҘ? +͏/p{s5 A)L y P{n`*0 oǎk #Ǭs!T'U3̇)NC" QJY[;X68#~J)JR)JWBω^d!aaFԶffGΧ`G9z +4ITnlSrj"FcÊ㺟;ƨm&98b.C{*"BbI#~ Wn/ÌdrLEZXes#v)XŜ:dK?+ovϿ1޻Ng9?e 4-4K.zI5L#@moD"i@#6sOb~#9*:*ԇЃ ҕQhn"AW>L'TX$q)4ʊ.- 1#l sX95 R]FW=qi&v +!ѷ8-u#5Ж y&z2QnȬU[$ +D)DtDN5bk6cTIF;|}n{Ƿ/OZ,|!3bIU8Պ)JR)JR)JR)JR)JR)JR)JR)JR)JR)JR)JRMJz]ۀckzKOG_m%ymP:鶯SҞ}?zKOG^}6jzS{~;OIihT@ϦڿWOJz]ۀci->m_ʟ(yWJOKpy=%ͣS>j^~=)wn=?q*|um_+֧=.mg6/Om%y +endstream +endobj +469 0 obj<>endobj +470 0 obj<>stream +JFIF,, ExifMM*X(12i,-'-'Adobe Photoshop CS2 Macintosh2009:04:30 15:07:01Xz( HHJFIFHH Adobe_CMAdobed    +         e"? +  +  3!1AQa"q2B#$Rb34rC%Scs5&DTdE£t6UeuF'Vfv7GWgw5!1AQaq"2B#R3$brCScs4%&5DTdEU6teuFVfv'7GWgw ?RI% )$IJI$RI$I%)$IJI$SRI]'0^44Zi{]o( kӓ] i$lh"z;:}vdiٗу9dhG4#|#oA$Z ?EH}6׍yO\J>q;|B}Q$HDAJRI$I$RI% )$IN:/O6cØ=+wncҺCǪʬ n&j7_snEu?.s~{ُkپ^c9^7z2VӒLEu"<AzFRF5/˳}}QQFn#5?G.(]#]Y(fFMlv#ּ/VuY>.ϳOn;}?CWޝv%Y ~M^P=k=7who]un5YNs݅e>wXmhkсk>w%uQ/-ȤQ{g_۟s>Tn}/eZUoBe6?&:Vo*w?[Z[X|11W %uX9ӑeWP]/KŻs2q)$I{RI% ) %$Jtb;3&[]2.8ݺ}ckhs=h!c6Ic&fSqߏ}VQCu +zw:z@s=.;s=%b:eYunsa3~^5XβK*uvͻ/w>Q[(q"L

UU8UmXTSj]۱~힧?Q>WV^>&;1E.ɠT)R˙\eM~z{*ʰV^4mfrrϽmo8k;\ᴝzLs?":wci w[G,Mãа=2^m{ ,wɯʭ57MeS;kq S-==š}nshܜfoR7k7KC%ֶ)/fC#plK2@H{>uNSn`ecڎ;P=k$xx 1#{ҒI$?QGnw%&`{߫naRo7oTR@_Rݸ`,9%Iq?oYgwookWȱ4nSELRY`gnu| &/8AKT]gqz !-8COZfr~ -;HUcq~ +:IXgw'7HYj{+=Oat 2FZn  % : O d y + +' += +T +j + + + + + + " 9 Q i  * C \ u & @ Z t .Id %A^z &Ca~1Om&Ed#Cc'Ij4Vx&IlAe@e Ek*Qw;c*R{Gp@j>i  A l !!H!u!!!"'"U"""# +#8#f###$$M$|$$% %8%h%%%&'&W&&&''I'z''( (?(q(())8)k))**5*h**++6+i++,,9,n,,- -A-v--..L.../$/Z///050l0011J1112*2c223 3F3334+4e4455M555676r667$7`7788P8899B999:6:t::;-;k;;<' >`>>?!?a??@#@d@@A)AjAAB0BrBBC:C}CDDGDDEEUEEF"FgFFG5G{GHHKHHIIcIIJ7J}JK KSKKL*LrLMMJMMN%NnNOOIOOP'PqPQQPQQR1R|RSS_SSTBTTU(UuUVV\VVWDWWX/X}XYYiYZZVZZ[E[[\5\\]']x]^^l^__a_``W``aOaabIbbcCccd@dde=eef=ffg=ggh?hhiCiijHjjkOkklWlmm`mnnknooxop+ppq:qqrKrss]sttptu(uuv>vvwVwxxnxy*yyzFz{{c{|!||}A}~~b~#G +k͂0WGrׇ;iΉ3dʋ0cʍ1fΏ6n֑?zM _ɖ4 +uL$h՛BdҞ@iءG&vVǥ8nRĩ7u\ЭD-u`ֲK³8%yhYѹJº;.! +zpg_XQKFAǿ=ȼ:ɹ8ʷ6˶5̵5͵6ζ7ϸ9к<Ѿ?DINU\dlvۀ܊ݖޢ)߯6DScs 2F[p(@Xr4Pm8Ww)Km0Photoshop 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spacing caused by using "\left" or "\right" +\mleftright + +% Sets the margins of the document. +% One can do this using plain TeX, but why? +\geometry{margin=1in} + +% Lines 43-46 set up my use of the cleveref package. +% This provides a "smarter" cross referencing style +\usepackage[colorlinks=true]{hyperref} +\usepackage[nosort,capitalise]{cleveref} +\newcommand{\creflastconjunction}{, and\nobreakspace} +\crefname{enumi}{item}{items} + +% My personal command macros that I use. +% Note that "\displaySpace" is use to create the correct spacing for displayed math puctuation (and can be easily updated throughout the entire document by changing line 50). +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\renewcommand{\i}{\boldsymbol{i}} +\renewcommand{\j}{\boldsymbol{j}} +\renewcommand{\k}{\boldsymbol{k}} +\renewcommand{\u}{\boldsymbol{u}} +\renewcommand{\v}{\boldsymbol{v}} +\newcommand{\w}{\boldsymbol{w}} +\newcommand{\pq}{\overrightarrow{PQ}} +\newcommand{\dotp}{\boldsymbol{\cdot}} +\newcommand{\displaySpace}{\,} +\newcommand{\F}{\boldsymbol{F}} + +% Creates my paired math delimiters for easy use. +\DeclarePairedDelimiter{\pr}{(}{)} +\DeclarePairedDelimiter{\br}{[}{]} +\DeclarePairedDelimiter{\cu}{\{}{\}} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\DeclarePairedDelimiter{\vt}{\langle}{\rangle} + +% Creates desired header and footer for document. +% Note that it uses the adaptively created date macro. +\setlength{\headheight}{15pt} +\usepackage{fancyhdr} +\pagestyle{fancy} +\fancyhf{} +\fancyhead[R]{} +\cfoot{} +\lfoot{\small{\textsc{Dissemination prohibited. \today}}} +\rfoot{\thepage} + +% Updates equation and figure numbering. +\counterwithin{equation}{section} +\counterwithin{figure}{section} + +% Defines theorem-like environments +\theoremstyle{definition} +\newtheorem{definition}{Definition}[section] +\newtheorem{theorem}[definition]{Theorem} +\newtheorem{problem}[definition]{Problem} +\newtheorem{remark}[definition]{Remark} +\renewcommand{\thetheorem}{\theenumi\arabic{theorem}} + +% Helps create better arrows in figures +\usetikzlibrary{arrows.meta} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic proof headers and endings, local labels +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentEnvironment {athm} {m m} {% +\phantomsection +\addcontentsline{toc}{subsection}{\texorpdfstring{\cref{#2}}{\ref*{#2}}} +\begin{#1}\label{#2}\global\def\loc{#2}% +}{% +\end{#1}% +} + +\NewDocumentEnvironment{aproof} {} {% +\begin{proof}[Proof~of~\cref{\loc}]% +}{% +\finishproofthus +\end{proof}% +} + +\ExplSyntaxOff + +\newcommand{\eqqref}[1]{\cref{eq:\loc.#1}} +\newcommand{\eqlabel}[1]{\label{eq:\loc.#1}} + +\newcommand{\finishproofthus}{The solution to \cref{\loc} is thus complete.} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Creating blank space with line breaks +%%%%%%%%%%%%%%%%%%%%%%%% + +% Note that the code in lines 137-147 is simply a method to +% create blank space that allows for proof environments to +% break across pages. This is something that TeX does not +% like by default, so this is my effort to bypass this. + +\newcommand{\linebreaks}[1]{{\mbox{}\par + \newcount\foo + \foo=#1 + \loop + \phantom{a} + + \advance \foo -1 + \ifnum \foo>0 + \repeat + \mbox{}\par} +\noindent} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Beginning of actual text. %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\title{A survey of direct methods for sparse linear systems} + +\author{Joshua Lee Padgett} + +\date{\today} + +\begin{document} + +\maketitle + +\begin{abstract} +The following problems are meant to be additional practice which may help students understand some of the more difficult material in \emph{A survey of direct methods for sparse linear systems} by Davis et al. +\end{abstract} + +\tableofcontents + +\section{Solving sparse triangular systems} + +We begin by introducing a few definitions from graph theory. +These definitions and concepts are in no way complete (in fact, there are much better ways of formulating these ideas if we have more background); however, they will suffice for our current needs. +Students who are interested in obtaining a deeper understanding of the material in Davis et al.\ should consult a textbook on introductory graph theory. + +\begin{athm}{definition}{def:graph} +Let $n \in \N = \{1,2,3,\ldots\}$. A \emph{graph} $G = (V,E)$ is a set of \emph{nodes} $V = \{v_1,v_2,\allowbreak \ldots,\allowbreak v_n\}$ and a set of \emph{edges} $E = \{(v_i,v_j) \colon v_i,v_j \in V \}$ connecting those nodes. +\end{athm} + +\begin{athm}{definition}{def:directed} +Let $n \in \N = \{1,2,3,\ldots\}$, let $V = \{v_1,v_2,\ldots,v_n\}$ be a set of vertices, let $E = \{(v_i,v_j) \colon v_i,v_j \in V \}$ be a set of edges for $V$, and let $G = (V,E)$ be a graph (cf.\ \cref{def:graph}). +We say that $G$ is an \emph{undirected} graph if for all $i,j \in \{1,2,\ldots,n\}$ it holds that $(v_i,v_j) = (v_j,v_i)$. +We say that $G$ is a directed graph, otherwise. +\end{athm} + +\begin{athm}{definition}{def:path} +Let $n \in \N = \{1,2,3,\ldots\}$, let $V = \{v_1,v_2,\ldots,v_n\}$ be a set of vertices, let $E = \{(v_i,v_j) \colon v_i,v_j \in V \}$ be a set of edges for $V$, and let $G = (V,E)$ be a graph (cf.\ \cref{def:graph}). +For every $i,j \in \{1,2,\ldots,n\}$ with $i \neq j$ a \emph{path} $v_i \rightsquigarrow v_j$ is a sequence of nodes $(v_i,v_{i+1},\ldots,v_j)$ which satisfies for all $k \in \{i,i+1,\ldots,j-1\}$ that $(v_k,v_{k+1}) \in E$. +Moreover, the \emph{length} of the path is $i-j+1$ (i.e., the number of vertices in the sequence). +\end{athm} + +Note that \cref{def:path} is not completely rigorous because our definition is enforcing that ``$v_i$ comes before $v_j$ in the path.'' This is not a requirement --- we have simply presented \cref{def:path} in this way for convenience. We can circumvent this issue by introducing sub-indices to allow for arbitrary ordering. However, this should not cause any confusion for our purposes (and interested students should attempt this generalization themselves). +Moreover, observe +that \cref{def:path} essentially states that a path is a sequence of vertices in a graph where one can ``trace'' the sequence by following consecutive edges. + +\begin{athm}{definition}{def:cycle} +Let $n \in \N = \{1,2,3,\ldots\}$, let $V = \{v_1,v_2,\ldots,v_n\}$ be a set of vertices, let $E = \{(v_i,v_j) \colon v_i,v_j \in V \}$ be a set of edges for $V$, and let $G = (V,E)$ be a graph (cf.\ \cref{def:graph}). +For every $i \in \{1,2,\ldots,n\}$ a \emph{cycle} is a path $v_i \rightsquigarrow v_i$ where the first and last vertex are the same (cf.\ \cref{def:path}). +If $G$ has no cycles, then $G$ is said to be \emph{acyclic}. +\end{athm} + +We first provide a worked example of a situation related to Section 3 of Davis et al. +Note that the situation in \cref{prob:0} is much simpler than that depicted in Figure 3.2 of Davis et al.\ (to provide more clarity). + +\begin{athm}{problem}{prob:0} +Let $A \in \R^{5 \times 5}$ satisfy +\begin{equation}\label{eq:01} +A = +\begin{pmatrix} +1 & & & & & \\ +& 1 & & & & \\ +& -2 & 1 & & \\ +& 4 & 7 & 1 & \\ +& & 3 & 9 & 1 \\ +\end{pmatrix} +\end{equation} +(where empty spaces in $A$ are assumed to represent entries of zero) +and let $b \in \R^{5 \times 1}$ satisfy +\begin{equation}\label{eq:02} +b = \begin{pmatrix} +0 & 4 & 0 & 1 & 0 +\end{pmatrix}^T \, . +\end{equation} +\begin{enumerate}[label=\alph*.] +\item\label{0_a} Does $A$ seem to be \emph{sparse}? Does $b$ seem to be \emph{sparse}? +\item\label{0_b} What is the compressed-column form of $A$ (cf.\ page 8 of Davis et al.)? +\item\label{0_c} Let $V = \{1,2,3,4,5\}$, let $E = \{(j,i) \colon A_{ij} \neq 0,\ i,j \in V \}$ (cf.\ \cref{eq:1}), and let $G_A$ be the directed acyclic graph which satisfies $G_A = (V,E)$ (cf.\ \cref{def:graph,def:directed,def:cycle}). +Draw the graph $G_A$ --- be sure to label all nodes appropriately and demonstrate the correct directionality of the edges. +\item\label{0_d} Let $\mathcal{B} \subseteq \{1,2,3,4,5\}$ be the set which satisfies +\begin{equation}\label{eq:03} +\mathcal{B} = \cu[\big]{ i \in \{1,2,3,4,5\} \colon b_i \neq 0 } +\end{equation} +(cf.\ \cref{eq:2}), let $\operatorname{Reach}_A \colon \{1,2,3,4,5\} \to \{1,2,3,4,5\}$ be the function which satisfies for all $i \in \{1,2,3,4,5\}$ that +\begin{equation}\label{eq:04} +\operatorname{Reach}_A(i) = \cu[\big]{ j \in \{1,2,3,4,5\} \colon \exists \, i \rightsquigarrow j,\ i,j \in G_A } +\end{equation} +(cf.\ \cref{def:path,1_c}), +and let $\mathcal{X} \subseteq \{1,2,3,4,5\}$ satisfy +\begin{equation}\label{eq:05} +\mathcal{X} = \cup_{i \in \mathcal{B}} \operatorname{Reach}_A(i) \, . +\end{equation} +Compute $\mathcal{X}$ using a ``depth-first'' ordering to determine the topology of the set (cf.\ page 11 of Davis et al.). +\end{enumerate} +\end{athm} + +\begin{aproof} +We solve each part of \cref{prob:0} separately. + +\begin{enumerate}[label=\alph*.] +\item Without ``muddying the water'' with unnecessary details, it should be clear that the objects in \cref{eq:01,eq:02} are indeed sparse. +The solution to \cref{0_a}\ is thus complete. + +\item The compressed-column form of $A$ is given by: +\begin{verbatim} +int p[] = { 0, 1, 4, 7, 10 } ; +int i[] = { 0, 1, 2, 3, 2, 3, 4, 3, 4, 4 } ; +double A[] = { 1, 1, -2, 4, 1, 7, 3, 1, 9, 1 } ; +\end{verbatim} +(again, see page 6 of Davis et al.). +\begin{quote} +\textbf{Aside:} Note that the above is written with a \texttt{C}-style formatting --- can you update the style to reflect the syntax of a different programming language? +\end{quote} +The solution to \cref{0_b}\ is thus complete. + +\item A possible depiction of the graph $G_A$ is given in \cref{fig:1} below. + +\begin{figure}[H] +\centering + +\begin{tikzpicture}[scale=2,vertex/.style={draw,circle}, arc/.style={draw,thick,->}] + \node[vertex] (2) at (0,0) {2}; + \node[vertex] (3) at (1,-0.5) {3}; + \node[vertex] (4) at (-1,-0.5) {4}; + \node[vertex] (5) at (1,-1.4) {5}; + \draw[-{Latex[width=3mm]}] (2) -- (3) ; + \draw[-{Latex[width=3mm]}] (2) -- (4) ; + \draw[-{Latex[width=3mm]}] (3) -- (4) ; + \draw[-{Latex[width=3mm]}] (3) -- (5) ; + \draw[-{Latex[width=3mm]}] (4) -- (5) ; +\end{tikzpicture} + +\captionsetup{format=hang,labelfont=bf,width=0.75\linewidth} +\caption{A sample depiction of the directed graph $G_A$.}\label{fig:1} +\end{figure} + +Note that we say that \cref{fig:1} is a \emph{possible} depiction of $G_A$ because the configuration of the vertices in a drawing of $G_A$ is arbitrary (i.e., not specified by \cref{def:graph}). +The solution to \cref{0_c}\ is thus complete. + +\item First, note that $\mathcal{B}$ is the set of indices associated to nonzero values in $b$ (cf.\ \cref{eq:01,eq:03}). +Thus, it holds that +\begin{equation} +\mathcal{B} = \{2,4\} +\end{equation} +(where here we use the actual position, rather than the, e.g., associated position in a \texttt{C}-style array). +Next, observe that \cref{eq:04} is the set of all $j \in \{1,2,3,4,5\}$ \emph{reachable} from $i \in \{1,2,3,4,5\}$ with $i \neq j$. +Moreover, \cref{eq:05} indicates that +\begin{equation}\label{eq:05a} +\mathcal{X} = \operatorname{Reach}_A(2) \cup \operatorname{Reach}_A(4) \, , +\end{equation} +where the topology of the set is determined via a depth-first search. +To clarify this issue, we have marked the elements of $\mathcal{B}$ in \cref{fig:2} below. + +\begin{figure}[H] +\centering + +\begin{tikzpicture}[scale=2,vertex/.style={draw,circle}, arc/.style={draw,thick,->}] + \node[vertex,red] (2) at (0,0) {2}; + \node[vertex] (3) at (1,-0.5) {3}; + \node[vertex,red] (4) at (-1,-0.5) {4}; + \node[vertex] (5) at (1,-1.4) {5}; + \draw[-{Latex[width=3mm]}] (2) -- (3) ; + \draw[-{Latex[width=3mm]}] (2) -- (4) ; + \draw[-{Latex[width=3mm]}] (3) -- (4) ; + \draw[-{Latex[width=3mm]}] (3) -- (5) ; + \draw[-{Latex[width=3mm]}] (4) -- (5) ; +\end{tikzpicture} + +\captionsetup{format=hang,labelfont=bf,width=0.75\linewidth} +\caption{A sample depiction of the directed graph $G_A$. +The elements of the set $\mathcal{B}$ are indicated by red nodes.}\label{fig:2} +\end{figure} + +In addition, note that \cref{fig:2,eq:04} ensure that +\begin{equation}\label{eq:06} +\operatorname{Reach}_A(2) = \{2,3,5\} \, . +\end{equation} +Moreover, observe that \cref{fig:2,eq:04} assure that +\begin{equation}\label{eq:07} +\operatorname{Reach}_A(4) = \{4,5\} \, . +\end{equation} +Combining \cref{eq:05a,eq:06,eq:07} with the depth-first search topology (i.e., that we order the ``lists'' by traversing the set $\mathcal{B}$ in reverse and we omit repeated indices) yields +\begin{equation} +\mathcal{X} = \{4,5,2,3\} \, . +\end{equation} +\begin{quote} +\textbf{Aside:} It is important to note that there is some confusion from Figure 3.2 of Davis et al. +In Davis et al.\ the graph is more complicated, hence, there are parts of the graph ``$G_L$'' which have no affect on $\mathcal{X}$ --- which is perfectly fine! +This is important to notice because it helps one understand that there is a difference between the graph of the matrix and the set $\mathcal{X}$. +\end{quote} +The solution to \cref{0_d}\ is thus complete. +\end{enumerate} +\end{aproof} + +\addcontentsline{toc}{subsection}{\texorpdfstring{\cref{rem1}}{Remark 1.6}} +\begin{remark}\label{rem1} +It is worth noting that in \cref{0_c}\ of \cref{prob:0} we could have potentially included edges of the form, e.g., $(1,1)$ since there are nonzero entries along the main diagonal of $A$. In some branches of graph theory this is done (and is represented by a ``looping'' edge). However, in numerical linear algebra, this edge is almost always omitted as it does not aid in the symbolic study of the underlying algorithms (think about why this is the case). +\end{remark} + +Now, attempt to complete \cref{prob:1} yourself. +The problem is fairly similar to \cref{prob:0} with an additional programming issue presented in \cref{1_e}. + +\begin{athm}{problem}{prob:1} +Let $A \in \R^{8 \times 8}$ satisfy +\begin{equation}\label{eq:1} +A = +\begin{pmatrix} +1 & & & & & & & & \\ +& 1 & & & & & & & \\ +& 3 & 1 & & & & & \\ +& 1 & 2 & 1 & & & & \\ +& & 2 & 3 & 1 & & & \\ +& & 1 & 2 & 3 & 1 & & \\ +& & & & 4 & & 1 & \\ +& & & 1 & & & & 1 +\end{pmatrix} +\end{equation} +(where empty spaces in $A$ are assumed to represent entries of zero) +and let $b \in \R^{8 \times 1}$ satisfy +\begin{equation}\label{eq:2} +b = \begin{pmatrix} +0 & 0 & 1 & 3 & 4 & 0 & 0 & 9 +\end{pmatrix}^T \, . +\end{equation} +\begin{enumerate}[label=\alph*.] +\item Does $A$ seem to be \emph{sparse}? Does $b$ seem to be \emph{sparse}? + +\item What is the compressed-column form of $A$ (cf.\ page 8 of Davis et al.)? + +\item\label{1_c} Let $V = \{1,2,\ldots,8\}$, let $E = \{(j,i) \colon A_{ij} \neq 0 \}$ (cf.\ \cref{eq:1}), and let $G_A$ be the directed acyclic graph which satisfies $G_A = (V,E)$ (cf.\ \cref{def:graph,def:directed,def:cycle}). +Draw the graph $G_A$ --- be sure to label all nodes appropriately and demonstrate the correct directionality of the edges. + +\item\label{1_d} Let $\mathcal{B} \subseteq \{1,2,\ldots,8\}$ be the set which satisfies +\begin{equation} +\mathcal{B} = \cu[\big]{ i \in \{1,2,\ldots,8\} \colon b_i \neq 0 } +\end{equation} +(cf.\ \cref{eq:2}), let $\operatorname{Reach}_A \colon \{1,2,\ldots,8\} \to \{1,2,\ldots,8\}$ be the function which satisfies for all $i \in \{1,2,\ldots,8\}$ that +\begin{equation} +\operatorname{Reach}_A(i) = \cu[\big]{ j \in \{1,2,\ldots,8\} \colon \exists \, i \rightsquigarrow j,\ i,j \in G_A } +\end{equation} +(cf.\ \cref{def:path,1_c}), +and let $\mathcal{X} \subseteq \{1,2,\ldots,8\}$ satisfy +\begin{equation} +\mathcal{X} = \cup_{i \in \mathcal{B}} \operatorname{Reach}_A(i) \, . +\end{equation} +Compute $\mathcal{X}$ using a ``depth-first'' ordering to determine the topology of the set (cf.\ page 11 of Davis et al.). + +\item\label{1_e} Write a computer program which finds the $x \in \R^{8 \times 1}$ (Do you understand that this $x$ is unique? Can you prove this?) which satisfies +\begin{equation} +Ax = b +\end{equation} +(cf.\ \cref{eq:1,eq:2}) using your work from \cref{1_d}\ and Algorithm 3.2 from Davis et al. +\end{enumerate} +\end{athm} + +\begin{aproof} +\linebreaks{20} +\end{aproof} + +\section{The Cholesky factorization method} + +Let $n \in \N$, $b \in \C^{n\times 1}$, $A \in \C^{n\times n}$ satisfy that $A$ is \emph{sparse}. +The problem of finding $x \in \C^{n\times 1}$ such that +\begin{equation}\label{2_1} +Ax = b +\end{equation} +using direct methods starts by factoring the matrix $A$. +This factorization typically is carried out by going through a symbolic phase (this phase typically depends only on the nonzero pattern of $A$) and the numerical phase (the phase which actually produces the desired factorization). +The solve phase uses standard triangular system solvers (i.e., is not the difficult part of solving \cref{2_1}). +It is worth noting that symbolic phase is much faster than the numerical phase. Moreover, a well spent symbolic phase increases the efficiency of the numerical phase. +Loosely speaking, one should think of the symbolic phase as the effort spend trying to understand the nonzero pattern of the solution vector by studying the nonzero pattern of the original matrix $A$ and the chosen factorization method. + +For simplicity of exposition, Davis et al.\ chooses to motivate the idea of symbolic analysis via the well-known Cholesky decomposition method. + +\begin{athm}{problem}{prob:2.1} +Let $A \in \R^{3\times 3}$ satisfy +\begin{equation} +A = +\begin{pmatrix} +25 & 15 & -5 \\ +15 & 18 & 0 \\ +-5 & 0 & 11 +\end{pmatrix} \, . +\end{equation} +\begin{enumerate}[label=\alph*.] +\item Does the Cholesky decomposition of $A$ exist? (One possible method to determine this is to show that $A$ is \emph{Hermitian} and \emph{positive-definite}.) +\item Compute that Cholesky decomposition of $A$. That is, find $L \in \R^{3\times 3}$ which satisfies for all $i , j \in \{1,2,3\}$ with $i < j$ that $L_{ij} = 0$ and +\begin{equation} +L L^T = A \, . +\end{equation} +\end{enumerate} +\end{athm} + +\begin{aproof} +\linebreaks{20} +\end{aproof} + +\begin{athm}{remark}{rem2} +Something mentioned in Davis et al., but not yet explored, is the fact that we can use a Cholesky factorization even if no such factorization exists for a given matrix $A$. This is done by permuting the matrix into a preferable form (cf.\ page 394 of Davis et al.). +For now, we will not worry with such issues (in fact, Davis et al.\ postpones such issues until Section 8 of the article). +\end{athm} + +We now attempt to better understand the concept of \emph{elimination trees}. +The algorithm for constructing elimination trees is provided in Section 4.1 of Davis et al. + +\begin{athm}{problem}{prob:2.2} +Let $A \in \R^{5 \times 5}$ satisfy +\begin{equation}\label{eq:2_2_1} +A = +\begin{pmatrix} +1 & & & & \\ + & 1 & -2 & 4 & \\ + & -2 & 5 & -1 & 3 \\ + & 4 & -1 & 66 & 30 \\ + & & 3 & 30 & 91 +\end{pmatrix} +\end{equation} +(where empty spaces in $A$ are assumed to represent entries of zero). +\begin{enumerate}[label=\alph*.] +\item What is the compressed-column form of $A$ (cf.\ page 8 of Davis et al.)? +\item Let $V = \{1,2,3,4,5\}$, let $E = \{(j,i) \colon A_{ij} \neq 0 \}$ (cf.\ \cref{eq:2_2_1}), and let $G_A$ be the directed acyclic graph which satisfies $G_A = (V,E)$ (cf.\ \cref{def:graph,def:directed,def:cycle}). +Draw the graph $G_A$ --- be sure to label all nodes appropriately and demonstrate the correct directionality of the edges. +\item Does the Cholesky decomposition of $A$ exist? (One possible method to determine this is to show that $A$ is \emph{Hermitian} and \emph{positive-definite}.) +\item Compute that Cholesky decomposition of $A$. That is, find $L \in \R^{5\times 5}$ which satisfies for all $i , j \in \{1,2,3,4,5\}$ with $i < j$ that $L_{ij} = 0$ and +\begin{equation} +L L^T = A \, . +\end{equation} +Be sure to mark the \emph{fill-in entries} of $L$ (or, at least, designate them in some way). +\item Construct the elimination tree of $A$. Do you understand how this aids in the computation of the Cholesky factorization? +\end{enumerate} +\end{athm} + +\begin{aproof} +\linebreaks{20} +\end{aproof} + +\begin{athm}{problem}{prob:2.3} +Let $A \in \R^{8 \times 8}$ satisfy +\begin{equation}\label{eq:2_3_1} +A = +\begin{pmatrix} +1 & & & & & & & \\ + & 1 & 3 & 1 & & & & \\ + & 3 & 10 & 5 & 2 & 1 & & \\ + & 1 & 5 & 6 & 7 & 4 & & 1 \\ + & & 2 & 7 & 14 & 11 & 4 & 3 \\ + & & 1 & 4 & 11 & 15 & 12 & 2 \\ + & & & & 4 & 12 & 17 & \\ + & & & 1 & 3 & 2 & & 2 +\end{pmatrix} +\end{equation} +(where empty spaces in $A$ are assumed to represent entries of zero). +\begin{enumerate}[label=\alph*.] +\item What is the compressed-column form of $A$ (cf.\ page 8 of Davis et al.)? +\item Let $V = \{1,2,\ldots,8\}$, let $E = \{(j,i) \colon A_{ij} \neq 0 \}$ (cf.\ \cref{eq:2_3_1}), and let $G_A$ be the directed acyclic graph which satisfies $G_A = (V,E)$ (cf.\ \cref{def:graph,def:directed,def:cycle}). +Draw the graph $G_A$ --- be sure to label all nodes appropriately and demonstrate the correct directionality of the edges. +\item Does the Cholesky decomposition of $A$ exist? (One possible method to determine this is to show that $A$ is \emph{Hermitian} and \emph{positive-definite}.) +\item Compute that Cholesky decomposition of $A$. That is, find $L \in \R^{8\times 8}$ which satisfies for all $i , j \in \{1,2,\ldots,8\}$ with $i < j$ that $L_{ij} = 0$ and +\begin{equation} +L L^T = A \, . +\end{equation} +Be sure to mark the \emph{fill-in entries} of $L$ (or, at least, designate them in some way). +\item Construct the elimination tree of $A$. Do you understand how this aids in the computation of the Cholesky factorization? +\end{enumerate} +\end{athm} + +\begin{aproof} +\linebreaks{20} +\end{aproof} + +\end{document} \ No newline at end of file diff --git a/Notes/Graph Theory Notes/Graph_Theory_Linear_Algebra.toc b/Notes/Graph Theory Notes/Graph_Theory_Linear_Algebra.toc new file mode 100644 index 0000000..3cee11b --- /dev/null +++ b/Notes/Graph Theory Notes/Graph_Theory_Linear_Algebra.toc @@ -0,0 +1,13 @@ +\contentsline {section}{\numberline {1}Solving sparse triangular systems}{1}{section.1}% +\contentsline {subsection}{\cref {def:graph}}{1}{section*.2}% +\contentsline {subsection}{\cref {def:directed}}{1}{section*.3}% +\contentsline {subsection}{\cref {def:path}}{2}{section*.4}% +\contentsline {subsection}{\cref {def:cycle}}{2}{section*.5}% +\contentsline {subsection}{\cref {prob:0}}{2}{section*.6}% +\contentsline {subsection}{\cref {rem1}}{4}{equation.1.10}% +\contentsline {subsection}{\cref {prob:1}}{5}{section*.9}% 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+\usepackage{amsmath, + amsfonts, + amssymb, + amsthm, + enumerate, + enumitem, + geometry, + mleftright, + nicefrac, + mathtools, + xparse, + ifthen, + tikz, + pgfplots, + floatrow, + multicol, + caption, + xurl, + relsize, + bbm + } + +% Set up PGF plots (used for graphing) +\pgfplotsset{compat=newest} + +% Fixes spacing caused by using "\left" or "\right" +\mleftright + +% Sets the margins of the document. +% One can do this using plain TeX, but why? +\geometry{margin=1in} + +% Lines 44-47 set up my use of the cleveref package. +% This provides a "smarter" cross referencing style +\usepackage[colorlinks=true]{hyperref} +\usepackage[sort,capitalise]{cleveref} +\newcommand{\creflastconjunction}{, and\nobreakspace} +\crefname{enumi}{item}{items} + +% My personal command macros that I use. +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\dpp}{\text{.}} +\newcommand{\dc}{\text{,}} +\newcommand{\dd}{{\rm d}} +\newcommand{\dx}{\, {\rm d}} +\newcommand{\lr}{\ensuremath{\mkern-1.5mu}} +\newcommand{\induct}{\dashrightarrow} +\newcommand{\with}{\curvearrowleft} +\newcommand{\medcup}[1]{{\raisebox{0.25ex}{$\mathsmaller{\ensuremath{\bigcup}}_{#1}$}}} + +% Creates my paired math delimiters for easy use. +\DeclarePairedDelimiter{\pr}{(}{)} +\DeclarePairedDelimiter{\br}{[}{]} +\DeclarePairedDelimiter{\cu}{\{}{\}} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\DeclarePairedDelimiter{\vt}{\langle}{\rangle} +\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} + +% Creates desired header and footer for document. +% Note that it uses the adaptively created date macro. +\setlength{\headheight}{15pt} +\usepackage{fancyhdr} +\pagestyle{fancy} +\fancyhf{} +\fancyhead[R]{} +\cfoot{} +\lfoot{\small{\textsc{Dissemination prohibited. \today}}} +\rfoot{\thepage} + +% Updates equation and figure numbering. +\counterwithin{equation}{section} +\counterwithin{figure}{section} + +% Defines theorem-like environments +\theoremstyle{definition} +\newtheorem{definition}[equation]{Definition} +\newtheorem{setting}[equation]{Setting} +\newtheorem{theorem}[equation]{Theorem} +\newtheorem{lemma}[equation]{Lemma} +\newtheorem{problem}[equation]{Problem} +\newtheorem{remark}[equation]{Remark} +\newtheorem{assumption}[equation]{Assumption} +\newtheorem{example}[equation]{Example} +\renewcommand{\thetheorem}{\theenumi\arabic{theorem}} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic proof headers and endings, local labels +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentEnvironment {athm} {m m} {% +\phantomsection +\addcontentsline{toc}{subsection}{\texorpdfstring{\cref{#2}}{\ref*{#2}}} +\begin{#1}\label{#2}\global\def\loc{#2}% +}{% +\end{#1}% +} + +\NewDocumentEnvironment{aproof} {} {% +\begin{proof}[Proof~of~\cref{\loc}]% +}{% +\finishproofthus +\end{proof}% +} + +\ExplSyntaxOff + +\newcommand{\eqqref}[1]{\cref{eq:\loc.#1}} +\newcommand{\eqlabel}[1]{\label{eq:\loc.#1}} + +\newcommand{\finishproofthus}{The proof of \cref{\loc} is thus complete.} + +\NewDocumentEnvironment{asol} {} {% +\begin{proof}[Solution~to~\cref{\loc}]% +}{% +\noindent\finishsolthus +\end{proof}% +} + +\newcommand{\finishsolthus}{The solution to \cref{\loc} is thus complete.} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Creating blank space with line breaks +%%%%%%%%%%%%%%%%%%%%%%%% + +% Note that the code in lines 145-154 is simply a method to +% create blank space that allows for proof environments to +% break across pages. This is something that TeX does not +% like by default, so this is my effort to bypass this. + +\newcommand{\linebreaks}[1]{{\mbox{}\par + \newcount\foo + \foo=#1 + \loop + \phantom{a} + + \advance \foo -1 + \ifnum \foo>0 + \repeat + \mbox{}\par}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Beginning of actual text. %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% +% +% + +\title{Some questions related to Iserles' textbook} + +% +% +% + +\author{Joshua Lee Padgett} + +% +% +% + +\date{\today} + +% +% +% + +\begin{document} + +% +% +% + +\maketitle + +% +% +% + +%\begin{abstract} +% +%\end{abstract} + +% +% +% + +\tableofcontents + +% +% +% + +\section{Euler's method and beyond} + +% +% +% + +The following questions are meant to help ensure you have a solid \emph{conceptual} understanding of the material from Chapter 1 of Iserles' textbook. +% +% + +% +% +% + +\bigskip + +% +% +% + +\begin{athm}{setting}{setting1} +% +Let $T \in (0,\infty)$, $d \in \N = \{1,2,3,\dots\}$, +let $\norm{\cdot} \colon \R^d \to [0,\infty)$ be a function which satisfies for all $u,v \in \R^d$, $s \in \R$ that $\norm{u+v} \le \norm{u}+\norm{v}$, $\norm{ su } = \abs{s} \norm{u}$, and $\norm{u} = 0$ if and only if $u = 0$, +let $\floor{\cdot}_h \colon [0,T] \to [0,T]$, $h \in (0,\infty)$, be the functions which satisfy for all $h \in (0,\infty)$, $t \in [0,T]$ that $\floor{t}_h = \max\pr[]{ [0,t] \cap \{0,h,2h,\dots\} }$, +let $f \colon \R^d \to \R^d$ be a function which satisfies that +% +\begin{equation} +\br*{ \sup_{v \in \R^d} \norm[\big]{ f(v) } } + \br*{ \sup_{v,w\in\R^d, v\neq w} \frac{ \norm{ f(v) - f(w) } }{ \norm{ v-w } } } < \infty \dc +\end{equation} +% +let $y \colon [0,T] \to \R^d$ be a measurable function which satisfies for all $t \in [0,T]$ that +% +\begin{equation}\label{def:y_prob} +y(t) = y(0) + \int_0^t f\pr[\big]{ y(s) } \dx s \dc +\end{equation} +% +and for every $h \in (0,\infty)$ let $Y_{0,h}, Y_{1,h} , \dots , Y_{\floor{T/h},h} \in \R^d$ satisfy for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-1\}$ that $Y_{0,h} = y(0)$ and +% +\begin{equation}\label{def:euler} +Y_{n+1,h} = Y_{n,h} + h f \pr[\big]{ Y_{n,h} } \dpp +\end{equation} +% +\end{athm} + +% +% +% + +\begin{athm}{problem}{prob1} +% +Do you understand \cref{setting1} above? +% +% +Do you understand what each individual component means and do you see why each component is necessary to present a well-defined numerical method (i.e., the method in \cref{def:euler})? +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{10} +\noindent +\end{aproof} + +% +% +% + +\begin{athm}{definition}{def:converge} +% +Assume \cref{setting1}. +% +% +We say that \cref{def:euler} is a convergent numerical method for \cref{def:y_prob} if and only if it holds that +% +\begin{equation} +\lim_{h \to 0^+} \br*{ \max_{n \in \{0,1,\dots,\floor{T/h} \} } \norm[\big]{ y(nh) - Y_{n,h} } } = 0 \dpp +\end{equation} +% +\end{athm} + +% +% +% + +\begin{athm}{problem}{prob2} +% +Do you understand conceptually what the notion of convergence is implying? +% +Can you see how the topology of the problem would come into play if we were not considering a problem posed in a finite-dimensional space? +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{10} +\noindent +\end{aproof} + +% +% +% + +\begin{athm}{lemma}{lem:gronwall} +% +Let $\alpha \in [0,\infty)$ and let $a_0,a_1,a_2,\ldots \in [0,\infty)$ and $b_0,b_1,b_2,\ldots \in [0,\infty)$ satisfy for all $n \in \N_0 = \N \cup \{0\}$ that +% +\begin{equation}\label{eq:gron1} +a_n \le \alpha + \sum_{k=0}^{n-1} b_k a_k \dpp +\end{equation} +% +Then it holds for all $n \in \N_0$ that +% +\begin{equation}\label{eq:gron2} +a_n \le \alpha \exp\pr*{ \sum_{k=0}^{n-1} b_k } \dpp +\end{equation} +\end{athm} + +% +% +% + +\begin{aproof} +First, we claim that for all $n \in \N_0$ it holds that +% +\begin{equation}\label{eq:induct_claim} +a_n \le \alpha \br*{ \prod_{k=0}^{n-1} \pr[\big]{ 1 + b_k } } \dpp +\end{equation} +% +We now prove \cref{eq:induct_claim} by mathematical induction on $n \in \N_0$. +% +% +For the base case $n=0$, note that \cref{eq:gron1} ensures that +% +\begin{equation} +a_0 \le \alpha + \sum_{k=0}^{-1} b_k a_k = \alpha + 0 = \alpha \dpp +\end{equation} +% +Combining this and the fact that $\prod_{k=0}^{-1} (1 + b_k) = 1$ establishes \cref{eq:induct_claim} in the base case $n=0$. +% +For the induction step $\N_0 \ni (n-1) \induct n \in \N$, let $n \in \N$ and assume that for every $m \in \{0,1,\dots,n-1\}$ it holds that +% +\begin{equation}\label{eq:induct_claim_intermediate} +a_m \le \alpha \br*{ \prod_{k=0}^{m-1} \pr[\big]{ 1 + b_k } } \dpp +\end{equation} +% +This and \cref{eq:gron1} assure that +% +\begin{equation} +a_n +\le +\alpha + \sum_{k=0}^{n-1} b_k a_k +\le +\alpha + \sum_{k=0}^{n-1} b_k \pr*{ \alpha \br*{ \prod_{j=0}^{k-1} \pr[\big]{ 1 + b_j } } } += +\alpha \pr*{ 1 + \sum_{k=0}^{n-1} \br*{ \prod_{j=0}^{k-1} \pr[\big]{ 1 + b_j } } b_k } +\dpp +\end{equation} +% +Next, observe that +% +\begin{equation}\label{eq:gron3} +\begin{split} +1 + \sum_{k=0}^{n-1} \br*{ \prod_{j=0}^{k-1} \pr[\big]{ 1 + b_j } } b_k +& = +1 + \sum_{k=0}^{n-1} \br*{ \prod_{j=0}^{k-1} \pr[\big]{ 1 + b_j } } \pr[\big]{ (1 + b_k) - 1 } +\\ +& = +1 + \sum_{k=0}^{n-1} \br*{ \prod_{j=0}^{k} \pr[\big]{ 1 + b_j } - \prod_{j=0}^{k-1} \pr[\big]{ 1 + b_j } } +\\ +& = +1 + \prod_{j=0}^{n-1} \pr[\big]{ 1 + b_j } - \prod_{j=0}^{-1} \pr[\big]{ 1 + b_j } += +\prod_{j=0}^{n-1} \pr[\big]{ 1 + b_j } +\dpp +\end{split} +\end{equation} +% +Combining this, \cref{eq:gron3}, and mathematical induction establishes \cref{eq:induct_claim}. +% +Moreover, note that the fact that for all $x \in [0,\infty)$ it holds that $1 + x \le \exp(x)$, the assumption that $b_0,b_1,b_2,\ldots \in [0,\infty)$, and \cref{eq:induct_claim} imply that for all $n \in \N_0$ it holds that +% +\begin{equation} +a_n \le \alpha \br*{ \prod_{k=0}^{n-1} \pr[\big]{ 1 + b_k } } +\le \alpha \br*{ \prod_{k=0}^{n-1} \exp(b_k) } +\le \alpha \exp\pr*{ \sum_{k=0}^{n-1} b_k } \dpp +\end{equation} +% +This establishes \cref{eq:gron2}. +% +\end{aproof} + +% +% +% + +\begin{athm}{problem}{prob3} +% +Assume \cref{setting1}. +% +Using \cref{lem:gronwall} above, prove that there exists $C \in [0,\infty)$ such that for all $h \in (0,\infty)$ +%%%, $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}\}$ +it holds that +% +\begin{equation}\label{eq:converge} +\max_{n \in \{0,1,\dots,\floor{T/h}\} } \norm[\big]{ y(nh) - Y_{n,h} } \le C h \dpp +\end{equation} +% +Explain how proving \cref{eq:converge} holds would relate to the notion of convergence (cf.\ \cref{def:converge}). +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{20} +\noindent +\end{aproof} + +% +% +% + +\begin{athm}{problem}{prob4} +% +Can you present the theta method from the textbook in the rigorous format used in \cref{setting1} above? +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{15} +\noindent +\end{aproof} + +% +% +% + +\newcommand{\T}{T_{\text{new}}} + +\begin{athm}{setting}{setting2} +% +Let $\T , p \in (0,\infty)$, $d \in \N$, +let $\norm{\cdot} \colon \R^d \to [0,\infty)$ be a function which satisfies for all $u,v \in \R^d$, $s \in \R$ that $\norm{u+v} \le \norm{u}+\norm{v}$, $\norm{ su } = \abs{s} \norm{u}$, and $\norm{u} = 0$ if and only if $u = 0$, +let $\floor{\cdot}_h \colon [0,\T] \to [0,\T]$, $h \in (0,\infty)$, be the functions which satisfy for all $h \in (0,\infty)$, $t \in [0,\T]$ that $\floor{t}_h = \max\pr[]{ [0,t] \cap \{0,h,2h,\dots\} }$, +let $g \colon \R^d \to \R^d$ be a function which satisfies that +% +\begin{equation} +%%%\br*{ \sup_{v \in \R^d} \norm[\big]{ f(v) } } + +%%%\br*{ +\sup_{v,w\in\R^d, v\neq w} \frac{ \norm{ g(v) - g(w) } }{ \pr[\big]{ 1 + \norm{v}^p + \norm{w}^p }\norm{ v-w } } +%%%} +< \infty \dc +\end{equation} +% +let $z \colon [0,\T] \to \R^d$ be a measurable function which satisfies for all $t \in [0,\T]$ that +% +\begin{equation}\label{def:z_prob} +z(t) = z(0) + \int_0^t g\pr[\big]{ z(s) } \dx s \dc +\end{equation} +% +and for every $h \in (0,\infty)$ let $Z_{0,h}, Z_{1,h} , \dots , Z_{\floor{\T/h},h} \in \R^d$ satisfy for all $n \in \{0,1,\dots,\allowbreak\floor{\nicefrac{\T}{h}}-1\}$ that $Z_{0,h} = z(0)$ and +% +\begin{equation}\label{def:euler_new} +Z_{n+1,h} = Z_{n,h} + h g \pr[\big]{ Z_{n,h} } \dpp +\end{equation} +% +\end{athm} + +% +% +% + +\begin{athm}{problem}{prob5} +% +Can we prove a result similar to that in \cref{prob3} under the assumptions outline in \cref{setting2} above? +% +If not, can we prove a result that is ``similar'' to the result in \cref{prob3}? +% +What additional assumptions (if any) are needed to prove either of the above results? +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{25} +\noindent +\end{aproof} + +% +% +% + +\subsection{An exploration of the linear case} + +% +% +% + +\begin{athm}{definition}{def:mat_exp} +% +We denote by +$ \exp \colon (\medcup{d \in \N} \C^{d\times d}) \to (\medcup{d \in \N} \C^{d\times d})$ +the function which satisfies for all +$d \in \N$, +$A \in \C^{d \times d}$ +that +% +$ +\exp(A) = \sum_{k=0}^\infty (\nicefrac{1}{k!}) A^k +$. +% +\end{athm} + +% +% +% + +\begin{athm}{definition}{def:mat_det} +% +\newcommand{\ppp}{\mathfrak{p}} +\newcommand{\nnn}{\mathfrak{N}} +% +For every +$ d \in \N $ +let +$ \nnn_d = \{1,2,\dots,d\} $, +for every +$ d \in \N $ +let +$S_d = \{ (\sigma \colon \nnn_d \to \nnn_d ) \colon \text{$\sigma$ is a bijection} \}$, +let +$ \ppp \colon (\medcup{d\in\N}S_d) \to \N_0 $ +be the function which satisfies for all +$ d \in \N $, +$ \sigma \in S_d $ +that +$ \ppp(\sigma) = \sum_{i=1}^d \sum_{j=i+1}^d \mathbbm{1}_{(0,\infty)} (\sigma_i - \sigma_j) $, +and let +$ \operatorname{sgn} \colon (\medcup{d\in\N}S_d) \to \{-1,1\} $ +be the function which satisfies for all +$ d \in \N $, +$ \sigma \in S_d $ +that +$ \operatorname{sgn}(\sigma) = (-1)^{\ppp(\sigma)} $. +% +% +Then we denote by +$\det \colon (\medcup{d\in\N}\R^{d\times d}) \to \R$ +the function which satisfies for all +$ d \in \N $, +$A = (a_{i,j})_{i,j \in \{1,2,\dots,d\}} \in \R^{d \times d}$ +that +$ \det(A) = \sum_{\sigma \in S} \br[]{ \operatorname{sgn}(\sigma) \prod_{i=1}^d a_{i,\sigma_i} } $. +% +\end{athm} + +% +% +% + +\begin{athm}{definition}{def:mat_trace} +% +We denote by +$ \operatorname{tr} \colon (\medcup{d\in\N}\R^{d\times d}) \to \R $ +the function which satisfies for all +$ d\in \N $, +$A = (a_{i,j})_{i,j \in \{1,2,\dots,d\}} \in \R^{d \times d}$ +that +$ \operatorname{tr}(A) = \sum_{i=1}^d a_{i,i} $. +% +\end{athm} + +% +% +% + +\begin{athm}{lemma}{lem:mat_exp1} +% +Let +$d \in \N$, +$A,B \in \R^{d\times d}$ +and +let $\norm{\cdot} \colon \R^d \to [0,\infty)$ be a function which satisfies for all $u,v \in \R^d$, $s \in \R$ that $\norm{u+v} \le \norm{u}+\norm{v}$, $\norm{ su } = \abs{s} \norm{u}$, and $\norm{u} = 0$ if and only if $u = 0$. +% +Then +% +\begin{enumerate}[label=(\roman*)] +% +\item +it holds that +$ \norm{ \exp(A) } \le \exp( \norm{A} ) < \infty$, +% +\item +it holds for all +$ s,t \in \R $ +that +$ \exp( sA + tA ) = \exp(sA) \exp(tA) $, +% +\item +it holds that +$ \exp(A) \exp(-A) = \operatorname{id}_{\R^{d\times d}} $, +% +\item +it holds that +$ \exp( A + B ) = \exp(A) \exp(B) $ +if and only if it holds that +$ AB = BA $, +and +% +\item +it holds that +$ \det(\exp(A)) = \exp(\operatorname{tr}(A)) $ +% +\end{enumerate} +% +(cf.\ \cref{def:mat_exp,def:mat_det,def:mat_trace}). +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{25} +\noindent +\end{aproof} + +% +% +% + +\begin{athm}{problem}{prob:mat_exp1} +% +Let +$ A \in \R^{2\times 2}$ +satisfy +% +\begin{equation} +A = +\begin{pmatrix} +-1 & 1 \\ -2 & -4 +\end{pmatrix} +\dpp +\end{equation} +% +\begin{enumerate}[label=(\roman*)] +% +\item \label{prob:mat_exp1_item1} +Show that there exist +$D = (d_{i,j})_{i,j\in\{1,2\}} \in \R^{2\times 2}$, +$P \in \R^{2\times 2}$ +with +$ \det(P) \neq 0 $, +$ d_{1,2} = d_{2,1} = 0 $, +and +$ A = PDP^{-1} $ +(cf.\ \cref{def:mat_det}). +% +\item +Use the results from \cref{prob:mat_exp1_item1} to show that +% +\begin{equation} +\exp(A) += +\begin{pmatrix} +2\exp(-2) - \exp(-3) & \exp(-2) - \exp(-3) \\ +2\exp(-3) - 2\exp(-2) & 2\exp(-3) - \exp(-2) +\end{pmatrix} +\end{equation} +% +(cf.\ \cref{def:mat_exp}). +% +\end{enumerate} +% +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{25} +\noindent +\end{aproof} + +% +% +% + +\begin{athm}{problem}{prob:mat_exp2} +% +Let +$ T \in (0,\infty) $, +% +let $\norm{\cdot} \colon \R^2 \to [0,\infty)$ be the function which satisfies for all +$ u = (u_1,u_2) \in \R^2 $ +that +$\norm{u} = [\abs{u_1}^2 + \abs{u_2}^2]^{\nicefrac{1}{2}}$, +% +let $\floor{\cdot}_h \colon [0,T] \to [0,T]$, $h \in (0,\infty)$, be the functions which satisfy for all $h \in (0,\infty)$, $t \in [0,T]$ that $\floor{t}_h = \max\pr[]{ [0,t] \cap \{0,h,2h,\dots\} }$, +% +let +$ A \in \R^{2\times 2}$, +$ y \in C([0,T],\R^2) $ +satisfy for all +$ t \in [0,T] $ +that +%%%$ y(0) = (1,1) \in \R^2 $ +%%%and +% +\begin{equation} +A = +\begin{pmatrix} +-1 & 1 \\ -2 & -4 +\end{pmatrix} +\qquad +\text{and} +\qquad +y(t) = (1,1)^* + \int_0^t A y(s) \dx s \dc +\end{equation} +% +and for every $h \in (0,\infty)$ let $Y_{0,h}, Y_{1,h} , \dots , Y_{\floor{T/h},h} \in \R^2$ satisfy for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-1\}$ that $Y_{0,h} = y(0)$ and +% +\begin{equation} +Y_{n+1,h} = Y_{n,h} + h A Y_{n,h} \dpp +\end{equation} +% +\begin{enumerate}[label=(\roman*)] +% +\item +Prove that for all +$ t \in [0,T] $ +it holds that +$ y(t) = \exp(tA)y(0) $ +(cf.\ \cref{def:mat_exp}). +% +\item +Prove that for all +$h \in (0,\infty)$, +$n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}\}$ +it holds that +% +\begin{equation} + Y_{n,h} = \pr[\big]{ \operatorname{id}_{\R^{2\times 2}} + hA }^{\!n} y(0) + \dpp +\end{equation} +% +\item +Prove that for all +$ h \in (0,\infty) $ +it holds that +% +\begin{equation} +\begin{split} +& +\norm[\Big]{ \exp(hA)y(0) - \pr[\big]{ \operatorname{id}_{\R^{2\times 2}} + hA } y(0) } += +\norm*{ \int_0^h (h-s) A^2 \exp\pr[\big]{ sA } y(0) \dx s } +\\ +& +\qquad +\le +\frac{h^2}{2} \br*{ \sup_{\mathfrak{h} \in (0,h)} \pr*{ \sup_{v \in \R^2 \backslash \{0\} } \frac{ \norm{ \exp(\mathfrak{h} A) v } }{ \norm{v} } } } \norm[\big]{ A^2 y(0) } +\le +\sqrt{17} \, h^2 +\end{split} +\end{equation} +% +(cf.\ \cref{def:mat_exp}). +% +\item +Prove that for all +$h \in (0,\infty)$, +$n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}\}$ +it holds that +% +\begin{equation} +\begin{split} +& +y(nh) - Y_{n,h} +\\ +& +\quad += +\sum_{k=0}^{n-1} \exp\pr[\big]{ khA } \br[\Big]{ \exp\pr[\big]{ hA } - \pr[\big]{ \operatorname{id}_{\R^{2\times 2}} + hA } } \pr[\big]{ \operatorname{id}_{\R^{2\times 2}} + hA }^{(n-k-1)} y(0) +\end{split} +\end{equation} +% +(cf.\ \cref{def:mat_exp}). +% +\item +Prove that +% +\begin{equation} +\sup_{h \in (0,\infty)} \br*{ \max_{ n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}\} } \norm[\big]{ y(nh) - Y_{n,h} } } \le T \exp\pr[\big]{ \nicefrac{9T}{2} } \sqrt{34} \, h +\dpp +\end{equation} +% +\end{enumerate} +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{25} +\noindent +\end{aproof} + +% +% +% + +\section{Multistep methods} + +% +% +% + +\newcommand{\U}{\mathbb{U}} +\newcommand{\A}{\mathcal{A}} +\newcommand{\D}{\mathcal{D}} +\newcommand{\E}{\mathcal{E}} + +\begin{athm}{setting}{setting_2_1} +% +Let $T \in (0,\infty)$, +$d,s \in \N$, +$a_0,a_1,\dots,a_s \in \R$, +$b_0,b_1,\dots,b_s \in \R$, +let $\norm{\cdot} \colon \R^d \to [0,\infty)$ be a function which satisfies for all $u,v \in \R^d$, $s \in \R$ that $\norm{u+v} \le \norm{u}+\norm{v}$, $\norm{ su } = \abs{s} \norm{u}$, and $\norm{u} = 0$ if and only if $u = 0$, +let $\floor{\cdot}_h \colon [0,T] \to [0,T]$, $h \in (0,\infty)$, be the functions which satisfy for all $h \in (0,\infty)$, $t \in [0,T]$ that $\floor{t}_h = \max\pr[]{ [0,t] \cap \{0,h,2h,\dots\} }$, +let $\A = \cu[]{ g \colon [0,T] \to \R^d \colon \text{$f$ is analytic in $[0,T]$} }$, +for every $h \in (0,\infty)$, $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-1\}$, $g \in \A$ let $\D \colon \A \to \A$ and $\E_h \colon \A \to \A$ satisfy +% +\begin{equation} +\pr[\big]{ \D g } \lr (nh) = \pr[\big]{ \tfrac{\dd}{\dd t} g } \lr (nh) +\qquad \text{and} \qquad +\pr[\big]{ \E_h g } \lr (nh) = g\pr[\big]{ (n+1)h } \dc +\end{equation} +% +let $f \colon \R^d \to \R^d$ be a function which satisfies that +% +\begin{equation} +\br*{ \sup_{v \in \R^d} \norm[\big]{ f(v) } } + \br*{ \sup_{v,w\in\R^d, v\neq w} \frac{ \norm{ f(v) - f(w) } }{ \norm{ v-w } } } < \infty \dc +\end{equation} +% +let $y \colon [0,T] \to \R^d$ be a measurable function which satisfies for all $t \in [0,T]$ that +% +\begin{equation}\label{eq:ode_multi} +y(t) = y(0) + \int_0^t f\pr[\big]{ y(s) } \dx s \dc +\end{equation} +% +and for every $h \in (0,\infty)$ let $Y_{0,h}, Y_{1,h} , \dots , Y_{\floor{T/h},h} \in \R^d$ satisfy for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-s\}$ that $Y_{0,h} = y(0)$ and +% +\begin{equation}\label{def:multi} +\sum_{m=0}^s a_m Y_{n+m,h} = h \sum_{m=0}^s b_m f \pr[\big]{ Y_{n+m,h} } \dpp +\end{equation} +% +\end{athm} + +% +% +% + +\begin{athm}{definition}{def:order} +% +Assume \cref{setting_2_1}. +% +We say that \cref{def:multi} is a numerical method of order $p \in \N_0$ if and only if there exists $C \in (0,\infty)$ such that for all $h \in (0,\infty)$, $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}\}$ with $h$ sufficiently close to zero it holds that +% +\begin{equation} +\norm[\Big]{ \textstyle \sum_{m=0}^s a_m y\pr[\big]{ (n+m)h } - h \sum_{m=0}^s b_m f\pr[\big]{ y\pr[\big]{ (n+m)h } } } \le C h^{p+1} \dpp +\end{equation} +% +\end{athm} + +% +% +% + +\begin{athm}{lemma}{lemma:multi} +% +Assume \cref{setting_2_1} and let $p \in \N$. +% +Then \cref{def:multi} is of order $p$ if and only if there exists $C \in (0,\infty)$ such that for all $z \in \R$ +with $z$ sufficiently close to one +it holds that +% +\begin{equation}\label{eq:lemma:multi} +\abs[\big]{ \textstyle\sum_{m=0}^s a_m z^m - \ln(z) \sum_{m=0}^s b_m z^m } \le C \abs{ z-1 }^{p+1} +\end{equation} +% +(cf.\ \cref{def:order}). +% +\end{athm} + +% +% +% + +\begin{aproof} +% +Throughout this proof let $h \in (0,\infty)$ be sufficiently small, +let $\rho \colon \R \to \R$ and $\sigma \colon \R \to \R$ be the functions which satisfy for all $z \in \R$ that +% +\begin{equation} +\rho(z) = \sum_{m=0}^s a_m z^m +\qquad \text{and} \qquad +\sigma(z) = \sum_{m=0}^s b_m z^m \dc +\end{equation} +and +without loss of generality assume that $y \in \A$. +%%% and that its radius of convergence is \emph{at least} $sh$. +% +Note that Taylor's theorem guarantees that for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}\}$, $k \in \N_0$ it holds that +% +\begin{equation} +\begin{split} +\pr[\Big]{ \E_h \pr[\big]{ \tfrac{\dd^k}{\dd t^k} y } } \lr (nh) += +\pr[\big]{ \tfrac{\dd^k}{\dd t^k} y } \lr \pr[\big]{ (n+1)h } +& = +\sum_{j=0}^\infty \frac{h^j}{j!} \pr[\big]{ \tfrac{\dd^{k+j}}{\dd t^{k+j}} y } \lr \pr[]{ nh } +\\ +& = +\sum_{j=0}^\infty \frac{h^j}{j!} \pr[\Big]{ \tfrac{\dd^j}{\dd t^j} \pr[\big]{ \tfrac{\dd^{k}}{\dd t^{k}} y } } \lr \pr[]{ nh } +\\ +& = +\sum_{j=0}^\infty \frac{h^j}{j!} \pr[\Big]{ \D^j \pr[\big]{ \tfrac{\dd^{k}}{\dd t^{k}} y } } \lr \pr[]{ nh } +\dpp +\end{split} +\end{equation} +% +Combining this and the fact that $\D$ is a bounded linear operator (something we have not shown, but which can be shown) ensures that +% +\begin{equation}\label{eq:shift_rep} +\E_h = \exp\pr[\big]{ h \D } \dpp +\end{equation} +% +Next, observe that \cref{eq:ode_multi} assures that for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-s\}$ it holds that +% +\begin{equation} +\begin{split} +& +\sum_{m=0}^s a_m y\pr[\big]{ (n+m)h } - h \sum_{m=0}^s b_m f\pr[\big]{ y( (n+m)h ) } +\\ +& \quad = +\sum_{m=0}^s a_m y\pr[\big]{ (n+m)h } - h \sum_{m=0}^s b_m \pr[\big]{ \tfrac{\dd}{\dd t} y } \lr \pr[\big]{ (n+m)h } +\\ +& \quad = +\sum_{m=0}^s a_m \pr[\big]{ \E_h^m y } \lr (nh) - h \sum_{m=0}^s b_m \pr[\Big]{ \E_h^m \pr[\big]{ \D y } } \lr (nh) +\dpp +\end{split} +\end{equation} +% +This, the fact that \cref{eq:shift_rep} implies that for all $g \in \A$ it holds that $(\D(\E_h g)) = (\E_h(\D g))$, the fact that $\D$ is a linear operator, and the so-called Borel functional calculus guarantee that for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-s\}$ it holds that +% +\begin{equation} +\begin{split} +& +\sum_{m=0}^s a_m y\pr[\big]{ (n+m)h } - h \sum_{m=0}^s b_m f\pr[\big]{ y( (n+m)h ) } +\\ +& \quad = +\sum_{m=0}^s a_m \pr[\big]{ \E_h^m y } \lr (nh) - h \pr*{ \D \sum_{m=0}^s b_m \pr[\big]{ \E_h^m y } } \lr (nh) +\\ +& \quad = +\pr[\Bigg]{ \pr[\bigg]{ \sum_{m=0}^s a_m \E_h^m - h \D \sum_{m=0}^s b_m \E_h^m } y } \lr (nh) += +\pr[\Big]{ \pr[\big]{ \rho(\E_h) - h \D \sigma(\E_h) } y } \lr (nh) +\dpp +\end{split} +\end{equation} +% +This shows that for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-s\}$ it holds that +% +\begin{align}\label{eq:2_15} +& +\abs*{ \sum_{m=0}^s a_m y\pr[\big]{ (n+m)h } - h \sum_{m=0}^s b_m f\pr[\big]{ y( (n+m)h ) } } +\\ +& \quad = +\abs*{ \pr[\Big]{ \pr[\big]{ \rho(\E_h) - h \D \sigma(\E_h) } y } \lr (nh) } +\le +\br*{ \sup_{g \in \A \backslash \{0\} } \frac{ \abs*{ \pr[\big]{ \pr[]{ \rho(\E_h) - h \D \sigma(\E_h) } g } \lr (nh) } }{ \abs{g(nh)} } } \abs[\big]{ y(nh) } +\dpp \nonumber +\end{align} +% +In addition, note that \cref{eq:shift_rep}, the fact that for all $g \in \A$, $t \in [0,T]$ it holds that $\lim_{z\to 0^+} (\E_z g)(t) = g(t)$ (can you see that this is true?), and the implicit function theorem demonstrate that for all $g \in \A$, $t \in [0,T]$ it holds that +% +\begin{equation} +\pr[\big]{ h\D g } \lr (t) = \pr[\big]{ \ln(\E_h) g } \lr (t) = \pr*{ \sum_{k=0}^\infty \frac{(-1)^{k}}{k+1} \pr[\big]{ \E_h - \operatorname{id} }^{k+1} g } \lr (t) \dpp +\end{equation} +% +This and the Borel functional calculus +%%%, and the fact that the spectrum of $\E_h$ is contained in the set $[-h,h] \subseteq \R$ (this is something we can prove if we know a little functional analysis) +yield that +there exists $\gamma_h \subseteq \C$ (with the spectrum of $\E_h$ contained inside of $\gamma_h$---we can discuss this, if desired) such that +for all $g \in \A$, $t \in [0,T]$ it holds that +% +\begin{equation}\label{eq:2_17} +\pr[\Big]{ \pr[\big]{ \rho(\E_h) - \ln(\E_h) \sigma(\E_h) } g } \lr (t) += +\frac{1}{2\pi i} \int_{\gamma_h} \br[\big]{ \rho(z) - \ln(z) \sigma(z) } \pr[\big]{ (z\operatorname{id} - \E_h)^{-1} g } \lr (t) \dx z +\dpp +\end{equation} +% +Combining \cref{eq:2_15,eq:2_17} hence proves \cref{eq:lemma:multi}. +% +\end{aproof} + +% +% +% + +\section{Runge-Kutta methods} + +% +% +% + + + +% +% +% + +\section{Stiff equations} + +% +% +% + +\begin{athm}{definition}{def:stable} +Let +$y_\lambda \colon [0,\infty) \to \C$, $\lambda \in \C$, be measurable functions which satisfy for all $\lambda \in \C$, $t \in [0,\infty)$ that +% +\begin{equation} +y_\lambda(t) = 1 + \lambda \int_0^t y(s) \dx s \dc +\end{equation} +% +let $h \in (0,\infty)$, +for every $\lambda \in \C$ let +$Y_{0,\lambda} , Y_{1,\lambda} , Y_{2,\lambda}, \ldots \in \R$ satisfy +$Y_{0,\lambda} = 1$, +and +assume there exists +$p,C \in (0,\infty)$ +such that for all $\lambda \in \C$ with $\lambda + \bar{\lambda} \in (-\infty,0)$ it holds that +% +\begin{equation} +\sup_{n \in \N_0} \abs[\big]{ y_\lambda(nh) - Y_{n,\lambda} } \le C h^p \dpp +\end{equation} +% +% +Then the set +% +\begin{equation} +\mathcal{D} = \cu[\big]{ h\lambda \in \C \colon \textstyle\lim_{n\to\infty} Y_{n,\lambda} = 0 } \subseteq \C +\end{equation} +% +is the \emph{linear stability domain} of the numerical method +$\{Y_{n,\lambda}\}_{(n,\lambda) \in \N_0 \times \C}$. +% +% +Moreover, we say that the numerical method +$\{Y_{n,\lambda}\}_{(n,\lambda) \in \N_0 \times \C}$ +is A-stable if it holds that +% +\begin{equation} +\cu[\big]{ z \in \C \colon z + \bar{z} \in (-\infty,0) } \subseteq \mathcal{D} \dpp +\end{equation} +\end{athm} + +% +% +% + +\begin{athm}{problem}{problem_stiff} +% +Let $T \in (0,\infty)$, $d \in \N$, +let $\norm{\cdot} \colon \R^d \to [0,\infty)$ be a function which satisfies for all $u,v \in \R^d$, $s \in \R$ that $\norm{u+v} \le \norm{u}+\norm{v}$, $\norm{ su } = \abs{s} \norm{u}$, and $\norm{u} = 0$ if and only if $u = 0$, +let $\floor{\cdot}_h \colon [0,T] \to [0,T]$, $h \in (0,\infty)$, be the functions which satisfy for all $h \in (0,\infty)$, $t \in [0,T]$ that $\floor{t}_h = \max\pr[]{ [0,t] \cap \{0,h,2h,\dots\} }$, +let $f \in C^1(\R^d,\R^d)$ satisfy +% +\begin{equation} +\br*{ \sup_{v \in \R^d} \norm[\big]{ f(v) } } + \br*{ \sup_{v,w\in\R^d, v\neq w} \frac{ \norm{ f(v) - f(w) } }{ \norm{ v-w } } } < \infty \dc +\end{equation} +% +let $y \colon [0,T] \to \R^d$ be a measurable function which satisfies for all $t \in [0,T]$ that +% +\begin{equation} +y(t) = y(0) + \int_0^t f\pr[\big]{ y(s) } \dx s \dc +\end{equation} +% +and for every $h \in (0,\infty)$ let $Y_{0,h}, Y_{1,h} , \dots , Y_{\floor{T/h},h} \in \R^d$ satisfy for all $n \in \{0,1,\dots,\floor{\nicefrac{T}{h}}-1\}$ that $Y_{0,h} = y(0)$ and +% +\begin{equation}\label{eq:3_4_theta} +Y_{n+1,h} = Y_{n,h} + \tfrac{h}{4} \br[\Big]{ f\pr[\big]{ Y_{n,h} } + 3 f\pr[\big]{ Y_{n+1,h} } } \dpp +\end{equation} +% +% +\begin{enumerate}[label=\alph*.] +% +\item +Determine whether or not \cref{eq:3_4_theta} is consistent (cf.\ \cref{def:order}). +% +% +If \cref{eq:3_4_theta} is consistent, determine its order. +% +% +\item +Determine whether or not \cref{eq:3_4_theta} is convergent (cf.\ \cref{def:converge}). +% +% +\item +Determine whether or not \cref{eq:3_4_theta} is A-stable (cf.\ \cref{def:stable}). +% +\end{enumerate} +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{25} +\noindent +\end{aproof} + +% +% +% + +\section{Geometric numerical integration} + +% +% +% + +\section{Error control} + +% +% +% + +\section{Nonlinear algebraic systems} + +% +% +% + +\section{Finite difference schemes} + +% +% +% + +\begin{athm}{problem}{prob:finite_difference} +% +Let $N \in \N_0$, $\alpha , \beta \in \R$, +let +$f \in C(\R,\R)$ +and +$u \in C^4([0,1],\R)$ +satisfy for all +$ x \in [0,1] $ +that +$u(0) = \alpha$, +$u(1) = \beta$, +and +% +\begin{equation}\label{eq:1d_laplace} +\pr[\big]{ \tfrac{\dd^2}{\dd x^2} u } \lr (x) = f(x) \dc +\end{equation} +% +and +let +$h_0,h_1,\dots,h_N , x_0,x_1,\dots,x_{N+1} \in [0,1]$ +satisfy for all +$n \in \{0,1,\dots,N\}$ +that +% +\begin{equation}\label{eq:non-uniform_grid} +0 = x_0 < x_1 < x_2 < \ldots < x_N < x_{N+1} = 1 +\qquad \text{and} \qquad +h_n = x_{n+1} - x_n \dpp +\end{equation} +% +% +\begin{enumerate}[label=\alph*.] +% +\item \label{prob:finite_difference_a} +Construct a three-point finite difference scheme for approximating the solution to \cref{eq:1d_laplace} on the non-uniform grid $\{ x_n \}_{n\in\{0,1,\dots,N+1\}} \subseteq [0,1]$ given by \cref{eq:non-uniform_grid}. +% +% +\item +Determine the order of the method constructed in \cref{prob:finite_difference_a}~above. +% +% +Determine what additional assumptions are necessary (if any) for guaranteeing this order. +% +% +Compare these results with the case from Section 8.2 of the textbook (i.e., the case when $h_0 = h_1 = \ldots = h_N$). +% +% +\item \label{prob:finite_difference_c} +Write the finite difference scheme constructed in \cref{prob:finite_difference_a}~above in the form of a linear system (i.e., as a matrix-vector equation). +% +% +\item +Determine whether the linear system obtained in \cref{prob:finite_difference_c}~is always nonsingular. +% +% +If the linear system is not always nonsingular, provide sufficient conditions to guarantee that the linear system is nonsingular. +% +% +\item +Implement your finite difference scheme (i.e., the difference equations from \cref{prob:finite_difference_a}\ above or the linear system from \cref{prob:finite_difference_c}~above) in Python. +% +% +Numerically compare the approximate solution with the true solution for some ``test case.'' +% +% +\end{enumerate} +\end{athm} + +% +% +% + +\begin{aproof} +\linebreaks{30} +\noindent +\end{aproof} + +% +% +% + +\end{document} \ No newline at end of file diff --git a/Notes/Iserles_Questions.toc b/Notes/Iserles_Questions.toc new file mode 100644 index 0000000..4df130d --- /dev/null +++ b/Notes/Iserles_Questions.toc @@ -0,0 +1,30 @@ +\contentsline {section}{\numberline {1}Euler's method and beyond}{2}{section.1}% +\contentsline {subsection}{\cref {setting1}}{2}{section*.2}% +\contentsline {subsection}{\cref {prob1}}{2}{section*.3}% +\contentsline {subsection}{\cref {def:converge}}{3}{section*.4}% +\contentsline {subsection}{\cref {prob2}}{3}{section*.5}% +\contentsline {subsection}{\cref {lem:gronwall}}{3}{section*.6}% +\contentsline {subsection}{\cref {prob3}}{4}{section*.7}% +\contentsline {subsection}{\cref {prob4}}{5}{section*.8}% +\contentsline {subsection}{\cref {setting2}}{5}{section*.9}% +\contentsline {subsection}{\cref {prob5}}{6}{section*.10}% 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enumitem, + geometry, + mleftright, + nicefrac, + mathtools, + xparse, + ifthen, + tikz, + pgfplots, + floatrow, + multicol, + caption, + xurl} + +% Set up PGF plots (used for graphing) +\pgfplotsset{compat=newest} + +% Fixes spacing caused by using "\left" or "\right" +\mleftright + +% Sets the margins of the document. +% One can do this using plain TeX, but why? +\geometry{margin=1in} + +% Lines 43-46 set up my use of the cleveref package. +% This provides a "smarter" cross referencing style +\usepackage[colorlinks=true]{hyperref} +\usepackage[sort,capitalise]{cleveref} +\newcommand{\creflastconjunction}{, and\nobreakspace} +\crefname{enumi}{item}{items} + +% My personal command macros that I use. +% Note that "\displaySpace" is use to create the correct spacing for displayed math puctuation (and can be easily updated throughout the entire document by changing line 50). +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\dpp}{\text{.}} +\newcommand{\dc}{\text{,}} +\newcommand{\M}{\mathcal{M}} +\newcommand{\dd}{{\rm d}} +\newcommand{\fX}{\mathfrak{X}} +\newcommand{\fg}{\mathfrak{g}} +\newcommand{\G}{\mathcal{G}} +\newcommand{\dexp}{\operatorname{dexp}} +\newcommand{\ad}{\operatorname{ad}} +\newcommand{\lr}{\ensuremath{\mkern-1.5mu}} + +% Creates my paired math delimiters for easy use. +\DeclarePairedDelimiter{\pr}{(}{)} +\DeclarePairedDelimiter{\br}{[}{]} +\DeclarePairedDelimiter{\cu}{\{}{\}} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\DeclarePairedDelimiter{\vt}{\langle}{\rangle} + +% Creates desired header and footer for document. +% Note that it uses the adaptively created date macro. +\setlength{\headheight}{15pt} +\usepackage{fancyhdr} +\pagestyle{fancy} +\fancyhf{} +\fancyhead[R]{} +\cfoot{} +\lfoot{\small{\textsc{Dissemination prohibited. \today}}} +\rfoot{\thepage} + +% Updates equation and figure numbering. +\counterwithin{equation}{section} +\counterwithin{figure}{section} + +% Defines theorem-like environments +\theoremstyle{definition} +\newtheorem{definition}{Definition}[section] +\newtheorem{theorem}[definition]{Theorem} +\newtheorem{lemma}[definition]{Lemma} +\newtheorem{problem}[definition]{Problem} +\newtheorem{remark}[definition]{Remark} +\newtheorem{assumption}[definition]{Assumption} +\newtheorem{example}[definition]{Example} +\renewcommand{\thetheorem}{\theenumi\arabic{theorem}} + +% Helps create better arrows in figures +\usetikzlibrary{arrows.meta} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic proof headers and endings, local labels +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentEnvironment {athm} {m m} {% +%\phantomsection +%\addcontentsline{toc}{subsection}{\texorpdfstring{\cref{#2}}{\ref*{#2}}} +\begin{#1}\label{#2}\global\def\loc{#2}% +}{% +\end{#1}% +} + +\NewDocumentEnvironment{aproof} {} {% +\begin{proof}[Proof~of~\cref{\loc}]% +}{% +\finishproofthus +\end{proof}% +} + +\ExplSyntaxOff + +\newcommand{\eqqref}[1]{\cref{eq:\loc.#1}} +\newcommand{\eqlabel}[1]{\label{eq:\loc.#1}} + +\newcommand{\finishproofthus}{The proof of \cref{\loc} is thus complete.} + +\NewDocumentEnvironment{asol} {} {% +\begin{proof}[Solution~to~\cref{\loc}]% +}{% +\noindent\finishsolthus +\end{proof}% +} + +\newcommand{\finishsolthus}{The solution to \cref{\loc} is thus complete.} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Creating blank space with line breaks +%%%%%%%%%%%%%%%%%%%%%%%% + +% Note that the code in lines 137-147 is simply a method to +% create blank space that allows for proof environments to +% break across pages. This is something that TeX does not +% like by default, so this is my effort to bypass this. + +\newcommand{\linebreaks}[1]{{\mbox{}\par + \newcount\foo + \foo=#1 + \loop + \phantom{a} + + \advance \foo -1 + \ifnum \foo>0 + \repeat + \mbox{}\par}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Beginning of actual text. %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\title{A survey of numerical integration techniques on manifolds} + +\author{Joshua Lee Padgett} + +\date{\today} + +\begin{document} + +\maketitle + +\begin{abstract} +The following set of notes provide a very rough overview and introduction to the ideas and concepts needed to approximate differential equations evolving on homogeneous manifolds. +This document mostly serves as a means to provide some basic ideas and then allow everyone else to inform me of which concepts need additional details in order to improve understanding. +\end{abstract} + +\tableofcontents + +\section{Introduction} + + + +\section{Some basic background material} + +The following are some basic definitions needed to discuss numerical integration on homogeneous manifolds (manifolds acted upon transitively by a Lie group). +Note that the following should not be viewed as a complete list of all needed (or even relevant) topics and concepts. +Any set of notes created in this way are usually incomplete, so we should either view them as a starting point from which we commence our journey to total understanding or we should discuss what further details are needed. + +\begin{athm}{definition}{def:manifold} +Let $m,d \in \N = \{1,2,3,\ldots\}$ with $m \le d$ and let $\M \subseteq \R^d$ with $\M \neq \emptyset$. +Then $\M$ is an $m$-dimensional smooth manifold if and only if for every $p \in \M$ there exist $\Omega \subseteq \R^d$, $U \subseteq \M$ with $p \in U$, and a smooth function $\varphi \colon \Omega \to \R^d$ such that $\varphi$ is a homeomorphism and $\varphi' \circ \varphi^{-1}$ is injective. +See, e.g., \url{https://en.wikipedia.org/wiki/Manifold} for more details. +\end{athm} + +\begin{athm}{definition}{def:tangent} +Let $\M$ be a manifold, let $p \in \M$, and let $\rho \colon [0,1] \to \M$ be a differentiable function with $\rho(0) = p$ (cf.\ \cref{def:manifold}). +Then the vector +\begin{equation} +\pr[\big]{ \tfrac{\dd }{\dd t} \rho } \lr (t) \Bigr\rvert_{t=0} +\end{equation} +is a tangent vector at the point $p$. +The set of all tangent vectors at $p$ is the tangent space at $p$ and is denoted by $T\M \rvert_p$. +The collection of all tangent spaces at all points $q \in \M$ is called the tangent bundle of $\M$ and is denoted by $T\M = \cup_{q\in\M} T\M\rvert_{q}$. +See, e.g., \url{https://en.wikipedia.org/wiki/Tangent_space} for more details. +\end{athm} + +\begin{athm}{definition}{def:vector_field} +Let $\M$ be a manifold and let $F \colon \M \to T\M$ be a differentiable function which satisfies for all $p \in \M$ that $F(p) \in T\M\rvert_p$ (cf.\ \cref{def:manifold,def:tangent}). +Then $F$ is a vector field on $\M$. +The collection of all vector fields on $\M$ is denoted by $\fX(\M)$. +See, e.g., \url{https://en.wikipedia.org/wiki/Vector_field} for more details. +\end{athm} + +\begin{athm}{definition}{def:lie_algebra} +Let $\fg$ be a vector space and let $[\cdot,\cdot] \colon \fg \times \fg \to \fg$ be a bilinear function which satisfies for all $a,b,c\in\fg$ that $[a,b] = -[b,a]$ and +\begin{equation} +[a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0 \dpp +\end{equation} +Then we say that $\fg$ is a Lie algebra and we call the function $[\cdot,\cdot] \colon \fg \times \fg \to \fg$ the Lie bracket on $\fg$. +See, e.g., \url{https://en.wikipedia.org/wiki/Lie_algebra} for more details. +\end{athm} + +\begin{athm}{problem}{prob:lie_alg} +Let $\fg = \mathfrak{so}(3) = \{ a \in \R^{3\times 3} \colon a^* = -a \}$ and let $[ \cdot , \cdot ] \colon \fg \times \fg \to \fg$ satisfy for all $a,b \in \fg$ that $[a,b] = ab - ba$. +Show that $\fg$ is a Lie algebra (cf.\ \cref{def:lie_algebra}). +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\begin{athm}{problem}{prob:lie_alg2} +Let $\M$ be a manifold, let $C^\infty(\M)$ be the set of smooth function on $\M$, let $\fg = \fX(\M)$, and let $[ \cdot , \cdot ] \colon \fg \times \fg \to \fg$ satisfy for all $X,Y \in \fX(\M)$, $f \in C^\infty(\M)$ that $[X,Y](f) = (X\circ Y)(f) - (Y\circ X)(f)$ (cf.\ \cref{def:manifold,def:vector_field}). +Show that $\fg$ is a Lie algebra (cf.\ \cref{def:lie_algebra}). +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\begin{athm}{definition}{def:adjoint} +Let $\fg$ be a Lie algebra (cf.\ \cref{def:lie_algebra}). +Then we define $\ad^n \colon \fg \times \fg \to \fg$, $n\in\N_0 = \{0\} \cup \N$, to be the functions which satisfy for all $n \in \N$, $u,v \in \fg$ that $\ad^0(u,v) = v$ and +\begin{equation} +\ad^n(u,v) = +[u,\ad^{n-1}(u,v)] +\dpp +\end{equation} +See, e.g., \url{https://en.wikipedia.org/wiki/Adjoint_representation} for more details. +\end{athm} + +%\begin{athm}{definition}{def:lie_alg_homo} +%Let $\fg,\mathfrak{h}$ be Lie algebras and let $\varphi \colon \fg \to \mathfrak{h}$ be a linear function which satisfies for all $u,v\in\fg$ that +%\begin{equation} +%\varphi\pr[\big]{ [u,v]_\fg } = \br[\big]{ \varphi(u) , \varphi(v) }_{\mathfrak{h}} +%\end{equation} +%(cf.\ \cref{def:lie_algebra}). +%Then we say that $\varphi$ is a Lie algebra homomorphism. +%\end{athm} + +\begin{athm}{definition}{def:lie_group} +Let $\G$ be a manifold (cf.\ \cref{def:manifold}). +We say that $\G$ is a Lie group if and only if there exists a function $\cdot \colon \G \times \G \to \G$ and $e \in \G$ such that +\begin{enumerate}[label=(\roman*)] +\item it holds for all $p,q,r \in \G$ that $p \cdot (q \cdot r) = (p \cdot r) \cdot q$, +\item it holds for all $p \in \G$ that $p \cdot e = e \cdot p = p$, +\item it holds for all $p \in \G$ that there exists $p^{-1} \in \G$ such that $p^{-1} \cdot p = e$, and +\item it holds that $\cdot \colon \G \times \G \to \G$ and $\G \ni p \mapsto p^{-1} \in \G$ are smooth. +\end{enumerate} +See, e.g., \url{https://en.wikipedia.org/wiki/Lie_group} for more details. +\end{athm} + +\begin{athm}{problem}{prob:lie_group} +Let $\G = \{ a \in \R^{3\times 3} \colon a^{-1} = a^* \}$ and let $\cdot \colon \G \times \G \to \G$ satisfy for all $a,b\in\G$ that $a\cdot b = ab$. +Show that $\G$ is a Lie group (cf.\ \cref{def:lie_group}). +\emph{Note:} We have not defined matrix manifolds, but we can easily update this definition to allow for such things (e.g., via representations). +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\begin{athm}{definition}{def:alg_of_group} +Let $\G$ be a Lie group and let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$ (cf.\ \cref{def:lie_group}). +Then we define the Lie algebra of a Lie group to be $\fg = T\G\rvert_{e}$ (cf.\ \cref{def:lie_algebra,def:tangent}). +The Lie bracket on $\fg$ is the function $[ \cdot , \cdot ] \colon \fg \times \fg \to \fg$ which satisfies for all $u,v \in \fg$, $g,h \colon [0,1] \to \G$ with $g$ and $h$ being differentiable, $g(0) = h(0) = e$, $g'(0) = u$, and $h'(0) = v$ that +\begin{equation} +[u,v] = \frac{\partial^2}{\partial t \partial s} g(t) \cdot h(s) \cdot g^{-1}(t) \bigg\rvert_{t=s=0} \dpp +\end{equation} +See, e.g., \url{https://en.wikipedia.org/wiki/Lie_group%E2%80%93Lie_algebra_correspondence} for more details. +\end{athm} + +\begin{athm}{problem}{prob:lie_group2} +Let $\G = \operatorname{SO}(3) = \{ a \in \R^{3\times 3} \colon a^{-1} = a^*,\ \det(a)=1 \}$ and let $\cdot \colon \G \times \G \to \G$ satisfy for all $a,b\in\G$ that $a\cdot b = ab$. +Show that $\fg = \mathfrak{so}(3)$ is the Lie algebra of $\G$ (cf.\ \cref{def:alg_of_group}). +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\begin{athm}{definition}{def:action} +Let $\M$ be a manifold, let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, and let $\Lambda \colon \G \times \M \to \M$ be a smooth function which satisfies for all $p,q\in\G$, $y\in\M$ that +\begin{equation} +\Lambda(e,y) = y +\qquad\text{and}\qquad +\Lambda\pr[\big]{p,\Lambda(q,y)} = \Lambda(p\cdot q,y) +\end{equation} +(cf.\ \cref{def:manifold,def:lie_group}). +Then we say that $\Lambda$ is an (left) action of $\G$ on $\M$. +See, e.g., \url{https://en.wikipedia.org/wiki/Lie_group_action} for more details. +\end{athm} + +\begin{athm}{definition}{def:right} +Let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, and let $\fg$ be the Lie algebra of $\G$ (cf.\ \cref{def:lie_group,def:alg_of_group}). +Then we define the function $R_v \colon \G \to \G$, $v \in \G$, to be the function which satisfies for all $u,v\in\G$ that $R_v(u) = u\cdot v$. +Moreover, we have that for all $v \in \G$ it holds that $R_v' = T\rvert_e R_v \colon \fg \to T\G\rvert_v$ (cf.\ \cref{def:tangent}). +See, e.g., \url{https://math.stackexchange.com/questions/1740179/differential-of-the-multiplication-and-inverse-maps-on-a-lie-group} for more details. +\end{athm} + +\begin{athm}{definition}{def:exp} +Let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, let $\fg$ be the Lie algebra of $\G$, and let $\sigma_a \colon [0,1] \to \G$, $a \in \fg$, be a differentiable function satisfying for all $a\in\fg$, $t \in [0,1]$ that $\sigma_a(0) = e$ and +\begin{equation} +\sigma_a'(t) = R_{\sigma_a(t)}'(a) +\end{equation} +(cf.\ \cref{def:lie_algebra,def:lie_group,def:right,def:alg_of_group}). +Then we define the exponential map $\exp \colon \fg \to \G$ to be the function which satisfies for all $a \in \fg$ that $\exp(a) = \sigma_a(1)$. +See, e.g., \url{https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#:~:text=In%20the%20theory%20of%20Lie,tool%20for%20studying%20Lie%20groups.}~for more details. +\end{athm} + +\begin{athm}{problem}{prob:exp} +Let $d \in \N$, let $\G = \operatorname{GL}(d;\R) = \{ a \in \R^{d\times d} \colon \det(a) \neq 0 \}$, and let $\cdot \colon \G \times \G \to \G$ satisfy for all $a,b\in\G$ that $a\cdot b = ab$. +\begin{enumerate}[label=\alph*.] +\item Show that $\mathfrak{gl}_d = \R^{d\times d}$ is the Lie algebra of $\G$ (cf.\ \cref{def:alg_of_group}). +\item Show that for all $a \in \G$ it holds that +%\begin{equation} +$\exp(a) = \sum_{k=0}^\infty \tfrac{1}{k!} a^n$ +%\end{equation} +(cf.\ \cref{def:exp}). +\item Determine an action of $\G$ on the manifold $\R^d$ (cf.\ \cref{def:action}). Is this action unique? +\end{enumerate} +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\begin{athm}{lemma}{lem:1} +Let $\M$ be a manifold, let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, let $\fg$ be the Lie algebra of $\G$, let $\Lambda \colon \G \times \M \to \fg$ be an action, let $T \in (0,\infty)$, $y_0 \in \M$, $a \in \fg$, let $\rho \colon [0,1] \to \G$ be a differentiable function satisfying that $\rho(0) = e$ and $\rho'(0) = a$, and let $y \colon [0,T] \to \M$ be a differentiable function satisfying for all $t\in [0,T]$ that $y(0) = y_0$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \tfrac{\dd}{\dd s} \Lambda\pr[\big]{ \rho(s) , y(t) } \Bigr\rvert_{s=0} +\end{equation} +(cf.\ \cref{def:manifold,def:lie_algebra,def:lie_group,def:action,% +def:alg_of_group}). +Then it holds for all $t\in[0,T]$ that +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \Lambda\pr[\big]{ \exp(ta) , y_0 } +\end{equation} +(cf.\ \cref{def:exp}). +\end{athm} + +\begin{aproof} + +\end{aproof} + +\section{Approximating differential equations on manifolds} + +We now begin our exploration of how we may approximate solutions to differential equations evolving on (homogeneous) manifolds. +To that end, let $\M$ be a manifold, let $T \in (0,\infty)$, $y_0 \in \M$, let $F \colon [0,T] \times \M \to T\M$, and let $y \colon [0,T] \to \M$ be a differentiable function which satisfies for all $t \in [0,T]$ that $y(0) = y_0$ and +\begin{equation}\label{eq:3_1} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = F(t,y(t)) +\end{equation} +(cf.\ \cref{def:manifold,def:tangent}). +Our goal is to approximate the solution to \cref{eq:3_1} in a manner which preserves the fact that for all $t \in [0,T]$ it holds that $y(t) \in \M$ (you should convince yourself that this is indeed true). +While this may seem trivial, it is easy to see that many \emph{classical} numerical methods fail to preserve this property. +For an example of such a failure, please see \cref{prob:num1} below. + +\begin{athm}{problem}{prob:num1} +Let $T \in (0,\infty)$, $a \in \R^3$, let $y_0 = (y_0^1,y_0^2,y_0^3) \in \R^3$ satisfy that $(y_0^1)^2 + (y_0^2)^2 + (y_0^3)^2 = 1$, let $(\cdot,\cdot) \colon \R^3 \times \R^3 \to \R^3$ satisfy for all $u = (u^1,u^2,u^3) \in \R^3$, $v = (v^1,v^2,v^3)\in\R^3$ that $(u,v) = ( u^2v^3-u^3v^2 , u^3v^1 - u^1v^3 , u^1v^2 - u^2v^1 )$, and let $y \colon [0,T] \to \R^3$ be a differentiable function which satisfies for all $t \in [0,T]$ that $y(0) = y_0$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \pr[\big]{ a, y(t) } \dpp +\end{equation} +\begin{enumerate}[label=\alph*.] +\item Show that for all $t \in [0,T]$ it holds that $y(t) \in \operatorname{SO}(3)$ (cf.\ \cref{prob:lie_group2}). +\item Let $w \colon [0,T] \to \R^3$ satisfy for all $t \in [0,T]$ that $w(0) = y_0$ and +\begin{equation}\label{eq:3_3} +w(t) = w_0 + t \pr[\big]{ a, w_0 } \dpp +\end{equation} +Show that for all $t \in (0,T]$ it holds that $w(t) \not\in \operatorname{SO}(3)$. +\emph{Note:} The function in \cref{eq:3_3} is actually known as Euler's method and is the basic tangent line approximation to a vector field. +\end{enumerate} +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +In order to circumvent the issues observed in \cref{prob:num1} above, we assume that there exist a Lie group $\G$ with Lie algebra $\fg$ and $e \in \G$ satisfying for all $p \in \G$ that $p \cdot e = e \cdot p = p$, an action $\Lambda \colon \G \times \M \to \fg$, a function $a \colon [0,T] \times \M \to \fg$, and a differentiable function $\rho_{t,x} \colon [0,1] \to \G$, $t\in [0,T]$, $x \in \fg$, which satisfies for all $s \in [0,1]$, $t\in[0,T]$, $x\in\fg$ that $\rho_{t,x}(0) = e$, $\rho_{t,x}'(0) = a(t,x)$, and +\begin{equation}\label{eq:2_2} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \tfrac{\dd}{\dd s} \Lambda\pr[\big]{ \rho_{t,a(t,y(t))}(s) , y(t) } \Bigr\rvert_{s=0} +\end{equation} +(cf.\ \cref{def:manifold,def:lie_algebra,def:lie_group,def:action,% +def:alg_of_group,eq:3_1}). + +\begin{athm}{remark}{rem:1} +While the above assumptions may seem restrictive, it is important to note that Sophus Lie's third fundamental theorem guarantees that \cref{eq:2_2} will always hold \emph{locally} (cf., e.g., \url{https://en.wikipedia.org/wiki/Lie%27s_third_theorem}). +Thus, we may construct such actions locally throughout the interval of interest and then use the semigroup property of the exponential map to ``piece these solutions together.'' +However, in practice, one rarely runs into such pathological issues (at least, when trying to utilize structure-preserving methods). +\end{athm} + +\begin{athm}{definition}{def:dexp} +Let $\G$ be a Lie group and let $\fg$ be the Lie algebra of $\G$ (cf.\ \cref{def:lie_group,def:lie_algebra}). +Then we define $\dexp \colon \fg \times \fg \to \fg$ to be the function which satisfies for all $u,v \in \fg$ that +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} \exp } \lr (u)(v) = R_{\exp(u)}' \circ \dexp_u(v) +\end{equation} +(cf.\ \cref{def:exp,def:right}). +See, e.g., \url{https://en.wikipedia.org/wiki/Derivative_of_the_exponential_map} for more details. +\end{athm} + +\begin{athm}{lemma}{lem:2} +Let $\G$ be a Lie group and let $\fg$ be the Lie algebra of $\G$ (cf.\ \cref{def:lie_group,def:lie_algebra}). +Then it holds for all $u,v \in \fg$ that +\begin{equation} +\dexp_u(v) = \pr[\big]{ \ad^1(u,v) }^{-1} \br[\Big]{ \exp\pr[\big]{ \ad^1(u,v) } - v } += \sum_{k=0}^\infty \frac{1}{(k+1)!} \ad^k(u,v) +\end{equation} +(cf.\ \cref{def:dexp,def:exp,def:adjoint}). +\end{athm} + +\begin{aproof} + +\end{aproof} + +\begin{athm}{lemma}{lem:2a} +Let $\G$ be a Lie group, let $\fg$ be the Lie algebra of $\G$, and let $B_0,B_1,B_2,\ldots \in \R$ satisfy for all $n \in \N_0$ that +$B_n = \sum_{i=0}^n \sum_{j=0}^i (-1)^j \binom{i}{j} \nicefrac{j^n}{(i+1)}$ +(cf.\ \cref{def:lie_group,def:lie_algebra}). +Then it holds for all $u,v \in \fg$ that +\begin{equation} +\begin{split} +\dexp_u^{-1}(v) +& = \sum_{k=0}^\infty \frac{B_k}{k!} \ad^k(u,v) \\ +& = v - \tfrac{1}{2} \ad^1(u,v) + \tfrac{1}{12} \ad^2(u,v) - \tfrac{1}{720} \ad^4(u,v) + \tfrac{1}{30240} \ad^6(u,v) - \ldots +\end{split} +\end{equation} +(cf.\ \cref{def:dexp,def:adjoint}). +\end{athm} + +\begin{aproof} + +\end{aproof} + +\begin{athm}{remark}{rem:b} +The sequence of rational numbers +$B_0,B_1,B_2,\ldots \in \R$ which satisfy for all $n \in \N_0$ that +\begin{equation}\label{eq:3_8} +B_n = \sum_{i=0}^n \sum_{j=0}^i (-1)^j \binom{i}{j} \frac{j^n}{(i+1)} +\end{equation} +are known as \emph{Bernoulli's numbers} (cf., e.g., \url{https://en.wikipedia.org/wiki/Bernoulli_number}). +\cref{eq:3_8} is just one of many representations of these numbers. +\end{athm} + +\begin{athm}{lemma}{lem:3} +Let $\M$ be a manifold, let $T \in (0,\infty)$, $y_0 \in \M$, let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, let $\fg$ be the Lie algebra of $\G$, let $\Lambda \colon \G \times \M \to \fg$ be an action, let $a \colon [0,T] \times \M \to \fg$, let $\rho_{t,x} \colon [0,1] \to \G$, $t\in [0,T]$, $x \in \fg$, be a differentiable function which satisfies for all $s \in [0,1]$, $t\in[0,T]$, $x \in \fg$ that $\rho_{t}(0) = e$ and $\rho_{t}'(0) = a(t,x)$, let $y \colon [0,T] \to \M$ be a differentiable function which satisfies for all $t\in[0,T]$ that $y(0) = y_0$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \tfrac{\dd}{\dd s} \Lambda\pr[\big]{ \rho_{t,a(t,x)}(s) , y(t) } \Bigr\rvert_{s=0} \dc +\end{equation} +and let $\Theta \colon [0,T] \to \fg$ satisfy for all $t \in [0,T]$ that $\Theta(0) = O$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} \Theta } \lr (t) = \dexp_{\Theta(t)}^{-1} \pr[\Big]{ a\pr[\big]{ t,y(t) } } += \dexp_{\Theta(t)}^{-1} \pr[\bigg]{ a\pr[\Big]{ t, \Lambda\pr[\big]{ \exp\pr[\big]{ \Theta(t) } , y_0 } } } +\end{equation} +(cf.\ \cref{def:manifold,def:lie_algebra,def:lie_group,def:action,% +def:alg_of_group,def:tangent,def:dexp,def:exp}). +Then there exists $T_* \in (0,T]$ such that for all $t \in [0,T_*]$ it holds that +\begin{equation} +y(t) = \Lambda\pr[\Big]{ \exp\pr[\big]{ \Theta(t) } , y_0 } \dpp +\end{equation} +\end{athm} + +\begin{aproof} + +\end{aproof} + +Note how the assumptions employed in \cref{lem:4} below differ from, e.g., \cref{lem:3} above. +Due to the particular structure we have encountered in our actual project, \cref{lem:4} below utilizes a slightly simplified setting in order to reduce the cumbersome of the ensuing notation. + +\begin{athm}{lemma}{lem:4} +Let $\M$ be a manifold, let $T \in (0,\infty)$, $y_0 \in \M$, let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, let $\fg$ be the Lie algebra of $\G$, let $\Lambda \colon \G \times \M \to \fg$ be an action, let $a \colon [0,T] \to \fg$, let $\rho_{t} \colon [0,1] \to \G$, $t\in [0,T]$, be a differentiable function which satisfies for all $s \in [0,1]$, $t\in[0,T]$ that $\rho_{t}(0) = e$ and $\rho_{t}'(0) = a(t)$, let $y \colon [0,T] \to \M$ be a differentiable function which satisfies for all $t\in[0,T]$ that $y(0) = y_0$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \tfrac{\dd}{\dd s} \Lambda\pr[\big]{ \rho_{t,a(t)}(s) , y(t) } \Bigr\rvert_{s=0} \dc +\end{equation} +let $B_0,B_1,B_2,\ldots \in \R$ satisfy for all $n \in \N_0$ that +$B_n = \sum_{i=0}^n \sum_{j=0}^i (-1)^j \binom{i}{j} \nicefrac{j^n}{(i+1)}$, +let $\Theta \colon [0,T] \to \fg$ satisfy for all $t \in [0,T]$ that $\Theta(0) = O$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} \Theta } \lr (t) = \dexp_{\Theta(t)}^{-1} \pr[\big]{ a\pr[]{ t } } += \dexp_{\Theta(t)}^{-1} \pr[\big]{ a\pr[]{ t } } \dc +\end{equation} +and let $\Theta^{[k]} \colon [0,T] \to \fg$, $k \in \N_0$, satisfy for all $k \in \N$, $t \in [0,T]$ that +$\Theta^{[0]}(t) = O$ and +\begin{equation} +\Theta^{[k]}(t) = \int_0^t \dexp_{\Theta^{[k-1]}(\xi)}^{-1} \pr[\big]{ a(\xi) } \, \dd \xi +\end{equation} +(cf.\ \cref{def:manifold,def:lie_algebra,def:lie_group,def:action,% +def:alg_of_group,def:tangent,def:dexp,def:exp}). +Then +\begin{enumerate}[label=(\roman*)] +\item it holds for all $k \in \N$, $t \in [0,T]$ that +\begin{equation} +\Theta^{[k]}(t) += \sum_{i=0}^\infty \frac{B_i}{i!} \int_0^t \ad^i\pr[\big]{ \Theta^{[k-1]}(\xi), a(\xi) } \, \dd \xi +\end{equation} +and +\item there exists $T_* \in (0,T]$ such that for all $t \in [0,T_*]$ it holds that +\begin{equation} +\Theta(t) = \lim_{k\to\infty} \Theta^{[k]}(t) +\end{equation} +\end{enumerate} +(cf.\ \cref{def:adjoint}). +\end{athm} + +\begin{aproof} + +\end{aproof} + +\begin{athm}{remark}{rem:3} +Armed with \cref{lem:4}, we are now in a position to construct structure-preserving numerical approximations to \cref{eq:3_1}. +These approximations will proceed via two steps. +First, we will choose some $m \in \N_0$ and utilize $\Theta^{[m]}$ in our approximation method: this requires proving a so-called \emph{convergence result} for the Picard iterates. +Next, we will construct an appropriate approximation to the exponential map: this will be accomplished via certain \emph{rational function} approximation techniques. +These two issues will be discussed further as these notes continue. +\end{athm} + +We now (informally) continue the line of thinking developed in \cref{lem:4} above. +First, note that direct calculations yield that for all $t \in [0,T]$ it holds that +\begin{equation}\label{expand1} +\Theta^{[1]}(t) = \int_0^t a(\xi_1) \, \dd \xi_1 \dc +\end{equation} +\begin{equation}\label{expand2} +\begin{split} +\Theta^{[2]}(t) +& = \int_0^t a(\xi_1) \, \dd \xi_1 +- \tfrac{1}{2} \int_0^t \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , a(\xi_1) \right] \, \dd \xi_1 \\ +& \qquad + \tfrac{1}{12} \int_0^t \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , a(\xi_1) \right] \right] \, \dd \xi_1 + \ldots \dc +\end{split} +\end{equation} +and +\begin{equation}\label{expand3} +\begin{split} +\Theta^{[3]}(t) +& = \int_0^t a(\xi_1) \, \dd \xi_1 +- \tfrac{1}{2} \int_0^t \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , a(\xi_1) \right] \, \dd \xi_1 \\ +& \qquad + \tfrac{1}{12} \int_0^t \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , a(\xi_1) \right] \right] \, \dd \xi_1 \\ +& \qquad + \tfrac{1}{4} \int_0^t \left[ \int_0^{\xi_1} \left[ \int_0^{\xi_2} a(\xi_3) \, \dd\xi_3 , a(\xi_2) \right] \, \dd\xi_2, a(\xi_1) \right] \, \dd\xi_1 \\ +& \qquad - \tfrac{1}{24} \int_0^t \left[ \int_0^{\xi_1} \left[ \int_0^{\xi_2} a(\xi_3) \, \dd\xi_3 , \left[ \int_0^{\xi_2} a(\xi_3) \, \dd\xi_3 , a(\xi_2) \right] \right] \, \dd\xi_2 , a(\xi_1) \right] \, \dd \xi_1 \\ +& \qquad - \tfrac{1}{24} \int_0^t \left[ \int_0^{\xi_1} \left[ \int_0^{\xi_2} a(\xi_3) \, \dd\xi_3 , a(\xi_2) \right] \, \dd \xi_2 , \left[ \int_0^{\xi_1} a(\xi_2) \, \dd\xi_2 , a(\xi_1) \right]\right] \, \dd\xi_1 \\ +& \qquad - \tfrac{1}{24} \int_0^t \left[ \int_0^{\xi_1} a(\xi_2) \, \dd \xi_2 , \left[ \int_0^{\xi_1} \left[ \int_0^{\xi_2} a(\xi_3) \, \dd\xi_3 , a(\xi_2) \right] \, \dd \xi_2 , a(\xi_1) \right] \right ] \, \dd \xi_1 + \ldots \dpp +\end{split} +\end{equation} +Observe that we could have continued these calculations, but things become increasingly tedious (however, things can be simplified considerably via the use of graph theory). +%%%Next, note that \cref{expand1,expand2,expand3} seem to indicate that one could construct $H_k \colon [0,T] \to \fg$, $k\in\N_0$, which satisfy for all $k \in \N_0$, $t\in[0,T]$ that $H_k$ is a linear combination of terms including exactly $k+1$ integrals (or, equivalently, $k$ Lie brackets) and +%%%\begin{equation} +%%%\Theta(t) = \sum_{k=0}^\infty H_k(t) \dpp +%%%\end{equation} +Next, note that even when we keep all (infinitely many) terms in \cref{expand2} or \cref{expand3}, we are only able to approximate the function $\Theta$ with limited accuracy (this is a direct consequence of the Picard-Lindel{\"o}f theorem (cf., e.g., \url{https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem}). +That is, let $\norm{\cdot} \colon \fg \to [0,\infty)$ be an appropriate norm. +Then, there exists $C_k \in \R$, $k\in\N_0$, and $f \colon \N_0 \to [0,\infty)$ such that for all $k\in\N_0$, $t \in [0,T_*]$ it holds that +\begin{equation} +\norm[\big]{ \Theta(t) - \Theta^{[k]}(t) } \le C_k t^{f(k)} \dpp +\end{equation} +This indicates that keeping \emph{all} terms in the low-level expansions is not ideal. +This motivates the result in \cref{lem:5} below. + +\begin{athm}{lemma}{lem:5} +Let $\M$ be a manifold, let $T \in (0,\infty)$, $y_0 \in \M$, let $\G$ be a Lie group, let $e \in \G$ satisfy for all $p \in \G$ that $p \cdot e = e \cdot p = p$, let $\fg$ be the Lie algebra of $\G$, let $\Lambda \colon \G \times \M \to \fg$ be an action, let $a \colon [0,T] \to \fg$, let $\rho_{t} \colon [0,1] \to \G$, $t\in [0,T]$, be a differentiable function which satisfies for all $s \in [0,1]$, $t\in[0,T]$ that $\rho_{t}(0) = e$ and $\rho_{t}'(0) = a(t)$, let $y \colon [0,T] \to \M$ be a differentiable function which satisfies for all $t\in[0,T]$ that $y(0) = y_0$ and +\begin{equation}\label{lem:5_ode} +\pr[\big]{ \tfrac{\dd}{\dd t} y } \lr (t) = \tfrac{\dd}{\dd s} \Lambda\pr[\big]{ \rho_{t,a(t)}(s) , y(t) } \Bigr\rvert_{s=0} \dc +\end{equation} +let $B_0,B_1,B_2,\ldots \in \R$ satisfy for all $n \in \N_0$ that +$B_n = \sum_{i=0}^n \sum_{j=0}^i (-1)^j \binom{i}{j} \nicefrac{j^n}{(i+1)}$, +let $\Theta \colon [0,T] \to \fg$ satisfy for all $t \in [0,T]$ that $\Theta(0) = O$ and +\begin{equation} +\pr[\big]{ \tfrac{\dd}{\dd t} \Theta } \lr (t) = \dexp_{\Theta(t)}^{-1} \pr[\big]{ a\pr[]{ t } } += \dexp_{\Theta(t)}^{-1} \pr[\big]{ a\pr[]{ t } } \dc +\end{equation} +and let $\Omega^{[k]} \colon [0,T] \to \fg$, $k \in \N_0$, satisfy for all $k \in \N$, $t \in [0,T]$ that +$\Omega^{[0]}(t) = O$ and +\begin{equation}\label{eq:3_23} +\Omega^{[k]}(t) = \sum_{i=0}^{k-1} \frac{B_i}{i!} \int_0^t \ad^i\pr[\big]{ \Omega^{[k-1]}(\xi), a(\xi) } \, \dd \xi +\end{equation} +(cf.\ \cref{def:manifold,def:lie_algebra,def:lie_group,def:action,% +def:alg_of_group,def:tangent,def:dexp,def:exp,def:adjoint}). +Then +\begin{enumerate}[label=(\roman*)] +\item\label{i1} there exists $T_* \in (0,T]$ such that for all $t \in [0,T_*]$ it holds that +\begin{equation} +\Theta(t) = \lim_{k\to\infty} \Omega^{[k]}(t) +\end{equation} +and +\item\label{i2} it holds for all $k \in \N_0$, $t \in [0,T_*]$, and $\norm{\cdot} \colon \fg \to [0,\infty)$ which satisfy for all $u,v \in \fg$, $c \in \R$ that $\norm{u+v} \le \norm{u} + \norm{v}$, $\norm{cu} = \abs{c}\norm{u}$, and $\norm{u} = 0$ if and only if $u=O$ that there exists $C \in \R$ such that +\begin{equation}\label{eq:3_25} +\norm[\big]{ \Theta(t) - \Omega^{[k]}(t) } \le C t^{k+1} \dpp +\end{equation} +\end{enumerate} +\end{athm} + +\begin{aproof} + +\end{aproof} + +It is now important to emphasize what is implied by \cref{lem:5} above. +\begin{enumerate}[label=\arabic*.] +\item \Cref{i1} in \cref{lem:5} above demonstrates that we can truncate the Picard iterates and still obtain the desired convergence to the true solution. \emph{Note:} We have purposely avoided the issue of discussing\slash defining what a limit means in a Lie algebra, but hopefully this is clear. +\item \Cref{i2} in \cref{lem:5} above shows that we may choose our Picard iterate based on the amount of accuracy we desire. +This can be seen in the right-hand side of \cref{eq:3_25}. +\item It should be clear that \cref{i2} in \cref{lem:5} holds true if we can prove the result for one such $\norm{\cdot} \colon \fg \to [0,\infty)$ as we are in a finite-dimensional setting. +\item Something which is hidden in the details (and often misunderstood) is the constant ``$C$'' in \cref{i2} in \cref{lem:5}. +This constant is a function of the underlying smoothness of the function $a$, as it will be the result of trying to bound the nested commutators within the Picard iterates. +For our project we will not need to concern ourselves with this constant too much; we only need to understand its role and how it can be controlled. +\end{enumerate} + +\begin{athm}{problem}{prob:1} +Verify \cref{i2} in \cref{lem:5} for the case $k=1$. Feel free to impose as much regularity (i.e., increase the assumptions on the objects introduced in \cref{lem:5}) if you feel this will help. +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\begin{athm}{problem}{prob:2} +Verify \cref{i2} in \cref{lem:5} in the case where $k=2$, $T = 1$, $\M = \G = \operatorname{SO}(3)$ and $\Lambda \colon \G \times \G \to \fg$ satisfies for all $a,b \in \G$ that $\Lambda(a,b) = ab$. +If you want to simplify things further, you may let $\alpha \in \fg$ and choose $a \colon [0,T] \to \fg$ to satisfy for all $t \in [0,T]$ that $a(t) = t\alpha$. +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\subsection{A first attempt at a numerical approximation} + +We will now (somewhat informally) motivate the construction of a numerical approximation to, e.g., \cref{lem:5_ode}. +To that end, we will commence by fixing $k=1$. +This and \cref{eq:3_23} yield that for all $t \in [0,T_*]$ it holds that +\begin{equation}\label{eq:3_26} +\Omega^{[1]}(t) = B_0 \int_0^t \ad^0\pr[\big]{ \Omega^{[0]}(\xi) , a(\xi) } \, \dd \xi += \int_0^t a(\xi) \, \dd \xi \dpp +\end{equation} +We now apply an approximation to the integral in \cref{eq:3_26} (since, in general, we will not be able to compute the integral exactly) to obtain that for all $t\in[0,T_*]$ it holds that +\begin{equation}\label{eq:3_27} +\Omega^{[1]}(t) \approx t a(0) \dpp +\end{equation} + +\begin{athm}{problem}{prob:3} +Let $T,L \in (0,\infty)$, let $\fg$ be a Lie algebra, let $\norm{\cdot} \colon \fg \to [0,\infty)$ satisfy for all $u,v \in \fg$, $c \in \R$ that $\norm{u+v} \le \norm{u} + \norm{v}$, $\norm{cu} = \abs{c}\norm{u}$, and $\norm{u} = 0$ if and only if $u=O$, and let $a \colon [0,T] \to \fg$ satisfy for all $s,t \in [0,T]$ that $\norm{a(t) -a(s)} \le L\abs{t-s}$ (cf.\ \cref{def:lie_algebra}). +Show that for all $t\in[0,T]$ it holds that +\begin{equation} +\norm*{ \int_0^t a(\xi) \, \dd \xi - t a(0) } \le \frac{Lt^2}{2} \dpp +\end{equation} +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +Combining \cref{lem:3,lem:5,eq:3_27} demonstrate that for all $t\in[0,T_*]$ it holds that +\begin{equation} +y(t) = \Lambda\pr[\Big]{ \exp\pr[\big]{ \Theta(t) } , y_0 } \approx \Lambda\pr[\Big]{ \exp\pr[\big]{ \Omega^{[1]}(t) } , y_0 } = \Lambda\pr[\Big]{ \exp\pr[\big]{ t a(0) } , y_0 } \dpp +\end{equation} +Provided that the group action, $\Lambda$, and $\exp(ta(0))$ can be computed with relative ease, we have arrived at a reasonable numerical approximation. +Moreover, the approximations employed thus far have been linear approximations performed on objects in the Lie algebra---thus, our approximation will still lie in the original manifold for all $t\in[0,T_*]$. + +\begin{athm}{problem}{prob:4} +Develop a similar approximation using $\Omega^{[2]}$ (cf.\ \cref{lem:5}). +Note that in this case, you will likely need to impose additional regularity assumptions (be sure to only impose precisely enough to derive your result). +\end{athm} + +\begin{asol} +\linebreaks{30} +\end{asol} + +\subsection{Approximating the exponential map} + + + +\end{document} \ No newline at end of file diff --git a/Notes/Numerical Methods on Manifolds/Numerical_Methods_on_Manifolds.toc b/Notes/Numerical Methods on Manifolds/Numerical_Methods_on_Manifolds.toc new file mode 100644 index 0000000..c91effc --- /dev/null +++ b/Notes/Numerical Methods on Manifolds/Numerical_Methods_on_Manifolds.toc @@ -0,0 +1,5 @@ +\contentsline {section}{\numberline {1}Introduction}{1}{section.1}% +\contentsline {section}{\numberline {2}Some basic background material}{1}{section.2}% +\contentsline {section}{\numberline {3}Approximating differential equations on manifolds}{8}{section.3}% +\contentsline {subsection}{\numberline {3.1}A first attempt at a numerical approximation}{16}{subsection.3.1}% +\contentsline {subsection}{\numberline {3.2}Approximating the exponential map}{18}{subsection.3.2}% diff --git a/Programming/A Primer on Scientific 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2011-03-28T13:13:15+02:00 + 2011-03-28T15:40:03+02:00 + gnuplot 4.2 patchlevel 0 + + + application/pdf + + + uuid:83ad863a-b3c7-46f3-a41d-6917348ede0c + uuid:11e6414a-9f36-4e5d-b8a9-aa52057e381b + + + dvips + + + + endstream endobj 1302 0 obj <> endobj 1305 0 obj <> endobj 530 0 obj <> endobj 529 0 obj <> endobj 528 0 obj <> endobj 527 0 obj <> endobj 526 0 obj <> endobj 525 0 obj <> endobj 524 0 obj <> endobj 523 0 obj <> endobj 522 0 obj <> endobj 521 0 obj <> endobj 585 0 obj <> endobj 584 0 obj <> endobj 583 0 obj <> endobj 582 0 obj <> endobj 581 0 obj <> endobj 580 0 obj <> endobj 579 0 obj <> endobj 578 0 obj <> endobj 577 0 obj <> endobj 576 0 obj <> endobj 575 0 obj <> endobj 574 0 obj <> endobj 573 0 obj <> endobj 572 0 obj <> endobj 571 0 obj <> endobj 570 0 obj <> endobj 569 0 obj <> endobj 568 0 obj <> endobj 567 0 obj <> endobj 566 0 obj <> endobj 565 0 obj <> endobj 564 0 obj <> endobj 563 0 obj <> endobj 562 0 obj <> endobj 561 0 obj <> endobj 560 0 obj <> endobj 559 0 obj <> endobj 558 0 obj <> endobj 557 0 obj <> endobj 556 0 obj <> endobj 555 0 obj <> endobj 554 0 obj <> endobj 618 0 obj <> endobj 617 0 obj <> endobj 616 0 obj <> endobj 615 0 obj <> endobj 614 0 obj <> endobj 613 0 obj <> endobj 1306 0 obj <> endobj 1356 0 obj <> stream + + + + + 2011-03-28T15:39:55+02:00 + 2011-03-28T13:13:15+02:00 + 2011-03-28T15:39:55+02:00 + gnuplot 4.2 patchlevel 0 + + + application/pdf + + + uuid:5257ced6-e4d7-45c5-b27f-88bd4753b569 + uuid:6902529f-0f69-459b-9c15-ba06f2d6a121 + + + dvips + + + + endstream endobj 1355 0 obj <> endobj 1358 0 obj <> endobj 612 0 obj <> endobj 611 0 obj <> endobj 610 0 obj <> endobj 609 0 obj <> endobj 608 0 obj <> endobj 607 0 obj <> endobj 606 0 obj <> endobj 605 0 obj <> endobj 604 0 obj <> endobj 603 0 obj <> endobj 602 0 obj <> endobj 601 0 obj <> endobj 600 0 obj <> endobj 599 0 obj <> endobj 598 0 obj <> endobj 597 0 obj <> endobj 596 0 obj <> endobj 595 0 obj <> endobj 594 0 obj <> endobj 593 0 obj <> endobj 592 0 obj <> endobj 591 0 obj <> endobj 590 0 obj <> endobj 589 0 obj <> endobj 588 0 obj <> endobj 587 0 obj <> endobj 651 0 obj <> endobj 650 0 obj <> endobj 649 0 obj <> endobj 648 0 obj <> endobj 647 0 obj <> endobj 646 0 obj <> endobj 645 0 obj <> endobj 644 0 obj <> endobj 643 0 obj <> endobj 642 0 obj <> endobj 641 0 obj <> endobj 1359 0 obj <> endobj 1398 0 obj <> stream + + + + + 2011-03-28T15:39:49+02:00 + 2011-03-28T13:13:15+02:00 + 2011-03-28T15:39:49+02:00 + gnuplot 4.2 patchlevel 0 + + + application/pdf + + + uuid:857c4e17-4eb5-46d4-b708-af01373140d2 + uuid:92b65002-8533-4708-ba57-318e8183fac1 + + + dvips + + + + endstream endobj 1397 0 obj <> endobj 1400 0 obj <> endobj 640 0 obj <> endobj 639 0 obj <> endobj 638 0 obj <> endobj 637 0 obj <> endobj 636 0 obj <> endobj 635 0 obj <> endobj 634 0 obj <> endobj 633 0 obj <> endobj 632 0 obj <> endobj 631 0 obj <> endobj 630 0 obj <> endobj 629 0 obj <> endobj 628 0 obj <> endobj 627 0 obj <> endobj 626 0 obj <> endobj 625 0 obj <> endobj 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uuid:ba95d11e-9f29-4703-ae9f-f2cbb749956c + uuid:f70fe281-b95c-40ed-81b0-03b7afdd61b4 + + + dvips + + + + endstream endobj 1495 0 obj <> endobj 1498 0 obj <> endobj 1499 0 obj <> stream + + + + + 2011-04-03T04:17:34+02:00 + 2011-04-03T04:17:34+02:00 + 2011-04-03T04:17:34+02:00 + Adobe Acrobat 9.4 + + + application/pdf + + + uuid:f6343111-626f-43be-b530-f8b526693399 + uuid:83dfd73f-aec8-4c80-9fa2-e38328559b6b + + + Adobe Acrobat 9.4 Image Conversion Plug-in + + + + endstream endobj 2 0 obj <> endobj 1502 0 obj <> endobj 960 0 obj <> endobj 1025 0 obj <> endobj 1092 0 obj <> endobj 1167 0 obj <> endobj 1234 0 obj <> endobj 1304 0 obj <> endobj 1357 0 obj <> endobj 1399 0 obj <> endobj 1443 0 obj <> endobj 1497 0 obj <> endobj 746 0 obj <> endobj 745 0 obj <> endobj 744 0 obj <> endobj 743 0 obj <> endobj 742 0 obj <> endobj 741 0 obj <> endobj 740 0 obj <> endobj 739 0 obj <> endobj 738 0 obj <> endobj 737 0 obj <> endobj 736 0 obj <> endobj 735 0 obj <> endobj 734 0 obj <> endobj 733 0 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files/.ipynb_checkpoints/NonUniformFiniteDifference-checkpoint.ipynb @@ -0,0 +1,1483 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 109, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "\n", + "def f(x):\n", + " return np.sin(np.pi * x)\n", + "\n", + "x = np.linspace(0,1,100)\n", + "x = x**2\n", + "\n", + "h = np.diff(x) # we define the vector h as the difference of successive x[i] using the built in diff function\n", + "\n", + "f = f(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 110, + "metadata": {}, + "outputs": [], + "source": [ + "A = np.zeros((len(f),len(f))) #we create the matrix A by initializing to 0" + ] + }, + { + "cell_type": "code", + "execution_count": 111, + "metadata": {}, + "outputs": [], + "source": [ + "for i in range(1,len(f)-1):\n", + " A[i,i-1] = 2/(h[i-1]*(h[i-1] + h[i])) #we populate the diagonal (minus the extreme ends), the one to the left and right with a,b,c respectively\n", + " A[i,i] = -2/(h[i-1] * h[i])\n", + " A[i,i+1] = 2/(h[i]*(h[i] + h[i-1]))\n", + "\n", + "A[0,0] = 1 # we denote the far corners of our matrix to be 1\n", + "A[len(f)-1,len(f)-1] = 1\n" + ] + }, + { + "cell_type": "code", + "execution_count": 112, + "metadata": {}, + "outputs": [], + "source": [ + "u = np.linalg.inv(A).dot(f) # we solve our system" + ] + }, + { + "cell_type": "code", + "execution_count": 113, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 113, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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    " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt #we plot the result\n", + "\n", + "plt.plot(x,u)" + ] + }, + { + "cell_type": "code", + "execution_count": 115, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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    " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 115, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from sympy import *\n", + "import math\n", + "\n", + "init_printing(use_unicode=False, wrap_line=False)\n", + "x = Symbol('x')\n", + "\n", + "plot(integrate(integrate(sin(pi*x),x),x))\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "interpreter": { + "hash": "a7b5a1112396ad393f5588006c146c6954be07cba002f50e558ad6b83bb9990c" + }, + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.4" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Programming/Python files/.ipynb_checkpoints/Nonuniform_JLP-checkpoint.ipynb b/Programming/Python files/.ipynb_checkpoints/Nonuniform_JLP-checkpoint.ipynb new file mode 100644 index 0000000..ccb3501 --- /dev/null +++ b/Programming/Python files/.ipynb_checkpoints/Nonuniform_JLP-checkpoint.ipynb @@ -0,0 +1,272 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 68, + "id": "1dc137ec", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "\n", + "def f(x):\n", + " return np.sin(np.pi * x)\n", + " #return 1 - 4*(x-0.5)**2\n", + "\n", + "def nonuniform_grid(num_points,left_pt,right_pt,is_non_unif=True):\n", + " grid = np.array([(float(is_non_unif)/(num_points+1-0.5*(num_points+2))**2)*((i+1)-0.5*(num_points+2))**2 + 1 for i in range(num_points+1)])\n", + " grid *= (right_pt-left_pt)/sum(grid)\n", + " final_grid = np.zeros(num_points+2)\n", + " final_grid[0] = left_pt\n", + " for i in range(1,num_points+1) :\n", + " final_grid[i] = final_grid[i-1]+grid[i-1]\n", + " final_grid[num_points+1] = right_pt\n", + " return grid, final_grid" + ] + }, + { + "cell_type": "code", + "execution_count": 69, + "id": "6391d3e7", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "\n", + "num_points = 100\n", + "h_vals, grid = nonuniform_grid(num_points,0,1)\n", + "#grid = np.linspace(0,1,num_points+2)\n", + "#h_vals = np.diff(grid)\n", + "\n", + "plt.figure(figsize=(12,7.5))\n", + "\n", + "# Plot 1\n", + "plt.subplot(1,2,1)\n", + "plt.plot(grid)\n", + "plt.title(\"Plot of nonuniform grid\")\n", + "plt.xlabel(\"Grid point index\")\n", + "plt.ylabel(\"Domain value\")\n", + "\n", + "# Plot 2\n", + "plt.subplot(1,2,2)\n", + "plt.plot(grid[:-1],h_vals)\n", + "plt.title(\"Comparison of grid spacing\")\n", + "plt.xlabel(\"Point in domain\")\n", + "plt.ylabel(\"Grid spacing\")\n", + "\n", + "plt.tight_layout()\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "id": "ac600ea8", + "metadata": {}, + "outputs": [], + "source": [ + "# Construct the coefficient matrix\n", + "A = np.zeros((num_points,num_points))\n", + "\n", + "for i in range(num_points) :\n", + " A[i,i] = -2./(h_vals[i]*h_vals[i+1])\n", + " \n", + "for i in range(num_points-1) :\n", + " A[i+1,i] = 2./(h_vals[i+1]*(h_vals[i+1]+h_vals[i+2]))\n", + " A[i,i+1] = 2./(h_vals[i+1]*(h_vals[i]+h_vals[i+1]))" + ] + }, + { + "cell_type": "code", + "execution_count": 71, + "id": "7343d63c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[ -9343.45827382, 4718.44170984, 0. , ...,\n", + " 0. , 0. , 0. ],\n", + " [ 4813.73429872, -9724.66514761, 4910.93084888, ...,\n", + " 0. , 0. , 0. ],\n", + " [ 0. , 5010.04771244, -10121.16880196, ...,\n", + " 0. , 0. , 0. ],\n", + " ...,\n", + " [ 0. , 0. , 0. , ...,\n", + " -10121.16880196, 5010.04771244, 0. ],\n", + " [ 0. , 0. , 0. , ...,\n", + " 4910.93084888, -9724.66514761, 4813.73429872],\n", + " [ 0. , 0. , 0. , ...,\n", + " 0. , 4718.44170984, -9343.45827382]])" + ] + }, + "execution_count": 71, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Check the resulting matrix's form\n", + "A" + ] + }, + { + "cell_type": "code", + "execution_count": 72, + "id": "be8ce9d4", + "metadata": {}, + "outputs": [], + "source": [ + "# Solve the discrete system\n", + "\n", + "bc_left = 0\n", + "bc_right = 0\n", + "\n", + "rhs = f(grid[1:-1])\n", + "rhs[0] -= (2/(h_vals[0]*(h_vals[0]+h_vals[1])))*bc_left\n", + "rhs[-1] -= (2/(h_vals[-1]*(h_vals[-2]+h_vals[-1])))*bc_right\n", + "\n", + "u = np.linalg.solve(A,rhs)" + ] + }, + { + "cell_type": "code", + "execution_count": 73, + "id": "ec5cd1a5", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Add boundary conditions\n", + "\n", + "U = np.append(np.array([bc_left]),np.append(u,bc_right))\n", + "RHS = np.append(np.array([bc_left]),np.append(rhs,bc_right))\n", + "\n", + "plt.figure(figsize=(12,7.5))\n", + "plt.plot(grid,U,label=\"$u(x)$\")\n", + "plt.plot(grid,RHS,label=\"$f(x)$\")\n", + "plt.title(\"Solution vs. right-hand side function\")\n", + "plt.xlabel(\"$x$\")\n", + "plt.ylabel(\"$u(x)$\")\n", + "plt.legend()\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "id": "6a80c620", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Check error\n", + "\n", + "def error_check(approx,true,grid) :\n", + " error_grid = np.append(0.5*(grid[:-1]+grid[1:]),grid[-1])\n", + " return np.sqrt(np.multiply(error_grid,(approx-true)**2))\n", + "\n", + "true_sol = -(1/np.pi**2)*np.sin(np.pi*grid)\n", + "\n", + "error = error_check(U,true_sol,grid)\n", + "\n", + "plt.figure(figsize=(12,7.5))\n", + "plt.plot(grid,error)\n", + "plt.title(\"Pointwise error\")\n", + "plt.ylabel(\"Error\")\n", + "plt.xlabel(\"$x$\")\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 75, + "id": "6ae3c077", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.000910607929436465" + ] + }, + "execution_count": 75, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# L2 error\n", + "sum(error)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "aee347c7", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.4" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/Programming/Python files/Backward_Euler.ipynb b/Programming/Python files/Backward_Euler.ipynb new file mode 100644 index 0000000..4b1b82b --- /dev/null +++ b/Programming/Python files/Backward_Euler.ipynb @@ -0,0 +1,157 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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2OwkJCfz73/+mb9++LVoW5X5aw8+4aqSDq+HTRyBzL8ROgBl/gi5Dmu3yxpgUEUmqaZvW0CvZs2cPffr0YerUqRrmSqlzZR+Ef14F784DWyEseA+uX9GsYV6fVvuhaEsYNGgQhw4dauliKKVak6Ic+PIZ2PwqePnCtCdh1O3W+1ZGA10ppWpiL4Ot78Dq30NBNgy/BqY8BoGRLV2yWmmgK6VUVYfXWd0Q03dB9GiYuRS6DWvpUtWr3jZ0Y0wPY8waY8weY8y3xph7athnkjEmxxiz3fF6rGmKq5RSTej0EfjgOnh7DhSdgcvfgp9+0ibCHJyroduA+0RkqzEmEEgxxnwuInuq7LdOROoeek8ppVqj4lxY9zxsfBk8PGHyb2DMneDdoaVLdl7qraGLyAkR2ep4nwvsBeofZk+xfPlyjDEVfZ3r0tjR/hYtWsSdd97Z4OOVapfsdtj2HryUCOufh8Hz4K4UmPjrNhfmcJ7dFo0xMcBwoPrz3DDaGLPDGPOJMabG5/aNMbcaY5KNMcmZmZnnX9o2ZvHixYwbN86pR811+FalmtkPm+H1KfDRLyCoB9zyBVz2KnSq/Qnw1s7pQDfGdASWAr8UkbNVNm8FeorIUOAlYHlN5xCR10QkSUSSwsPDG1jkpvPss89WDOf6q1/9iilTrGEuV69ezTXXXHNe58rLy2P9+vW88cYbFY+EgzWo1v3338+QIUOIj4/npZdeqnH41soTJCxZsqRiCNiPP/6YkSNHMnz4cKZNm1YxrKtSykk5qbDkZnjzQshNh3mvwc2fQ1SNz+q0KU71cjHGeGOF+Xsisqzq9soBLyIrjTF/M8aEiUhWg0v2yUNwcleDD69RlziY9XStm8ePH89f/vIX7r77bpKTkykuLqa0tJR169ZVjKDorI8++oiZM2fSr18/QkNDSUlJITExkddee40jR46wfft2vLy8OHXqFCEhITz//POsWbOGsLCwOs87btw4Nm3ahDGG119/nWeeeYa//OUv51U2pdqlkgLY8IL1QmDCAzDul+Dj/MB2rV29gW6sIeTeAPaKyPO17NMFSBcRMcaMwKr5Z9e0b2uWmJhISkoKZ8+exdfXl4SEBJKTk1m3bt05EzE4Y/Hixdxzj9Uh6KqrrmLx4sUkJiayatUqbrvttopRGENCQs7rvKmpqSxYsIATJ05QUlJCbGzseR2vVLsjAruWwKrH4WwaDL4Mpj8JnaObvSh2u/DZnpPEhAUwoEsnl5/fmRr6WOA6YJcxZrtj3SNANICIvAJcDtxujLEBhcBV0thBYuqoSTcVb29vYmNjWbRoEWPGjCE+Pp41a9Zw4MCBauNvbN68mZ///OcA/O53vztnnO5Tp06xevVqdu3ahTGGsrIyjDE8++yzTpel8lCslYe2veuuu7j33nuZO3cua9eu5Yknnmjg3SrVDqSlWH/tp34DXYfC/DegZ81j4DelYlsZy7el8eqXhziUlc/1o3vyu0tcP2RAvYEuIuuBOseBFJGFwEJXFaoljR8/nueee44333yTuLg47r33XhITE6uNdT1y5Ei2b99e4zmWLFnCddddx6uvvlqxbuLEiaxbt47p06fz6quvMnny5HOaXMqHby1vcomMjGTv3r3079+fDz/8kMDAQABycnIq5vJ8++23m+AroJQbOHsCvngSdiyGgAiYuxCGXQMezTt8VW5RKYu/+YE31h8m/WwxQ7p3YuFPhjNrSNf6D24AHZyrivHjx3PixAlGjx5NZGQkfn5+FWNeO2vx4sUVswSVmz9/PosXL+aWW24hOjqa+Ph4hg4dyj//+U+g+vCtTz/9NHPmzGHMmDHnzEz0xBNPcMUVV5CYmFhve7tS7U5pEXz1nNUNcfdSGPtLqxtiwnXNGuaZucU8+799jHl6NX9cuY8+ER159+YRfHznOObEd8PTo2nGStfhc5VqRvoz3kREYM9H8Plv4cwPMGAOXPgUhPRq1mL8kF3Aa+sO8q/kVErL7Mwa0oWfT+jN0B6dXXaNuobP1bFclFJt24md8OlDcHQDRA6BGz62xilvRt8ez+GVLw/x353H8fLwYH5id342vhe9wjvWf7ALaaArpdqmvAxY/RRsfRf8Q2DO/0HCDdaj+81ARNh4KJu/rz3Iuu+z6Ojrxc8m9OKmsbFEdvJrljJUpYGulGpbbCWw+RVrjHJboTXt24RfQ4fOzXL58q6Hf197kB2pOYR19OWBmf25ZmRPgjp4N0sZaqOBrpRqG0Rg/yfw2aNw6hD0nQEz/gBhzTO7WNWuhz1D/fnDvCHMT4jCz7t5/iqojwa6Uqr1S98D/3sYDq2FsP5w7VLoM61ZLl1X18Om6q3SUBroSqnWq+AUrPkDJL8Jvp1g1jOQdBN4Nn3TRmZuMYu+Psw7G4+SW2RjbJ9QnrtiKOP6hFV7LqW10EB3IU9PT+Li4iqWr7rqKh566KEWLJFSbVRZKWx5Hdb+CYrz4IJbYNLD1oefTaw5uh42FQ10F+rQoUOtT4+WKysrw9PTs9ZlZ49Tym19v8pqXsn6DnpPgRl/hIim77vfWroeNoY+KVrJK6+8wrBhwxg2bBixsbEVT202VkxMDA8++CAJCQn8+9//rra8ePFi4uLiGDJkCA8++GDFcR07duS+++5j6NChbNy40SVlUarVyvoe3rsC3ptvTdB89Qdw7bImDXMR4euDWVz3xmYuenE9a/Zl8LMJvVj/4GT+dFl8mwpzaMU19D9/82f2nap/pp/zMSBkAA+OeLDW7bfddhu33XYbpaWlTJkyhXvvvfe8zl9YWMiwYcMqlh9++GEWLFgAQGhoKFu3bgXgoYceqlg+fvw4o0aNIiUlheDgYC688EKWL1/OpZdeSn5+PiNHjtThcZV7KzxtdUH85jXw9ocLfw8jfg5ePk12ydbc9bAxWm2gt6R77rmHKVOmcPHFF5/XcXU1uZQHe9XlLVu2MGnSJMon/Ljmmmv46quvuPTSS/H09GT+/PnnfwNKtQVlNti6CFb/wQr1xBusuTw7Nt3kN22h62FjtNpAr6sm3ZQWLVrE0aNHWbiw+uCRdQ2ZW5+AgIA6l2vi5+en7ebKPR1aC58+DBl7IGY8zPyTNQFNE2lLXQ8bo9UGektISUnhueeeY926dXjUMDJbXUPmNtSIESO4++67ycrKIjg4mMWLF3PXXXe59BpKtRrZB+Hzx2Dff6BzT7jyXRh4MTRRN8C22PWwMTTQK1m4cCGnTp2q+DA0KSmJ119/3enjq7ahz5w5k6efrnuijq5du/L0008zefJkRISLLrqISy65pEHlV6rVKjoL656DTX8HTx+Y+hiMugO8m2bMk5q6Ht42sTfxUZ2b5HqthQ6fq1Qzanc/4/Yy2P4efPEU5GdYk0xMfQwCuzTJ5dyh62F9dPhcpVTzO/o1fPIgnNwJPUbBTz6A7gkuv0xtox7ePDaWiBYa9bClaKArpVzr9FGrnXzPcugUZc3jOWS+y9vJ3bXrYWO0ukAXEbf8sEKplmrebDbFebDhr7DhRTAeMOkRGHMX+Pi79jJu3vWwMVpVoPv5+ZGdnU1oaKiGunIrIkJ2djZ+fm7YBGC3w65/waonIPcExF0J056AoO4uvUx76XrYGK0q0KOiokhNTSUzM7Oli6KUy/n5+REVFdXSxXAdexns+y+s/z84vhW6JcCV70CPES69THvretgYrSrQvb29iY2NbeliKKXqUpxn9VzZ9Dc4fcTqT37pKxC/AGp4fqOh2mvXw8ZoVYGulGrFctKs8VZS3oKiHOgxEqY/BQMucuk8nu2h62FT0UBXStXtxA7Y+DLsXgpih4FzYfSd0OMCl11Cux66hga6Uqo6ux2+/ww2LoQj68CnI4y4FUbeBsE9XXgZ7XroShroSqkflRbCjvet9vGs76BTd6tZJeF66NDZZZfRrodNo95AN8b0AN4BIgEBXhORF6rsY4AXgNlAAXCjiGx1fXGVUk0iL8Oa8m3L61CQDV2HWQ8EDbrEpfN3atfDpuVMDd0G3CciW40xgUCKMeZzEdlTaZ9ZQF/HayTwd8e/SqnWLGOv1T6+819QVgL9Z1nt4z3HuPTJzpq6Hv7limGM7aPPnLhSvYEuIieAE473ucaYvUB3oHKgXwK8I9ajcJuMMZ2NMV0dxyqlWhMRazzyjQvhwCrw6gDDr4VRv4CwPi69lHY9bF7n1YZujIkBhgObq2zqDhyrtJzqWHdOoBtjbgVuBYiOjj7PoiqlGsVWbPVU2fgypO+GgAiY8htIvAkCQl16Ke162DKcDnRjTEdgKfBLETnbkIuJyGvAa2ANn9uQcyilzlPBKUh+E775B+SdhIhBcMnfIO5y8PJ12WW062HLcyrQjTHeWGH+nogsq2GXNKBHpeUoxzqlVEvJPmj1Vtn+TygtgN5T4dK/Qe8pLm0f166HrYczvVwM8AawV0Ser2W3FcCdxpj3sT4MzdH2c6VagAj8sBG+Xgj7V1o9VOKvtGYHihzk0kudyClk2dY0lqSkctjR9fCP8+K4LKG7dj1sIc7U0McC1wG7jDHbHeseAaIBROQVYCVWl8UDWN0Wf+rykiqlaldms8Yf37gQjm+DDiEw4ddwwS0QGOmyyxSWlPHZnpMsSUll/YEsRGBETAj3XdhPux62As70clkP1PldcvRuucNVhVJKOakoB7a+A5tfhZxjENoHLnoehl7tsnHIRYSUo6dZkpLKf3eeILfYRlRwB+6e0pf5CVFEh7p2vHPVcPqkqFJt0ZkfYNMrVpiX5ELMeJj9LPSd4bIRD1NPF/Dh1jSWbk3lSHYB/j6ezI7ryvyEKEbGhuChtfFWRwNdqbYkNQU2vgR7VlgfbA6+DEb/AroNd8npC0psfLrbalL5+mA2AKN7hXLnlL7MGtKFAF+NjNZMvztKtXb2MusDzo0vWx94+gbB6Dtg5M8hqPETZtjtwjdHTrE0JZWVu06QX1JGdIg/907vx7zh3ekRok0qbYUGulKtVUk+bCufSOIwdI6GmU9bT3X6Bjb69D9kF7B0ayrLtqVy7FQhHX29mBPfjcuTokjqGayP5LdBGuhKtTZnT8A3r0LyW1B0BqIusOboHDAHPBv3Xzav2MbKXSdYkpLKN4dPYQyM7R3GfdP7M2NwFzr4aHfDtkwDXanW4uQuq1ll1xKQMivAx9zV6Dk67XZh06FslqSk8snukxSWltErLIBfz+jPvOHd6da5g4tuQLU0DXSlWpLdbg2QtXEhHP4SvAPggputiSRCGje/7pGsfKtJZWsaaWcKCfTzYl5Cd+YnRJEQ3VmbVNyQBrpSLaG0EHZ+ABv/Bln7IbAbTHsSEm9s1EQSZ4tKWbnTalJJPnoaDwPj+4bz0KwBTB8UqU9wujkNdKWaU15mpYkksqBLPFz2Dxh0KXj5NOiUZXZhw4Eslm5N5dPdJym22ekT0ZGHZg1g3vDuROrAWO2GBrpSzSFzv9WssuMDKCuGfjOtiSRixjV4oKwDGXks3ZrKh1vTOHm2iKAO3lyZ1IPLE6OIjwrSJpV2SANdqaYiYrWLf70QDnwOXn4w7CdWH/Kwvg06ZU5BKR/vPM7Srals++EMnh6Gif3CeeziQUwdGIGvlzaptGca6Eq5mq2k0kQSuyAgHCY/Ckk3QUDY+Z+uzM66A1ksSUnl8z3plNjs9I8M5NHZA7lkeDciArVJRVk00JVylYJTkPIWbH7NmkgifCDMXQhxV4D3+Yfud+m5LE1JZdm2NDJziwn29+YnI6K5PDGKwd06aZOKqkYDXanGyj4Im/4O29+zJpLoNRkufdmaUOI8Q/d0fgkf7zzOkpRUdqbm4OVhmDwggvkJUUwZEIGPl2sG3lLuSQNdqYYQgR82WR907vsveHg5JpL4BXQZcl6nKi2z8+X+TJZuTWXV3nRKy4RBXTvx2JxBzB3WjbCOrpsmTrk3DXSlzkeZDfZ+ZLWPp6VAh2AYfx+M+BkEdjmvU+09cZYlKal8tD2NrLwSQgN8uH50DPMTohjUrVMT3YByZxroSjmj6KxjIolXrIkkQnrDRX9xTCQR4PRpsvOK+Wi71Uvl2+Nn8fY0TB0QyeWJUUzsH463pzapqIbTQFeqLmeOWSG+9R0oPgs9x8KsP0O/WU5PJFFis7NmfwZLUlJZsy8Dm12I6x7Ek3MHM3doN4IDGvZAkVJVaaArVZO0FKtZ5dvl1vLgeVb/8e4JTh0uInx73GpSWbHjOKfySwgP9OWmcbHMT4iif5fGD3+rVFUa6EqVKzxtzQS0Y7FjIolO1mxAI34OnXs4dYrM3GI+2p7GkpRU9p3MxcfTg+mDrCaV8X3D8NImFdWENNBV+1aSD/s/sYasPbAK7KVW+/iMP8Lw68Cv/g8ni21lfLE3g6Upqaz9LpMyuzCsR2eeunQIF8d3pbO/Nqmo5qGBrtofWwkc/MIK8f0rrb7jgd2sKd3iLoeuw+rtPy4i7EzNqWhSySksJbKTL7dO6MX8hCj6RHRsnntRqhINdNU+2Mvg6AbY9W+rWaXoDHQIgfgF1pOc0aOd+pAz/WwRH25LY2lKKt9n5OHr5cGMwV2YnxjFuD5heHro05uq5WigK/clAmlbYfcS2L3MehzfpyMMuAiGXA69J4Ond72nKSot4/M96SxJSWXd95nYBRJ7BvOny+K4KL4rnfzqP4dSzUEDXbmfjH1WTXz3UmtyZU8f6HshDJlvDVvrU/8s9iLCtmNnWJKSysc7jpNbZKNbkB+/mNSHyxK60ytcm1RU66OBrtzD6aNWgO9eCum7wXhA7ASYcL81N6cTswCJCAcy8vjftydZtjWNQ1n5+Hl7MGtIVy5PjGJ0r1A8tElFtWIa6Krtysuw+onvXgLHNlvrokbArGesGYACI+s9RYnNzubD2XyxN4Mv9qVz7FQhACNiQ7htYm9mxXUhUJtUVBuhga7alqIc2Psfq0nl8JcgdogYDFMfs5pUgmPqPUVWXjFr92fyxd501n2fRV6xDV8vD8b2CeO2ib2ZMiCCrkEdmv5elHKxegPdGPMmMAfIEJFqw8gZYyYBHwGHHauWicjvXFhG1d6VFsJ3n1rdDL//3JrCrXNPGHev1c0wYmCdh4sI+07msnpfBl/sTWfbsTOIQGQnXy4e2o1pAyMY0zuMDj46249q25ypoS8CFgLv1LHPOhGZ45ISKQVQVgqH1lohvu8/UJIHHSOtWX/iLofuiXX2FS8qLWPTIaspZfW+DNLOWE0p8VFB/HJqP6YOjNBJIpTbqTfQReQrY0xMM5RFtXd2u/XI/e4lsOcjKMgGvyBrHJW4K6wJlT1qr0Vn5BaxZl8GX+zNYP2BLApKyujg7cm4vmHcNaUPUwZEENFJp2tT7stVbeijjTE7gOPA/SLybU07GWNuBW4FiI6OdtGlVZsmAid2/NhX/GwaeHWAAbOtvuJ9poJXzRM8lA+AZdXC09mRmgNAtyA/a4afgRGM7hWKn7c2paj2wRWBvhXoKSJ5xpjZwHKgxinNReQ14DWApKQkccG1VVuVdcAK8V1LIPt7a8afPtNg2pPQfxb41tzPu6i0jA0HsvhiXwar92Zw8mwRxsCwHp25/8J+TB0YyYAugdqUotqlRge6iJyt9H6lMeZvxpgwEclq7LmVm8lJc/QVX2LVyjFWM8qYO2HgXPAPqfGwkzlFFR9objiYRVGpnQAfT8b3DWfqwAgm9Y8gPFCnaVOq0YFujOkCpIuIGGNGAB5AdqNLptxDfjbsWW4F+dGvAYFuCdZohoPnQadu1Q6x24Xdx3NY5WhK2Z1m1Rmigjtw1QXRTBkQwcheIfh6aVOKUpU5021xMTAJCDPGpAKPA94AIvIKcDlwuzHGBhQCV4mINqe0Z8W5sG+l1Vf80Bqw2yCsP0x+xOorHtq72iEFJTbWf59l1cT3ZZCZW4yHgYToYB6cOYCpAyPoG9FRm1KUqoMzvVyurmf7Qqxujao9Ky2CA59bbeLf/Q9shRDUA0bfaXUzjBxSrZth2plCVu9N54t9GXx9MJsSm51AXy8m9A9n6gCrKSVEp2dTymn6pKhquDIbHPkKdi2FvR9DcQ74h8Hwa60QjxpxzpC0ZXZhR+oZVu/NYNXedPadzAWgZ6g/147sybSBESTFhODjpbP6KNUQGujq/IhA6harOeXbDyE/05qqbcAciJsPsZPA88cfq7xiG+u+y+SLfRms2ZdBdn4Jnh6GpJ7BPDJ7AFMHRtIrLECbUpRyAQ105Zz0b38ckvbMD+DpC/1nWn3F+14I3j8+sHPsVAFfOJpSNh3KprRM6OTnxaT+EUwdGMHEfuE6LZtSTUADXdXu1GFHX/GlkLkXjKc1KcSkR6xJIhzzbZbZhW1HTlX0SvkuPQ+A3uEB/HRsLFMHRJDYM1gnSFaqiWmgq3PlnrSe2Ny9BNJSrHXRo2H2c1Y3w4AwAM4WlfLVzuOs3pvBmv0ZnC4oxcvDMCI2hAUXRDN1QAQxYQEteCNKtT8a6AoKT1vzbO5eAofXAQJd4mH672DwZdC5BwBHsvJZtfUQq/dl8M3hU9jsQrC/N5P7RzBlYAQT+oXrdGxKtSAN9PaqJB/2f2J1MzywCuylENIbJj5gtYuH98NWZif56GlWf72XVXvTOZSZD0C/yI7cMr4X0wZGMDw6WCdGVqqV0EBvT2wlcPALK8T3r4TSAgjsBiN/bnUz7DqMnEIba7/L4ItV21i7P4OzRTa8PQ2jeoVy/aieTBkQSXRo/XNyKqWanwa6uysrtYak3eUYkrboDHQIhvgFEHc5Ej2ag1mFrN6XzqqPN5Fy9DRldiE0wIcLB3dh2sAIxvUNp6Ov/qgo1drp/1J3IwLZB+Hgauux+8ProCQXvAOsnilxV1AaM5EtP+SyalcGq//9FUeyCwAY0CWQ2yf2ZsrACIZFddYJkZVqYzTQ3UHBKWt2n4OrrX9zjlnrg2Mg/groPYXTXcez5lAeXyRn8NV7a8kttuHj5cGY3qHcPL4XUwZE0L2zzqOpVFumgd4W2Yrh2Dc/1sKPbwcEfIOg1wQY9yvye0wg+Wxnthw+xca12Wz7YQN2gfBAXy6K78qUARGM6xuGv4/+CCjlLvR/c1sgApn7fwzwI+utDzSNJ/QYAZMe5ky3cWwuimbz0Vy+2ZTNnmXfYRfw9DAM6daJO6f0ZdrACIZ0C9KmFKXclAZ6a5WXaTWfHFpjBXnuCWt9aB9k2DVkRY7la9tANqaV8E3KKQ59egY4g6+XB8OjO3Pn5D5cEBtCQnQwAfqBplLtgv5Pby1Ki6zeKOUBfnKXtb5DMBI7kfTwMXxNPGtP+rFl5ylO5BQBB+nk50VSTAhXJvXggpgQ4roH6WiFSrVTGugtRcQa8Ko8wI9+DbYi8PDG3mMkJxN/zSYzlE+ywtmy7yxntpYCp4js5MsFMSGMiA3hgpgQ+kcGahOKUgrQQG9euSfh4BorxA+thbx0AOxh/TnZ5yo2m2GsOBPDpkPFFO4vAyA2rIgLB0VWhHh0iL8ONauUqpEGelMqKbBq3uW18Iw9ANj9w0gPG0Vy+DCW5/Tjy+Pe2FIFY2BgFy8WXBDBiNgQkmKCiQj0q+ciSill0UB3JbsdTu78McB/2ARlJYinL1khw0npejsf5fbj06xw5JQHPp4eDO0RxK0TQrggNoTEnsE6uJVSqsE00BsrJ+3HAD+0FgqyATgT2JdtQfP4OK8/K8/GUpTvS0dfLxJ6BnNfQjAXxIQwtEdn/Lx15nqllGtooJ+v4jw4usEK8INrIGs/AAU+oez0SeC/ZgCfFg4gsyiY0AAfLogJ4dexIYyMDWFAl0Cd5EEp1WQ00OtjL4MT2x0BvhY5thljL8Xm4csenzj+J9exqmQw+4t6EBXsz4iBIdwXYzWh6FyZSqnmpIFek9NHHc0oa7Af+hKPotMAHPLqzarSmawtiyPF3o+YwFAuGBDMLxw9ULoG6VgoSqmWo4EOUHQWjqyDg2uwHfgCr9OHAMgyoayxxbGuLJ5NDKFbeDQjYkP4aUwIL/cMJjhAJzpWSrUe7TPQy2xwfCtycDXF+1fhc3IrHlJGIb5sLBvIOvt1bPEYRlD0YC6IDeWqmBCeju6sA1kppVq19pNQpw5hP7CGvD2f4Ze6AR9bLoJhvz2WdfY5bPcejlfPkST0imRuTAiPdA/CWz/AVEq1Ie4b6IVnKD34JWd2fYrP0a8IKkrFAzgrYfy3LIlv/RKwxUxgcJ9YpseE8IuIjvoIvVKqTas30I0xbwJzgAwRGVLDdgO8AMwGCoAbRWSrqwtar7JSCg9vJnP7J3gdWUuXvD14Y6eD+LHRPpi9/rMp6TmRXv2HMi42lKuCO2gPFKWUW3Gmhr4IWAi8U8v2WUBfx2sk8HfHv01LhNPH9nJi60o8j6ylR04y/lJIdzHskN58FXAlhdGT6DZ4LIm9Ipne0bfJi6SUUi2p3kAXka+MMTF17HIJ8I6ICLDJGNPZGNNVRE64qpCVHdixgdNr/0b0mc1ESibBwDGJYEPAZAqiJhIWP434Pj1J0EfolVLtjCva0LsDxyotpzrWVQt0Y8ytwK0A0dHRDbpY/qkT9D+1mu/8h7M36hZC42bQb1A8Pbz0EXqlVPvWrB+KishrwGsASUlJ0pBzDB43FzP+EpK8tAaulFKVuSLQ04AelZajHOuahJe3PsyjlFI1cUVH6xXA9cYyCshpqvZzpZRStXOm2+JiYBIQZoxJBR4HvAFE5BVgJVaXxQNY3RZ/2lSFVUopVTtnerlcXc92Ae5wWYmUUko1iD7brpRSbkIDXSml3IQGulJKuQkNdKWUchMa6Eop5SY00JVSyk1ooCullJvQQFdKKTehga6UUm5CA10ppdyEBrpSSrkJDXSllHITGuhKKeUmNNCVUspNaKArpZSb0EBXSik3oYGulFJuQgNdKaXchAa6Ukq5CQ10pZRyExroSinlJjTQlVLKTWigK6WUm9BAV0opN6GBrpRSbkIDXSml3IQGulJKuQmnAt0YM9MYs98Yc8AY81AN2280xmQaY7Y7Xre4vqhKKaXq4lXfDsYYT+BlYDqQCmwxxqwQkT1Vdv1ARO5sgjIqpZRygjM19BHAARE5JCIlwPvAJU1bLKWUUufLmUDvDhyrtJzqWFfVfGPMTmPMEmNMD5eUTimllNNc9aHox0CMiMQDnwNv17STMeZWY0yyMSY5MzPTRZdWSikFzgV6GlC5xh3lWFdBRLJFpNix+DqQWNOJROQ1EUkSkaTw8PCGlFcppVQtnAn0LUBfY0ysMcYHuApYUXkHY0zXSotzgb2uK6JSSiln1NvLRURsxpg7gf8BnsCbIvKtMeZ3QLKIrADuNsbMBWzAKeDGJiyzUkqpGhgRaZELJyUlSXJycotcWyml2ipjTIqIJNW0TZ8UVUopN6GBrpRSbkIDXSml3IQGulJKuQkNdKWUchMa6Eop5SY00JVSyk1ooCullJvQQFdKKTehga6UUs2szF7WJOetdywXpZRSzhERzhSfIb0gnfT8dOtfx/uMgoyK5esGXccdw+5w+fU10JVSygk2u42swqwfg9kR0icLTlrrHMsl9pJzjvMwHoR3CCfSP5LenXszptsYhoYPbZIyaqArpdq94rJiMvJ/rEGnF6RXhHT5clZhFnaxn3Ocj4cPkQGRRPpHEh8eX/G+/BXhH0Foh1C8PJonajXQlVJuLa8kr3oTSJXAPlN8ptpxHb07WsEcYNWsy99XDuwg3yCMMc1/U7XQQFdKtUkiwuni0xWhnFGQwcn8kz+GtSPEC2wF1Y4N8Qsh0j+SrgFdGRo+tCKoI/wjKt4HeAe0wF01jga6UqrVKW+vrlyzrtoEklGQQam99JzjPI0n4f7hRPhH0KdzH8Z2G3tOzTrCP4II/wh8PH1a6M6alga6UqpZFdmKzunxcU4PEMf7rKLq7dW+nr4VoTwsYljF+y7+XYgMcLRX+4Xi6eHZQnfW8jTQlVIuISLklead2wRScPKcwM4oyKixvTrQO7CiFt03uO+5TSD+kXQJ6EInn06tqr26NdJAV0rVyC52zhaf5UzxGc4UnyGnOKfW9+WBXWgrrHae8vbqbh27MTxieEUTSHlYR/pH4u/t3wJ36H400JVqB4psRfWGctX3Z4vPItQ857Cn8STIN4gg3yA6+3amX3A/xkeNP6e7XmRAJBEdIvD29G7mu22/NNCVakPsYie3JLd6EBedu1w1oIvKimo9ZwevDnT27Uxn384E+QbRLaBbRVCXryt/39m3M0F+QXT07oiH0ZFDWhsNdKVaSHFZcbUgru19Ra255Gy1DwvLeRgPgnx+rDV3DejKgJABVhD7da41pN21x0d7pIGuVCOV15rPpzkjpzinxvbmch28OlSEbpBvEF0CulQL4qrvA30CtdbczmmgK+VgFzuFtkLySvLIKcmpHsS11KZzSnLqrDV38ulUEbqR/pH0C+5XayiX16Z9PX2b+e6VO9BAV22WXewUlBaQX5pPvi2fwtJC671juaC0wNpus9bVuGwrqDimrhozgJ+n3zkBXB7MlYO4akhrrVk1Jw101WzK7GUVAVpgK/gxjCsFqrPLBbaCegO4Mn8vfwK8A/D39q94H+Efgb+XP/7e1nKAd0DFck01Zz8vvyb86ijVeBroqlZl9rJza7qVar7lNdzaars11X4bEsCVQzjCP+LH8PUKqHjv7+1f53IHrw5aS1btggZ6GyYi2MRGSVkJRbYiSspKKC4rrv6yFZ9TM64I40q13ZqW6+rqVpnBnBOi5UHaxb/Lj8telcK5ynJ5zbg8wP28/DSAlWoApwLdGDMTeAHwBF4XkaerbPcF3gESgWxggYgccW1RW68yexnFZcVWsJbVEqy2YortxbWGb7VjbdWDuabz1/ZhXF2qBnB5kHYN6Fpj80P5cnltt/Kyv5e/BrBSrUS9gW6M8QReBqYDqcAWY8wKEdlTabebgdMi0scYcxXwZ2BBUxS4NiJSZ/BVrLcVObVPvYFb6WWz2xpVdm8Pb3w9fStePp4++Hn54ePpg6+nL8HewT+u9/xx/TkvL98696lcE+7g1UHHxFDKDTlTQx8BHBCRQwDGmPeBS4DKgX4J8ITj/RJgoTHGiEjNzw03wvq09Tyz5ZlqNdmq0z6dLy/jVS1IKwdsoE8goZ6hdQZvTa9z9vGoHr6+nr5au1VKuYQzgd4dOFZpORUYWds+ImIzxuQAoUBW5Z2MMbcCtwJER0c3qMAdvTvSt3Nfp2qlzoavj6dPs00RpZRSTaVZU0xEXgNeA0hKSmpQ7X1YxDCGRQxzZbGUUsotOPO3fhrQo9JylGNdjfsYY7yAIKwPR5VSSjUTZwJ9C9DXGBNrjPEBrgJWVNlnBXCD4/3lwOqmaD9XSilVu3qbXBxt4ncC/8PqtvimiHxrjPkdkCwiK4A3gHeNMQeAU1ihr5RSqhk51YYuIiuBlVXWPVbpfRFwhWuLppRS6nxofzmllHITGuhKKeUmNNCVUspNaKArpZSbMC3Vu9AYkwkcbeDhYVR5CrUd0HtuH/Se24fG3HNPEQmvaUOLBXpjGGOSRSSppcvRnPSe2we95/ahqe5Zm1yUUspNaKArpZSbaKuB/lpLF6AF6D23D3rP7UOT3HObbENXSilVXVutoSullKpCA10ppdxEqw50Y8xMY8x+Y8wBY8xDNWz3NcZ84Ni+2RgT0wLFdCkn7vleY8weY8xOY8wXxpieLVFOV6rvnivtN98YI8aYNt/FzZl7NsZc6fhef2uM+Wdzl9HVnPjZjjbGrDHGbHP8fM9uiXK6ijHmTWNMhjFmdy3bjTHmRcfXY6cxJqHRFxWRVvnCGqr3INAL8AF2AIOq7PML4BXH+6uAD1q63M1wz5MBf8f729vDPTv2CwS+AjYBSS1d7mb4PvcFtgHBjuWIli53M9zza8DtjveDgCMtXe5G3vMEIAHYXcv22cAngAFGAZsbe83WXEOvmJxaREqA8smpK7sEeNvxfgkw1bTt6ezrvWcRWSMiBY7FTVgzSLVlznyfAZ4C/gwUNWfhmogz9/wz4GUROQ0gIhnNXEZXc+aeBejkeB8EHG/G8rmciHyFNT9EbS4B3hHLJqCzMaZrY67ZmgO9psmpu9e2j4jYgPLJqdsqZ+65spuxfsO3ZfXes+NP0R4i8t/mLFgTcub73A/oZ4zZYIzZZIyZ2WylaxrO3PMTwLXGmFSs+Rfuap6itZjz/f9eL53qvo0yxlwLJAETW7osTckY4wE8D9zYwkVpbl5YzS6TsP4K+8oYEyciZ1qyUE3samCRiPzFGDMaaxa0ISJib+mCtRWtuYbeHienduaeMcZMAx4F5opIcTOVranUd8+BwBBgrTHmCFZb44o2/sGoM9/nVGCFiJSKyGHgO6yAb6ucueebgX8BiMhGwA9rECt35dT/9/PRmgO9PU5OXe89G2OGA69ihXlbb1eFeu5ZRHJEJExEYkQkButzg7kiktwyxXUJZ362l2PVzjHGhGE1wRxqxjK6mjP3/AMwFcAYMxAr0DObtZTNawVwvaO3yyggR0RONOqMLf1JcD2fEs/GqpkcBB51rPsd1n9osL7h/wYOAN8AvVq6zM1wz6uAdGC747Wipcvc1PdcZd+1tPFeLk5+nw1WU9MeYBdwVUuXuRnueRCwAasHzHbgwpYucyPvdzFwAijF+ovrZuA24LZK3+OXHV+PXa74udZH/5VSyk205iYXpZRS50EDXSml3IQGulJKuQkNdKWUchMa6Eop5SY00JVSyk1ooCullJv4/3Bz1+8V2oIzAAAAAElFTkSuQmCC", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import math\n", + "\n", + "#We solve the same problem as before\n", + "\n", + "\n", + "\n", + "t0 = 0\n", + "tf = 1\n", + "N = 5\n", + "\n", + "t = np.linspace(t0,tf,N)\n", + "y = np.zeros(N)\n", + "z = np.zeros(N)\n", + "w = np.zeros(N)\n", + "h = (tf-t0)/N\n", + "\n", + "y[0] = 1\n", + "\n", + "for i in range(0,N-1):\n", + " y[i+1] = y[i]/(1-h)\n", + "\n", + "w = np.exp(t)\n", + "z = abs(y-w)\n", + "\n", + "plt.plot(t,y, label = \"y - Numerical Method method\") #very simple plot of y,z,w\n", + "plt.plot(t,w,label = \"w - Actual\")\n", + "plt.plot(t,z,label = \"z - Error\")\n", + "plt.title(\"Method with N=%d\"%N)\n", + "plt.legend()\n", + "plt.show()\n", + "\n", + "\n", + " \n", + "\n", + "\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([0. , 0.25, 0.5 , 0.75, 1. ])" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "t" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.0\n", + "0.25\n", + "0.5\n", + "0.75\n", + "1.0\n" + ] + } + ], + "source": [ + "for i in t:\n", + " print(i)\n" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([0.25, 0.5 , 0.75, 1. ])" + ] + }, + "execution_count": 39, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "t[1:]" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "interpreter": { + "hash": "7e1998ff7f8aa20ada591c520b972326324e5ea05489af9e422744c7c09f6dad" + }, + "kernelspec": { + "display_name": "Python 3.10.1 64-bit", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.10" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Programming/Python files/Forward_Euler.py b/Programming/Python files/Forward_Euler.py new file mode 100644 index 0000000..0e8f3fb --- /dev/null +++ b/Programming/Python files/Forward_Euler.py @@ -0,0 +1,110 @@ +## The following will produce a very simple implementation of Euler's +## method for solving the following ODE: +## +## y'(t) = y(t), t > 0, +## y(0) = 0 +## +## For now, the code will explicit use the forcing term f(y) = y, but will +## later be generalized. +## + +import math +import numpy as np # shortcut name for numpy +from matplotlib import pyplot as plt # shortcut name for pyplot +from matplotlib.animation import FuncAnimation + +N = 100 # Number of time steps +t0 = 0 # Initial time +tf = 1 # Final time +h = (tf - t0)/(N-1) # Numerical step size +t = np.linspace(t0,tf,N) # Temporal grid +y = np.zeros([N]) # Empty vector for storing solution values +z = np.zeros([N]) # Empty vector for storing error values +w = np.zeros([N]) # Empty vector for true solution +y[0] = 1 # Initial value of the problem + +# The following is the actual implementation of Euler's method +for i in range(1,N): + y[i] = y[i-1] + h*y[i-1] + +# The following constructs the true solution +for i in range(0,N): + w[i] = math.exp(t[i]) + +# The following computes the error between the approximation +# and the true solution, y = exp(t) +for i in range(0,N): + z[i] = abs( y[i] - w[i] ) + +# The following creates strings to generate the titles in the plots +s1 = "Euler Approximation \n for h = " +s = "%1.2f" % h +title1 = s1 + s +s2 = "Absolute Error \n for h = " +title2 = s2 + s + +# The following generates the desired plots + +fig, axs = plt.subplots(ncols=2,nrows=2) +gs = axs[1,0].get_gridspec() +# remove the underlying axes +for ax in axs[-1,0:]: + ax.remove() +axbig = fig.add_subplot(gs[-1,0:]) + +axs[0,0].plot(t,y,'--',label="Approx.") +axs[0,0].plot(t,w,label="True") +axs[0,0].set_title(title1) +axs[0,0].set_xlabel('t') +axs[0,0].set_ylabel('y') +axs[0,0].legend() + +axs[0,1].plot(t,z) +axs[0,1].set_title(title2) +axs[0,1].set_xlabel('t') +axs[0,1].set_ylabel('y') + +# The following is a (fairly complicated) bit of code that allows +# you to make a video of the pointwise error at each of the points +# in an ever refining domain. + +xdata, ydata = [], [] +ln, = plt.plot([], [], 'ro') + +def out(ii): + NN = 5*ii + tt = np.linspace(t0,tf,NN) + hh = tt[1]-tt[0] + ww = np.zeros([NN]) + yy = np.zeros([NN]) + zz = np.zeros([NN]) + yy[0] = 1 + for i in range(1,NN): + yy[i] = yy[i-1] + hh*yy[i-1] + for i in range(0,NN): + ww[i] = math.exp(tt[i]) + for i in range(0,NN): + zz[i] = abs( yy[i] - ww[i] ) + return hh,tt,zz + +def init(): + axbig.set_xlim(t0,tf) + axbig.set_ylim(0,0.15) + axbig.set_xlabel('t') + axbig.set_ylabel('Error') + return ln, + +def update(ii): + hh,xdata,ydata = out(ii) + ln.set_data(xdata, ydata) + ss = "Error for h = " + ss1 = "%1.4f" % hh + title3 = ss + ss1 + axbig.set_title(title3) + return ln, + +ani = FuncAnimation(fig,update,frames=range(1,20),interval=150,init_func=init) + +fig.tight_layout() + +plt.show() diff --git a/Programming/Python files/NonUniformFiniteDifference.ipynb b/Programming/Python files/NonUniformFiniteDifference.ipynb new file mode 100644 index 0000000..b8ec3bb --- /dev/null +++ b/Programming/Python files/NonUniformFiniteDifference.ipynb @@ -0,0 +1,1483 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "\n", + "def f(x):\n", + " return np.sin(np.pi * x)\n", + "\n", + "x = np.linspace(0,1,100)\n", + "x = x**2\n", + "\n", + "h = np.diff(x) # we define the vector h as the difference of successive x[i] using the built in diff function\n", + "\n", + "f = f(x)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "A = np.zeros((len(f),len(f))) #we create the matrix A by initializing to 0" + ] + }, + { + "cell_type": "code", + "execution_count": 111, + "metadata": {}, + "outputs": [], + "source": [ + "for i in range(1,len(f)-1):\n", + " A[i,i-1] = 2/(h[i-1]*(h[i-1] + h[i])) #we populate the diagonal (minus the extreme ends), the one to the left and right with a,b,c respectively\n", + " A[i,i] = -2/(h[i-1] * h[i])\n", + " A[i,i+1] = 2/(h[i]*(h[i] + h[i-1]))\n", + "\n", + "A[0,0] = 1 # we denote the far corners of our matrix to be 1\n", + "A[len(f)-1,len(f)-1] = 1\n" + ] + }, + { + "cell_type": "code", + "execution_count": 112, + "metadata": {}, + "outputs": [], + "source": [ + "u = np.linalg.inv(A).dot(f) # we solve our system" + ] + }, + { + "cell_type": "code", + "execution_count": 113, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 113, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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    " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt #we plot the result\n", + "\n", + "plt.plot(x,u)" + ] + }, + { + "cell_type": "code", + "execution_count": 115, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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    " + ] + }, + "metadata": {}, + "output_type": "display_data" + }, + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 115, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "from sympy import *\n", + "import math\n", + "\n", + "init_printing(use_unicode=False, wrap_line=False)\n", + "x = Symbol('x')\n", + "\n", + "plot(integrate(integrate(sin(pi*x),x),x))\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "interpreter": { + "hash": "a7b5a1112396ad393f5588006c146c6954be07cba002f50e558ad6b83bb9990c" + }, + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.7" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Programming/Python files/Nonuniform_JLP.ipynb b/Programming/Python files/Nonuniform_JLP.ipynb new file mode 100644 index 0000000..ccb3501 --- /dev/null +++ b/Programming/Python files/Nonuniform_JLP.ipynb @@ -0,0 +1,272 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 68, + "id": "1dc137ec", + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "\n", + "def f(x):\n", + " return np.sin(np.pi * x)\n", + " #return 1 - 4*(x-0.5)**2\n", + "\n", + "def nonuniform_grid(num_points,left_pt,right_pt,is_non_unif=True):\n", + " grid = np.array([(float(is_non_unif)/(num_points+1-0.5*(num_points+2))**2)*((i+1)-0.5*(num_points+2))**2 + 1 for i in range(num_points+1)])\n", + " grid *= (right_pt-left_pt)/sum(grid)\n", + " final_grid = np.zeros(num_points+2)\n", + " final_grid[0] = left_pt\n", + " for i in range(1,num_points+1) :\n", + " final_grid[i] = final_grid[i-1]+grid[i-1]\n", + " final_grid[num_points+1] = right_pt\n", + " return grid, final_grid" + ] + }, + { + "cell_type": "code", + "execution_count": 69, + "id": "6391d3e7", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import matplotlib.pyplot as plt\n", + "\n", + "num_points = 100\n", + "h_vals, grid = nonuniform_grid(num_points,0,1)\n", + "#grid = np.linspace(0,1,num_points+2)\n", + "#h_vals = np.diff(grid)\n", + "\n", + "plt.figure(figsize=(12,7.5))\n", + "\n", + "# Plot 1\n", + "plt.subplot(1,2,1)\n", + "plt.plot(grid)\n", + "plt.title(\"Plot of nonuniform grid\")\n", + "plt.xlabel(\"Grid point index\")\n", + "plt.ylabel(\"Domain value\")\n", + "\n", + "# Plot 2\n", + "plt.subplot(1,2,2)\n", + "plt.plot(grid[:-1],h_vals)\n", + "plt.title(\"Comparison of grid spacing\")\n", + "plt.xlabel(\"Point in domain\")\n", + "plt.ylabel(\"Grid spacing\")\n", + "\n", + "plt.tight_layout()\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 70, + "id": "ac600ea8", + "metadata": {}, + "outputs": [], + "source": [ + "# Construct the coefficient matrix\n", + "A = np.zeros((num_points,num_points))\n", + "\n", + "for i in range(num_points) :\n", + " A[i,i] = -2./(h_vals[i]*h_vals[i+1])\n", + " \n", + "for i in range(num_points-1) :\n", + " A[i+1,i] = 2./(h_vals[i+1]*(h_vals[i+1]+h_vals[i+2]))\n", + " A[i,i+1] = 2./(h_vals[i+1]*(h_vals[i]+h_vals[i+1]))" + ] + }, + { + "cell_type": "code", + "execution_count": 71, + "id": "7343d63c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "array([[ -9343.45827382, 4718.44170984, 0. , ...,\n", + " 0. , 0. , 0. ],\n", + " [ 4813.73429872, -9724.66514761, 4910.93084888, ...,\n", + " 0. , 0. , 0. ],\n", + " [ 0. , 5010.04771244, -10121.16880196, ...,\n", + " 0. , 0. , 0. ],\n", + " ...,\n", + " [ 0. , 0. , 0. , ...,\n", + " -10121.16880196, 5010.04771244, 0. ],\n", + " [ 0. , 0. , 0. , ...,\n", + " 4910.93084888, -9724.66514761, 4813.73429872],\n", + " [ 0. , 0. , 0. , ...,\n", + " 0. , 4718.44170984, -9343.45827382]])" + ] + }, + "execution_count": 71, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# Check the resulting matrix's form\n", + "A" + ] + }, + { + "cell_type": "code", + "execution_count": 72, + "id": "be8ce9d4", + "metadata": {}, + "outputs": [], + "source": [ + "# Solve the discrete system\n", + "\n", + "bc_left = 0\n", + "bc_right = 0\n", + "\n", + "rhs = f(grid[1:-1])\n", + "rhs[0] -= (2/(h_vals[0]*(h_vals[0]+h_vals[1])))*bc_left\n", + "rhs[-1] -= (2/(h_vals[-1]*(h_vals[-2]+h_vals[-1])))*bc_right\n", + "\n", + "u = np.linalg.solve(A,rhs)" + ] + }, + { + "cell_type": "code", + "execution_count": 73, + "id": "ec5cd1a5", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Add boundary conditions\n", + "\n", + "U = np.append(np.array([bc_left]),np.append(u,bc_right))\n", + "RHS = np.append(np.array([bc_left]),np.append(rhs,bc_right))\n", + "\n", + "plt.figure(figsize=(12,7.5))\n", + "plt.plot(grid,U,label=\"$u(x)$\")\n", + "plt.plot(grid,RHS,label=\"$f(x)$\")\n", + "plt.title(\"Solution vs. right-hand side function\")\n", + "plt.xlabel(\"$x$\")\n", + "plt.ylabel(\"$u(x)$\")\n", + "plt.legend()\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 74, + "id": "6a80c620", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "# Check error\n", + "\n", + "def error_check(approx,true,grid) :\n", + " error_grid = np.append(0.5*(grid[:-1]+grid[1:]),grid[-1])\n", + " return np.sqrt(np.multiply(error_grid,(approx-true)**2))\n", + "\n", + "true_sol = -(1/np.pi**2)*np.sin(np.pi*grid)\n", + "\n", + "error = error_check(U,true_sol,grid)\n", + "\n", + "plt.figure(figsize=(12,7.5))\n", + "plt.plot(grid,error)\n", + "plt.title(\"Pointwise error\")\n", + "plt.ylabel(\"Error\")\n", + "plt.xlabel(\"$x$\")\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "code", + "execution_count": 75, + "id": "6ae3c077", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.000910607929436465" + ] + }, + "execution_count": 75, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# L2 error\n", + "sum(error)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "aee347c7", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.4" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/Programming/Python files/Some numerical method.ipynb b/Programming/Python files/Some numerical method.ipynb new file mode 100644 index 0000000..6cf4d6f --- /dev/null +++ b/Programming/Python files/Some numerical method.ipynb @@ -0,0 +1,88 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", 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    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "t0 = 0\n", + "tf = 1\n", + "N = 100\n", + "\n", + "#Every variable is the same as the one given in Forward_Euler.py\n", + "\n", + "t = np.linspace(t0,tf,N)\n", + "y = np.zeros(N)\n", + "z = np.zeros(N)\n", + "w = np.zeros(N)\n", + "h = (tf-t0)/N\n", + "\n", + "y[0] = 1\n", + "y[1] = y[0] + h*y[0]\n", + "\n", + "for i in range (0,N-2):\n", + " y[i+2] = y[i+1] + h*(1.5*y[i+1]+0.5*y[i])\n", + "\n", + "w = np.exp(t) #we vectorized the subtraction instead of iterating through w\n", + "z = abs(y-w) #same here\n", + "\n", + "plt.plot(t,y, label = \"y - Numerical Method method\") #very simple plot of y,z,w\n", + "plt.plot(t,w,label = \"w - Actual\")\n", + "plt.plot(t,z,label = \"z - Error\")\n", + "plt.title(\"Method with N=%d\"%N)\n", + "plt.legend()\n", + "plt.show()\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "interpreter": { + "hash": "b0fa6594d8f4cbf19f97940f81e996739fb7646882a419484c72d19e05852a7e" + }, + "kernelspec": { + "display_name": "Python 3.9.10 64-bit", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.10" + }, + "orig_nbformat": 4 + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/Programming/Python files/eulermethod.ipynb b/Programming/Python files/eulermethod.ipynb new file mode 100644 index 0000000..1aa280f --- /dev/null +++ b/Programming/Python files/eulermethod.ipynb @@ -0,0 +1,83 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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    " + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "import numpy as np #import appropriate libraries\n", + "import matplotlib.pyplot as plt\n", + "\n", + "start = 0 #start point of the domain over which the diffeq will be solved\n", + "end = 1 #end-point\n", + "steps = 30 # number of steps\n", + "t = np.linspace(start, end, steps) # t is a uniformly spaced vector spanning the start and endpoints\n", + "y = np.zeros(len(t)) # y is a vector initialized to 0 as long as t\n", + "h = (end - start)/steps # h is the width of each step\n", + "t = t*h # t is h times the number of t's gone before\n", + "\n", + "y[0] = 1 # initial value\n", + "\n", + "for i in range (0,len(t)-1): # initiate loop\n", + " y[i+1] = y[i] + h * (-6*y[i]) # euler method\n", + "plt.plot(t,y) # plot the graph" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + 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+\numberwithin{equation}{section} +\usepackage[]{amssymb} +\usepackage[margin=1in]{geometry} +\usepackage[]{soul} +\usepackage[]{bbm} +\usepackage[]{cancel} +\usepackage[]{xcolor} +\usepackage[]{enumitem} +\usepackage{mathrsfs} +\usepackage{hyperref} +\usepackage[capitalise]{cleveref} +\usepackage{natbib} +\usepackage{neuralnetwork} +\usepackage{witharrows} + +\usepackage{listings} +\usepackage{graphicx} + +\DeclareMathAlphabet{\mymathbb}{U}{BOONDOX-ds}{m}{n} + +% \usepackage[]{enumerate} +\setlength\parindent{0pt} + +\DeclareMathOperator{\Trace}{Trace} +\DeclareMathOperator{\Hess}{Hess} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\domain}{Domain} +\DeclareMathOperator{\lip}{Lip} +\DeclareMathOperator{\diag}{diag} + +\usepackage{tikz-cd} + +\DeclareMathOperator{\rect}{Rect} +\DeclareMathOperator{\param}{Param} +\DeclareMathOperator{\inn}{In} +\DeclareMathOperator{\out}{Out} +\DeclareMathOperator{\neu}{NN} +\DeclareMathOperator{\hid}{Hid} +\DeclareMathOperator{\lay}{Lay} +\DeclareMathOperator{\dep}{Dep} +\DeclareMathOperator{\we}{Weight} +\DeclareMathOperator{\bi}{Bias} +\DeclareMathOperator{\aff}{Aff} +\DeclareMathOperator{\act}{Act} +\DeclareMathOperator{\real}{Rlz} +\DeclareMathOperator{\id}{Id} +\DeclareMathOperator{\mult}{Mult} +\DeclareMathOperator{\wid}{Wid} +\DeclareMathOperator{\sm}{Sum} +\DeclareMathOperator{\trn}{Trn} +\DeclareMathOperator{\cpy}{Copy} +\DeclareMathOperator{\ex}{Ex} +\DeclareMathOperator{\lin}{Lin} +\DeclareMathOperator{\relu}{ReLU} +\DeclareMathOperator{\zero}{Zr} + +\DeclareMathOperator{\obj}{obj} +\DeclareMathOperator{\dom}{dom} +\DeclareMathOperator{\cod}{cod} + +\newcommand{\bbP}{\mathbb{P}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\mft}{\mathfrak{t}} +\newcommand{\f}{\mathfrak{f}} +\newcommand{\C}{\mathfrak{C}} +\newcommand{\n}{\mathscr{N}} +\newcommand{\lp}{\left(} +\newcommand{\rp}{\right)} +\newcommand{\rb}{\right]} +\newcommand{\lb}{\left[} +\newcommand{\lv}{\left|} +\newcommand{\rv}{\right|} +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\ve}{\varepsilon} +\newcommand{\les}{\leqslant} +\newcommand{\ges}{\geqslant} + +% Input layer neurons'number +\newcommand{\inputnum}{3} + +% Hidden layer neurons'number +\newcommand{\hiddennum}{5} + +% Output layer neurons'number +\newcommand{\outputnum}{2} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{corollary}{Corollary}[theorem] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{claim}[theorem]{Claim} + +\title{Reformulation without f} +\author{Shakil Rafi} +\begin{document} +\maketitle + +\begin{lemma} + Let $T \in (0,\infty)$, let $\lp \Omega, \mathcal{F}, \mathbb{P}\rp $ be a probability space, let $\alpha_d \in \R^d \rightarrow \R$, $d \in \N$, be infinitely often differentiable functions, let $u_d \in C^{1,2} \lp \lb 0,T \rb \times, \R^d, \R \rp$, $d \in \N$, satisfy for all $d \in \N$, $t \in \lb 0,T\rb$, $x \in \R^d$ that: + \begin{align} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + \lp \Delta_x u_d \rp \lp t,x \rp + \alpha_x \lp x \rp u_d \lp t,x \rp = 0 + \end{align} + Let $\mathcal{W}^d: \lb 0,T \rb \times \Omega \rightarrow \R^d$, $d \in \N$ be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: \lb t,T\rb \times \Omega \rightarrow \R^d$, $d \in \N$, $t\in \lb 0,T \rb$, $s \in \lb t,T\rb$, $x\in \R^d$ we have $\mathbb{P}$-a.s. that: + \begin{align} + \mathcal{X}^{d,t,x}_s = x+ \int^t_s \sqrt{2} d\mathcal{W}^d_r + \end{align} + Then for all $d \in \N$, $t\in \lb 0,T\rb$, $x\in \R^d$ it holds that: + \begin{align} + u_d\lp t,x \rp + \mathbb{E} \lb \exp \lp \int^T_t \alpha_x \lp \mathcal{X}^{d,t,x}_r \rp dr \rp u_d \lp T, \mathcal{X}^{d,t,x}_T \rp \rb + \end{align} +\end{lemma} + +\begin{proof} + Let $T \in [0,\infty)$, and let $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$ be a probability space. For all $d \in \N$, let $V \in C^{1,1} \lp \R^d \times [0,T],\R \rp $ be $V(x,t) = \alpha_d(x)$, let $\sigma_d : \R^d \rightarrow \R^{d\times d}$ be given by $\sigma_d(x) = \diag_d \lp \sqrt{2} \rp$, let $\mu_d: \R^d \rightarrow \R^d$ be given by $\mu_d(x) = \mymathbb{0}_d$, and finally let $f(t,x) = 0$. By Feynman-Kac and substituting the above, the following expression: + \begin{align}\label{0.0.4} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + \frac{1}{2}\Trace\lp \sigma(t,x) \lb \sigma(t,x) \rb^* \lp \Hess_x(u_d \rp \lp t,x \rp \rp + \la \mu(t,x), \lp \nabla_x u_d \rp \lp t,x\rp \ra +V(t,x)u_d(t,x) \nonumber\\ +f(t,x) = 0 + \end{align} + is rendered: + \begin{align}\label{0.0.5} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp + \lp \Delta_x u_d \rp \lp t,x\rp + \alpha_d(x)u_d(x) = 0 + \end{align} + Note then that Feyman-Kac sates that the solution to (\ref{0.0.4}) can be written as: + \begin{align} + u(t,x) = \E \lb \int^T_t e^{\int^r_t V (\mathcal{X}_t,\tau) d\tau}f(\mathcal{X}_r, r )dr + e^{-\int^T_t V(\mathcal{X}_\tau,\tau)d\tau } u(\mathcal{X}_T,T) \rb + \end{align} + Where $\mathcal{X}$ is an $\lp \Omega, \mathcal{F}, \mathbb{P} \rp$-adapted stochastic process given by: + \begin{align} + \mathcal{X}_t = x + \int^t_s \mu_d \lp \mathcal{X} \rp dr+\int^t_s\sqrt{2}d\mathcal{W}_r^d + \end{align} + Note then that the substitutions then yield that the solution to (\ref{0.0.5}) is given by: + \begin{align} + u(t,x) = \E \lb \exp \lp \int^T_t \alpha_d \lp \mathcal{X} \rp dr \rp u_d \lp T,\mathcal{X}^{d,t,x}_T \rp \rb + \end{align} +\end{proof} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +\end{document} \ No newline at end of file diff --git a/Scratch/Feynman-Kac.tex b/Scratch/Feynman-Kac.tex new file mode 100644 index 0000000..98acd27 --- /dev/null +++ b/Scratch/Feynman-Kac.tex @@ -0,0 +1,118 @@ +\documentclass[11pt]{report} +\usepackage{setspace} +\doublespacing +\usepackage[toc,page]{appendix} + +\usepackage[]{amsmath} +\usepackage[]{amsthm} +\usepackage{mathtools} +\numberwithin{equation}{section} +\usepackage[]{amssymb} +\usepackage[margin=1in]{geometry} +\usepackage[]{soul} +\usepackage[]{bbm} +\usepackage[]{cancel} +\usepackage[]{xcolor} +\usepackage[]{enumitem} +\usepackage{mathrsfs} +\usepackage{hyperref} +\usepackage[capitalise]{cleveref} +\usepackage{natbib} +\usepackage{neuralnetwork} +\usepackage{witharrows} + +\usepackage{listings} +\usepackage{graphicx} + +\DeclareMathAlphabet{\mymathbb}{U}{BOONDOX-ds}{m}{n} + +% \usepackage[]{enumerate} +\setlength\parindent{0pt} + +\DeclareMathOperator{\Trace}{Trace} +\DeclareMathOperator{\Hess}{Hess} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\domain}{Domain} +\DeclareMathOperator{\lip}{Lip} + +\usepackage{tikz-cd} + +\DeclareMathOperator{\rect}{Rect} +\DeclareMathOperator{\param}{Param} +\DeclareMathOperator{\inn}{In} +\DeclareMathOperator{\out}{Out} +\DeclareMathOperator{\neu}{NN} +\DeclareMathOperator{\hid}{Hid} +\DeclareMathOperator{\lay}{Lay} +\DeclareMathOperator{\dep}{Dep} +\DeclareMathOperator{\we}{Weight} +\DeclareMathOperator{\bi}{Bias} +\DeclareMathOperator{\aff}{Aff} +\DeclareMathOperator{\act}{Act} +\DeclareMathOperator{\real}{Rlz} +\DeclareMathOperator{\id}{Id} +\DeclareMathOperator{\mult}{Mult} +\DeclareMathOperator{\wid}{Wid} +\DeclareMathOperator{\sm}{Sum} +\DeclareMathOperator{\trn}{Trn} +\DeclareMathOperator{\cpy}{Copy} +\DeclareMathOperator{\ex}{Ex} +\DeclareMathOperator{\lin}{Lin} +\DeclareMathOperator{\relu}{ReLU} +\DeclareMathOperator{\zero}{Zr} + +\DeclareMathOperator{\obj}{obj} +\DeclareMathOperator{\dom}{dom} +\DeclareMathOperator{\cod}{cod} + +\newcommand{\bbP}{\mathbb{P}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\mft}{\mathfrak{t}} +\newcommand{\f}{\mathfrak{f}} +\newcommand{\C}{\mathfrak{C}} +\newcommand{\n}{\mathscr{N}} +\newcommand{\lp}{\left(} +\newcommand{\rp}{\right)} +\newcommand{\rb}{\right]} +\newcommand{\lb}{\left[} +\newcommand{\lv}{\left|} +\newcommand{\rv}{\right|} +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\ve}{\varepsilon} +\newcommand{\les}{\leqslant} +\newcommand{\ges}{\geqslant} + +% Input layer neurons'number +\newcommand{\inputnum}{3} + +% Hidden layer neurons'number +\newcommand{\hiddennum}{5} + +% Output layer neurons'number +\newcommand{\outputnum}{2} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{corollary}{Corollary}[theorem] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{claim}[theorem]{Claim} + +\title{Reformulation without f} +\author{Shakil Rafi} +\begin{document} +\maketitle + +\begin{theorem} + Consider the following diffeq: + \begin{align} + \lp \frac{\partial }{\partial t} u \rp \lp t,x \rp + \la \mu \lp t,x \rp, \nabla_x \lp t,x \rp \ra + \frac{1}{2} \lp \Trace \lp \Hess_x u_d \rp \lp x \rp \rp -V\lp t,x \rp u \lp t,x\rp +f\lp t,x \rp = 0 + \end{align} +\end{theorem} + +\end{document} \ No newline at end of file diff --git a/Scratch/Neural network diffeq.pdf b/Scratch/Neural network diffeq.pdf new file mode 100644 index 0000000..0c8b2a9 Binary files /dev/null and b/Scratch/Neural network diffeq.pdf differ diff --git a/Scratch/Neural network diffeq.tex b/Scratch/Neural network diffeq.tex new file mode 100644 index 0000000..03654f1 --- /dev/null +++ b/Scratch/Neural network diffeq.tex @@ -0,0 +1,179 @@ +\documentclass[11pt]{report} +\usepackage{setspace} +\doublespacing +\usepackage[toc,page]{appendix} + +\usepackage[]{amsmath} +\usepackage[]{amsthm} +\usepackage{mathtools} +\numberwithin{equation}{section} +\usepackage[]{amssymb} +\usepackage[margin=1in]{geometry} +\usepackage[]{soul} +\usepackage[]{bbm} +\usepackage[]{cancel} +\usepackage[]{xcolor} +\usepackage[]{enumitem} +\usepackage{mathrsfs} +\usepackage{hyperref} +\usepackage[capitalise]{cleveref} +\usepackage{natbib} +\usepackage{neuralnetwork} +\usepackage{witharrows} + +\usepackage{listings} +\usepackage{graphicx} + +\DeclareMathAlphabet{\mymathbb}{U}{BOONDOX-ds}{m}{n} + +% \usepackage[]{enumerate} +\setlength\parindent{0pt} + +\DeclareMathOperator{\Trace}{Trace} +\DeclareMathOperator{\Hess}{Hess} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\domain}{Domain} +\DeclareMathOperator{\lip}{Lip} + +\usepackage{tikz-cd} + +\DeclareMathOperator{\rect}{Rect} +\DeclareMathOperator{\param}{Param} +\DeclareMathOperator{\inn}{In} +\DeclareMathOperator{\out}{Out} +\DeclareMathOperator{\neu}{NN} +\DeclareMathOperator{\hid}{Hid} +\DeclareMathOperator{\lay}{Lay} +\DeclareMathOperator{\dep}{Dep} +\DeclareMathOperator{\we}{Weight} +\DeclareMathOperator{\bi}{Bias} +\DeclareMathOperator{\aff}{Aff} +\DeclareMathOperator{\act}{Act} +\DeclareMathOperator{\real}{Rlz} +\DeclareMathOperator{\id}{Id} +\DeclareMathOperator{\mult}{Mult} +\DeclareMathOperator{\wid}{Wid} +\DeclareMathOperator{\sm}{Sum} +\DeclareMathOperator{\trn}{Trn} +\DeclareMathOperator{\cpy}{Copy} +\DeclareMathOperator{\ex}{Ex} +\DeclareMathOperator{\lin}{Lin} +\DeclareMathOperator{\relu}{ReLU} +\DeclareMathOperator{\zero}{Zr} + +\DeclareMathOperator{\obj}{obj} +\DeclareMathOperator{\dom}{dom} +\DeclareMathOperator{\cod}{cod} + +\newcommand{\bbP}{\mathbb{P}} +\newcommand{\E}{\mathbb{E}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\mft}{\mathfrak{t}} +\newcommand{\f}{\mathfrak{f}} +\newcommand{\C}{\mathfrak{C}} +\newcommand{\n}{\mathscr{N}} +\newcommand{\lp}{\left(} +\newcommand{\rp}{\right)} +\newcommand{\rb}{\right]} +\newcommand{\lb}{\left[} +\newcommand{\lv}{\left|} +\newcommand{\rv}{\right|} +\newcommand{\la}{\langle} +\newcommand{\ra}{\rangle} +\newcommand{\ve}{\varepsilon} +\newcommand{\les}{\leqslant} +\newcommand{\ges}{\geqslant} + +% Input layer neurons'number +\newcommand{\inputnum}{3} + +% Hidden layer neurons'number +\newcommand{\hiddennum}{5} + +% Output layer neurons'number +\newcommand{\outputnum}{2} + + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{corollary}{Corollary}[theorem] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{claim}[theorem]{Claim} + +\title{Reformulation without f} +\author{Shakil Rafi} +\begin{document} +\maketitle + +\begin{lemma} + Let $\Theta = \bigcup_{n\in\N}$, $d,m,\mathfrak{d} \in \N$, $T \in (0,\infty)$, $\act \in C \lp \R,\R\rp$, $\mathfrak{I}, \mathbf{F}, \mathbf{G} \in \neu$ satisfy $\lay\lp \mathfrak{I} \rp = \lp 1,\mathfrak{d},1\rp$, $\real_{\act} \lp \mathfrak{I} \rp = \mathbb{I}_1$, $\real_{\act} \lp \mathbf{F} \rp \in C\lp \R, \R \rp$, and $\real_{\act} \lp \mathbf{G} \rp \in C \lp \R^d, \R \rp$, for every $\theta \in \Theta$ let $\mathcal{U}^\theta: \lb 0,T \rb \rightarrow \lb 0,T \rb$ and $\mathcal{W}^\theta : \lb 0,T \rb \rightarrow \R^d$, be functions, for every $\theta \in \Theta$, $n\in \N_0$ let $U^\theta_n: [0,T] \times \R^d \rightarrow \R$ satisfy for all $t\in [0,T]$, $x \in \R^d$, that: + \begin{align*} + &U^\theta_n(t,x) = \frac{\mymathbb{1}_\N (n)}{M^n} \lb \sum^{m^n}_{k=1} \lp \real_{\act} \lp \mathbf{G}\rp \rp \lp x+\mathcal{W}^{\lp \theta,0,-k \rp}_{T-t} \rp \rb \\ + &+\sum^{n-1}_{i=1} \frac{T-t}{M^{n-i}} \lb \sum^{M^{n-i}}_{k=1} \lp \lp \real_{\act} \lp \mathbf{F}\rp \circ U_i^{(\theta,i,k)} \rp -\mymathbb{1}_\N \lp i \rp \lp \real_{\act} \lp \mathbf{F}\rp \circ U^{\lp \theta,-i,k \rp} \rp \rp \lp \mathcal{U}_t^{\lp \theta,i,k \rp},x+\mathcal{W}^{\lp\theta,i,k\rp}_{\mathcal{U}_t^{\lp \theta,i,k\rp}-t} \rp \rb + \end{align*} + and let $\mathbf{U}^\theta_{n,t} \in \neu$, $t \in [0,T]$, $n\in \Z$, $\theta \in \Theta$, satisfy for all $\theta \in \Theta$, $n\in \N$, $t\in [0,T]$ that $\mathbf{U}^\theta_{0,t} = \lp \lp 0 \:0\:\cdots \:0 \rp,0 \rp \in \R^{1 \times d} \times \R$ and: + \begin{align*} + \mathbf{U}^\theta_{n,t} &= \lb \oplus^{M^n}_{k=1} \lp \frac{1}{M^n} \circledast \lp \mathbf{G}\bullet \aff_{\mathbb{I}_d, \mathcal{W}^{\lp \theta,0,-k \rp }_{T-t}} \rp \rp \rb \\ &\boxplus_{\mathfrak{I}} \lb \boxplus_{i=0,\mathfrak{I}} \lb \lp \frac{T-t}{M^{n-i}}\rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathfrak{I}}\lp \lp \mathbf{F}\bullet \mathbf{U}^{\lp \theta,i,k \rp}_{i,\mathcal{U}_t^{\lp \theta,i,k \rp}}\rp \bullet \aff_{\mathbb{I}_d,\mathcal{W}^{\lp \theta,i,k \rp}_{\mathcal{U}^{\lp \theta,i,k \rp}-t}} \rp \rp \rb \rb \\ + &\boxplus_{\mathfrak{I}}\lb \boxplus^{n-1}_{i=0,\mathfrak{I}}\lb \lp \frac{(t-T)\mymathbb{1}_\N \lp i \rp }{M^{n-i}} \rp \circledast \lp \boxplus^{M^{n-i}}_{k=1,\mathfrak{I}}\lp \lp \mathbf{F}\bullet \mathbf{U}^{\lp \theta,-i,k \rp }_{\max\{i-1,0\},\mathcal{U}_t^{\lp \theta,i,k \rp}} \rp \bullet \aff_{\mathbb{I}_d,\mathcal{W}^{\lp \theta,i,k \rp }_{\mathcal{U}_t^{\lp \theta,i,k \rp}-t}}\rp \rp \rb \rb + \end{align*} +\end{lemma} +\begin{theorem}\label{thm1} + Let $p,q,r,L,C,\alpha_0,\alpha_1,\beta_0,\beta_1, T \in [0,\infty)$, $\mathfrak{q} \in [2,\infty)$, $\act \in C(\R,\R)$, $\mathfrak{I} \in \neu$. $\lp \mathbf{F}_{d,\ve} \rp _{\lp d,\ve \rp \in \N_0 \times (0,1]} \subsetneq \neu$. For every $d \in \N_0$ let $f_d \in C \lp \R^{\max\{d,1\}},\R \rp$, for every $d \in \N$ let $\nu_d: \mathcal{B} \lp \R^d \rp \rightarrow [0,1]$ be a probability measure, and assume for all $d \in \N_0$, $v,w \in \R$, $x \in \R^{\max\{d,1\}}$, $\ve \in (0,1]$ that $\lp \int_{\R^d}\left\|y\right\|^{pq\mathfrak{q}} \nu_d \lp dy\rp \rp ^{\frac{1}{pq\mathfrak{q}}}\les Cd^r$, $\hid\lp \mathfrak{I} \rp=1$, $\real_{\act} \lp \mathfrak{I} \rp = \id_\R $, $\real_{\act} \lp \mathbf{F}_{d,\ve} \rp \in C \lp \R^{\max\{d,1\}},\R \rp $, $\max \{ \left|f_0(v)-f_0(w)\right|, \left|\lp \real_{\act} \lp \mathbf{F}_{0,\ve} \rp \rp \lp x \rp - \lp \real_{\act} \lp \mathbf{F}_{0,\ve} \rp \rp \lp x \rp \right|\} \les L \left|v-w\right|$, $\ve^{\alpha_{\min\{d,1\}}}\dep \lp \mathbf{F}\rp_{d,\ve}+\ve^{\beta_{\min\{d,1\}}}\left\| \lay \lp \mathbf{F}_{d,\ve} \rp \right\|_{\max} \les C \lp \max \{x,1\}^p \rp$, and: + \begin{align} + \ve \left| \lp \real_{\act} \lp \mathbf{F}_{d,\ve}\rp \rp \lp x \rp \right| + \left|f_d\lp x\rp - \lp \real_{\act}\lp \mathbf{F}_{d,\ve} \rp \rp \lp x \rp \right| \les \ve C \lp \max \{x,1\}\rp ^p \lp 1+ \left\|x \right\| \rp ^{pq} + \end{align} + It is then the case that for every $d \in \N$, there exists a $u_d \in C \lp \lb 0,T \rb \in \R^d,\R \rp$ with the following properties: + \begin{enumerate}[label = (\roman*)] + \item $u_d$ is polynomially growing. + \item $u_d$ is a viscosity solution. + \item $u_d$ is a solution to: + \begin{align} + \lp \frac{\partial}{\partial t} u_d \rp \lp t,x \rp +\frac{1}{2}\Trace\lp \sigma_d \lp x \rp \lb \sigma_d \lp x \rp \rb ^*\lp \Hess_x u_d \rp \lp t,x\rp \rp + \la u_d \lp x \rp, \lp \nabla_xu_d \rp \lp t,x \rp \ra + \alpha_d(x)u_d(t,x)= 0 \nonumber + \end{align} + with $u_d \lp T,x \rp =g_d \lp x \rp$ for $\lp t,x \rp \in \lp 0,T \rp \times \R^d$, and + \item there exist $\lp \mathbf{U}_{d,t,\ve} \rp _{ \lp d,t,\ve \rp \in \N \times \lb 0,T \rb \times \lp 0,1\rb}$ and $\eta \in \lp \eta_\delta \rp _{\delta \in \lp 0,\infty \rp}: \R \rightarrow \R$ such that for all $d \in \N$, $t \in \lb 0,T \rb$, $\ve \in (0,1]$, $\delta \in \lp 0,\infty \rp $ it holds that $\real_{\act} \lp \mathbf{U}_{d, t,\ve} \rp \in C \lp \R^d, \R \rp$, $\param \lp \mathbf{U}_{d,t,\ve} \rp \les \eta_\delta d^{p\lp 7+4q+ \lp 2+q \rp \delta \rp } \ve ^{-\lp 4+2\delta + \max\{\alpha_0,\alpha_1\}+2\max\{\beta_0,\beta_1\} \rp}$ and: + \begin{align} + \lp \int_{\R^d} \left|u_d\lp t,x \rp -\lp \real_{\act} \lp \mathbf{U}_{d,t,\ve} \rp \rp \lp x \rp \right|^\mathfrak{q} \nu_d \lp dx\rp \rp ^{\frac{1}{\mathfrak{q}}} \les \ve + \end{align} + \end{enumerate} + +\end{theorem} + +\begin{proof} + The proof of Theorem \ref{thm1} is thus complete +\end{proof} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +\end{document} \ No newline at end of file diff --git a/Scratch/Scratch for Lemma 3.2 b/Scratch/Scratch for Lemma 3.2 new file mode 100644 index 0000000..e69de29 diff --git a/Scratch/Scratch for Lemma 32 b/Scratch/Scratch 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+\newcommand{\N}{\mathbb{N}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\mft}{\mathfrak{t}} +\newcommand{\f}{\mathfrak{f}} +\newcommand{\C}{\mathfrak{C}} +\newcommand{\n}{\mathscr{N}} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{corollary}{Corollary}[theorem] +\newtheorem{lemma}[theorem]{Lemma} + +\title{Reformulation without f} +\author{Shakil Rafi} +\begin{document} +\maketitle + + + + + + + + + + + + +\end{document} \ No newline at end of file diff --git a/Scratch/Theorems with f = 0.tex b/Scratch/Theorems with f = 0.tex new file mode 100644 index 0000000..ea77b73 --- /dev/null +++ b/Scratch/Theorems with f = 0.tex @@ -0,0 +1,414 @@ +\documentclass{article} +\usepackage[]{amsmath} +\numberwithin{equation}{section} +\usepackage[]{amssymb} +\usepackage[a4paper, total={6in, 8in}]{geometry} +\usepackage[]{soul} +\usepackage[]{bbm} +\usepackage[]{cancel} +\usepackage[]{xcolor} +\usepackage[]{enumitem} +\usepackage{mathrsfs} +% \usepackage[]{enumerate} +\setlength\parindent{0pt} +\newcommand{\E}{\mathbb{E}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\mft}{\mathfrak{t}} +\newcommand{\f}{\mathfrak{f}} +\newcommand{\C}{\mathfrak{C}} +\newcommand{\n}{\mathscr{N}} +\title{Reformulation without f} +\author{Shakil Rafi} +\begin{document} +\maketitle +\begin{abstract} + \noindent This document aims to reformulate, rather naively, the theorems and lemmas of HJKP21 with $f=0$. Things that can be outright dismissed with will be typed with a \st{strikethrough}. Those that are to be modified will be given with the color \textcolor{red}{red}, but only the first instance of that variable will be colored. Note that \textcolor{blue}{blue} represents confusion and doubt. Reformulated theorems and lemmas will be referred to by their corresponding number in HJKP21 but the with suffix R. For example Lemma 3.5 from HJKP21, once reformulated will be referred to as Lemma R3.5 +\end{abstract} + +\textbf{Theorem 1.1 Reformulation} Let $T,\kappa,\delta, p \in (0, \infty)$ and $\Theta = \bigcup_{n\in \mathbb{N}}\mathbb{Z}^n$ be Lipschitz continuous. Let $u_d\in C^{1,2}([0,T] \times \mathbb{R}^d,\mathbb{R}),d\in \mathbb{N}$, satisfy $\forall d\in \mathbb{N}, t\in[0,T], x = (x_1,x_2,...,x_d)\in \mathbb{R}^d$ that +$|u_d(t,x)|\leq \kappa d^{\kappa}(1+\sum^d_{k=1}|x_k|)^\kappa$ and: +\begin{align} + \left(\frac{\partial}{\partial t} u_d\right)(t,x) = (\Delta_x u_d)(t,x) +\end{align} +Let $(\Omega, \mathcal{F},\mathbb{P})$, represent a probability space and let $W^{d,\theta} : [0,T] \times \Omega \rightarrow \mathbb{R}^d, d\in \mathbb{N}, \theta \in \Theta$ be independent standard Brownian motions, assume that $(W^{d,\theta})_{(d,\theta) \in \mathbb{N}\times \Theta}$ are independent, let $\phi: \mathbb{N} \rightarrow \mathbb{N}$ and $U^{d,\theta}_{n,m}:[0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R},d,n,m\in \mathbb{Z}, \theta \in \Theta$, satisfy $\forall n \in \mathbb{N}_0$$d,m \in \mathbb{N}, \theta \in \Theta, t\in [0,T], x\in \mathbb{R}^d$ that $\phi(m) = \max\{k\in \mathbb{N}: k \leq \exp(|ln(m)|^{\frac{1}{2}})\}$ and: +\begin{align*} + U^{d,\theta}_{n,m}(t,x)=\frac{}{(\phi(m))^n}\left[\sum^{(\phi(m))^n}_{k=1} u_d(0,x+\sqrt{2} W_t^{d,(\theta,0,-k)})\right] +\end{align*} +\medskip +and for every $d,n,m\in \mathbb{N}$, let $\mathfrak{C}_{d,n,m} \in \mathbb{N}$ be the number of function evaluatons of $f$ and of $u_d(0,\cdot)$ the number of realizations of scalar random variables that are used to compute one realization of $U^{d,0}_{n,m}(T,0):\Omega \rightarrow \mathbb{R}$. Then there exists $c\in \mathbb{R}$ and $\mathcal{n}: \mathbb{N}\times (0,1] \rightarrow \mathbb{N}$ such that $\forall d \in \mathbb{N}, \epsilon \in (0,1]$ it holds: +\begin{align} + (\mathbb{E}[|u_d(T,0)-U^{d,0}_{\mathcal{n}(d,\epsilon),\mathcal{n}(d,\epsilon)}(T,0)|^p])^{\frac{1}{p}} \leq \epsilon \quad \text{ and } \quad \textcolor{red}{\mathfrak{C}_{d,n(d,\epsilon),n(d,\epsilon)} \leq cd^c \epsilon^{-(2+\delta)}} +\end{align} + +\textbf{Reformulation of (1.5)}. Feynman-Kac proves that the solution functions $u_d:[0,T] \times \mathbb{R}^d \rightarrow \mathbb{R},d\in \mathbb{N}$ to (1) are the unique at most polynomially growing functions which satisfy $\forall d\in \mathbb{N}, \theta \in \Theta, t\in [0,T], x\in \mathbb{R}^d$ that: +\begin{align*} + u_d(t,x) = \mathbb{E}\left[u_d(0,x+\sqrt{2}W^{d,\theta}_t)\right] +\end{align*} + +\textbf{Reformulation of (1.6)}{Since in our case we have that $f=0$, we can then say from Lemmas 3.3,3.5, and (3.24) in HJKP21 that: +\begin{align*} + \mathbb{E}[U^{d,\theta}_{n,m}(t,x)] - \mathbb{E}[u_d(0,x+\sqrt{2}W^{d,\theta}_t)] &= 0 +\end{align*} +This follows from Lemmas R3.3, R3.5, and (R3.24) +\medskip +\section{Reformulation for the case without a forcing term}\label{sec:formulanof} +\textbf{Setting 1.1}\label{setting11} Let $d,m \in \mathbb{N}$ $T,L, \mathfrak{L},p \in [0,\infty)$, $\mathfrak{p} \in [2,\infty)$ $\mathfrak{m} = \mathfrak{k}_{\mathfrak{p}}\sqrt{\mathfrak{p}-1}$, $\Theta = \bigcup_{n\in \mathbb{N}}\mathbb{Z}^n$, $g \in C(\mathbb{R}^d,\mathbb{R})$, assume $\forall t \in [0,T],x\in \mathbb{R}^d$ that: +\begin{align} + \max\{|g(x)|\} \leq \mathfrak{L}(1+||x||^p) +\end{align} + +and let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space. Let $W^{\theta}: [0,T] \times \Omega \rightarrow \mathbb{R}^d$, $\theta \in \Theta$ be independent standard brownian motions, $(W^\theta)_{\theta \in \Theta}$ are independent, let $u \in C([0,T] \times \mathbb{R}^d,\mathbb{R})$ satisfy $\forall t \in [0,T]$,$x\in \mathbb{R}^d$, that $\mathbb{E}[|g(x+W^0_{T-t})] < \infty$ and: +\begin{align} + u(t,x) &= \mathbb{E}[g(x+W^0_{T-t})] +\end{align} +and let let $U^\theta_n:[0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$, $\theta \in \Theta$ satisfy $\forall n \in \mathbb{N}_0$,$\theta \in \Theta, t \in [0,T],x\in \mathbb{R}^d$, that: +\begin{align} + U^\theta_n(t,x) = \frac{1}{m^n}\left[\sum^{m^n}_{k=1}g(x+W^{(\theta,0,-k)})\right] +\end{align} + +\textbf{Lemma R3.3:} Assume Setting \ref{seting11}. Then: +\begin{enumerate}[label = (\roman*)] + \item it holds $\forall n \in \mathbb{N}_0$ $\theta \in \Theta$ that $U^\theta_n:[0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$ is a consinuous random field. + \item it holds $\forall n \in \mathbb{N}_0$, $\theta \in \Theta$ that $\sigma(U^\theta_n) \subseteq \sigma ((W^{(\theta, \mathcal{V})})_{\mathcal{V} \in \Theta})$ + \item it holds $\forall n \in \mathbb{N}_0$ that $(U^\theta_n)_{\theta \in\Theta}$,$(W^\theta)_{\theta \in \Theta}$, are independent + \item it holds $\forall n,m \in \mathbb{N}_0$, $i,k,\mathfrak{i},\mathfrak{k}\in \mathbb{Z}$, with $(i,k) \neq (\mathfrak{i},\mathfrak{k})$ that $(U_n^{(\theta,i,k)})_{\theta \in \Theta}$ and $(U_n^{(\theta,\mathfrak{i},\mathfrak{k})})_{\theta \in \Theta}$ are independent and, + \item it holds $\forall n \in \mathbb{N}_0$ that $(U^\theta_n)_{\theta \in \Theta}$ are identically distributed random variables +\end{enumerate} +\textit{Proof of Lemma R3.3:} For (i) Consider that $W^{(\theta,0,-k)}_{T-t}$ are continuous random fields and that $g\in C(\mathbb{R}^d,\mathbb{R})$, we have that $U^\theta_n(t,x)$ is the composition of continuous functions with $m^n > 0$ by hypothesis, ensuring no singularities. Thus $U^\theta_n: [0,T] \times \mathbb{R}^d\times \Omega \rightarrow \mathbb{R}$. + +\medskip +For (ii) observe that $\forall \theta \in \Theta$ it holds that $W^\theta$ is $\mathcal{B}([0,T] \otimes \sigma(W^\theta))/\mathcal{B}(\mathbb{R}^d)$-measurable, this, and induction on $\mathbb{N}_0$ prove item \textit{ii}. + +\medskip +Moreover observe that item (ii) and the fact that $\forall \theta \in \Theta$ it holds that $(W^{(\theta, \vartheta)}_{\vartheta \in \Theta})$, $W^\theta$ are independent establish item (iii). + +\medskip +Furthermore, note that (ii) and the fact that $\forall i,k,\mathfrak{i},\mathfrak{k} \in \mathbb{Z}$,$\theta \in \Theta$, with $(i,k) \neq (\mathfrak{i},\mathfrak{k})$ it holds that $(W^{(\theta,i,k,\vartheta)})_{\vartheta \in \Theta}$ and $(W^{(\theta,\mathfrak{i},\mathfrak{k},\vartheta)})_{\vartheta \in \Theta}$ are independent establish item (iv). + +\medskip +Hutzenhaler [30, Corollary 2.5] and induction on $\mathbb{N}_0$ establish item (v). This completes the proof of Lemma R3.3. +$\hfill \square$ + +\medskip +\textbf{Lemma R3.4} Assume setting R3.2. Then it holds $\forall n \in \mathbb{N}_0,\theta \in \Theta, s \in [0,T], t\in [s,T],x\in \mathbb{R}^d$ that: +\begin{align} + \mathbb{E}[|U^\theta_n(t,x+W^\theta_{t-s})|] +\mathbb{E}[|g(x+W^\theta_{t-s})|]+ \int^T_s \mathbb{E}[|U^\theta_n(r,x+W^\theta_{r-s})|]dr < \infty +\end{align} +\textit{Proof or Lemma R3.4:} Note that (3), the fact that $\forall r,a,b \in [0,\infty)$ it holds that $(a+b)^r \leq 2^{\max\{r-1,0\}}(a^r+b^r)$, and the fact that $\forall \theta \in \Theta$ it holds that $\mathbb{E}[||W^\theta_T||] < \infty$ assure that for all $s \in [0,T],t\in[s,T],\theta \in \Theta$ it holds that: +\begin{align} + \mathbb{E}[|g(x+W^\theta_{t-s})|] &\leq \mathbb{E}[\mathfrak{L}(1+||x+W^\theta_{t-s}||^p)] \\ + &\leq \mathfrak{L}[1+2^{\max\{p-1,0\}}(||x||^p+\mathbb{E}[||W^\theta_T||^p])]<\infty +\end{align} + +We next claim that $\forall n \in \mathbb{N}_0, s\in [0,T],t\in[s,T],\theta \in \Theta$ it holds that: +\begin{align} + \mathbb{E}[|U^\theta_n(t,x+W^\theta_{t-s})|]+ \int^T_s \mathbb{E}[|U^\theta_n(r,x+W^\theta_{r-s})|]dr < \infty \label{foo} +\end{align} + +To prove this claim observe the triangle inequality and \textbf{Setting R3.2}, demonstrate that for all $s\in[0,T]$ $t\in[s,T]$ $\theta \in \Theta$ it holds that: +\begin{align} + \mathbb{E}[|U^\theta_n(t,x+W^\theta_{t-s})|] \leq \frac{}{m^n}\left[ \sum^{m^n}_{i=1}\mathbb{E}[|g(x+W^\theta_{t-s}+W^{(\theta,0,-i)}_{T-t})]\right] +\end{align} + +Now observe that (8) and the fact that $(W^\theta)_{\theta \in \Theta}$ are independent imply that for all $s \in [0,T]$$t\in [s,T]$$\theta \in \Theta$ $i\in \mathbb{Z}$ it holds that: +\begin{align} + \mathbb{E}[|g(x+W^\theta_{t-s}+W^{(\theta,0,i)}_{T-t})|] = \mathbb{E}[|g(x+W^\theta_{(t-s)+(T-t)})|] = \mathbb{E}[|g(x+W^\theta_{T-s})|] <\infty +\end{align} +\medskip +Combining (10) and (11) demonstrate that for all $s \in [0,T]$ $t\in[s,T]$ $\theta \in \Theta$ it holds that: +\begin{align} + \mathbb{E}[|U^\theta_n(t,x+W^\theta_{t-s})|] < \infty +\end{align} + +Finally observe that for all $s\in [0,T]$ $\theta \in \Theta$ it holds that: +\begin{align} +\int^T_s \mathbb{E}[|U^\theta_n(r,x+W^\theta_{r-s})|] &\leq (T-s) \sup_{r\in [s,T]} \mathbb{E} [|U^\theta_n(r,x+W^\theta_{r-s})|] \\ +&\leq (T-s)(\mathbb{E}[|g(x+W^\theta_{T-s})]) <\infty +\end{align} + +Combining (9), (12), and (13) prove Lemma R 3.4. This completes the proof of Lemma R3.4. + +$\hfill \square$ +\medskip + +\textbf{Lemma R3.5} Assume reformulated setting 3.2, then: +\begin{enumerate}[label = (\roman*)] + \item it holds $\forall n \in \mathbb{N}_0, t \in [0,T],x\in \mathbb{R}^d$ that: + \begin{align} + \mathbb{E}[|U^0_n(t,x)|]+ \mathbb{E}[|g(x+W^{(0,0,-1)}_{T-t})] < \infty + \end{align} + \item it holds $\forall n\in \mathbb{N}_0, t\in [0,T],x\in \mathbb{R}^d$ that: + \begin{align*} + \mathbb{E}[U^0_n(t,x)] = \mathbbm{1}_{\mathbb{N}}(n) \mathbb{E}[g(x+W^{(0,0,-1)}_{T-t})] + \end{align*} +\end{enumerate} +\textit{Proof of Lemma R3.5}. (i) is a restatement of Lemma R3.4 in that for all $n\in \mathbb{N}_0$ $t\in [0,T]$: +\begin{align} + \mathbb{E}[|(U^0_n)t,x)|] + \mathbb{E}[|g(x+W^{(0,0,-1)}_{T-t})|] &< \\ + \mathbb{E}[|U^\theta_n(t,x+W^\theta_{t-s})|] +\mathbb{E}[|g(x+W^\theta_{t-s})|]+ \int^T_s \mathbb{E}[|U^\theta_n(r,x+W^\theta_{r-s})|]dr &< \infty +\end{align} + +Furthermore (ii) is a restatement of (5) with $\theta = 0$ +This completes the proof of Lemma R3.5 $\hfill \square$ + +\medskip +\textbf{Monte Carlo Approximations:} Let $p \in (2,\infty)$,$n\in \mathbb{N}$, let $(\Omega, \mathcal{F}, \mathbb{P})$, be a probability space and let $X_i: \Omega \rightarrow \mathbb{R}$, $i \in \{1,2,...,n\}$ be i.i.d. random variables with $\mathbb{E}[|X_1|]<\infty$. Then it holds that: +\begin{align} + (\E[|\E[X_1]-\frac{1}{n}(\sum^n_{i=1}X_i)|^p])^{\frac{1}{p}} \leq \left[\frac{p-1}{n}\right]^{\frac{1}{2}}\left(\E[|X_1-\E[X_1]|^p)]\right)^{\frac{1}{p}} +\end{align} +The hypothesis that for all $i \in \{1,2,...,n\}$ it holds that $X_i:\Omega \rightarrow \mathbb{R}$ ensures that: +\begin{align} + \E[|\E[X_1]-\frac{1}{n}\left(\sum^n_{i=1}X_i \right)|^p] = \E \left[\frac{1}{n} \left(\sum^n_{i=1}\left(\E[X_1]-X_i\right) \right)\right] +\end{align} +\medskip +\textbf{Lemma R3.9}: Assume reformualated Setting 3.2. Then it holds for all $n\in \mathbb{N}_0$,$t\in [0,T]$,$x\in \mathbb{R}^d$, that: +\begin{align} + &\left(\E\left[\left|U^0_n(t,x+W^0_t)-\E \left[U^0_n \left(t,x+W^0_t \right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \\ + &\leq \frac{\mathfrak{m}}{m^{\frac{n}{2}}} \left[\left(\E\left[|g(x+W^0_T)|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}}\right] \\ +\end{align} +\textit{Proof of Lemma R3.9}: For notational simplicity, let $G_k: [0,T] \times \mathbb{R}^d \times \Omega \rightarrow \mathbb{R}$,$k\in \mathbb{Z}$, satisfy for all $k\in \mathbb{Z}$,$t\in[0,T]$,$x\in \mathbb{R}^d$ that: +\begin{align} + G_k(t,x) = g\left(x+W^{(0,0,-k)}_{T-t}\right) +\end{align} +\medskip +Observe that the hypothesis that $(W^\theta)_{\theta \in \Theta}$ are independent Brownian motions and the hypothesis that $g \in C(\mathbb{R}^d,\mathbb{R})$ assure that for all $t \in [0,T]$,$x\in \mathbb{R}^d$ it holds that $(G_k(t,x))_{k\in \mathbb{Z}}$ are i.i.d. random variables. This and Corollary 3.8 (applied for every $n\in \mathbb{N}$, $t\in [0,T]$, $x\in \mathbb{R}^d$ with $p \curvearrowleft \mathfrak{p}$, $n \curvearrowleft m^n$,$(X_k)_{k\in \{1,2,...,m^n\}} \curvearrowleft (G_k(t,x))_{k\in \{1,2,...,m^n\}}$), with the notation of Corollary 3.8 ensure that for all $n\in \mathbb{N}_0$, $t\in [0,T]$, $x \in \mathbb{R}^d$, it holds that: +\begin{align} + \left( \E \left[ \left| \frac{1}{m^n} \left[ \sum^{m^n}_{k=1} G_k(t,x) \right] - \E \left[ G_1(t,x) \right] \right| ^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} \leq \frac{\mathfrak{m}}{m^{\frac{n}{2}}}\left(\E \left[|G_1(t,x)|^\mathfrak{p} \right] \right)^{\frac{1}{\mathfrak{p}}} +\end{align} +\medskip +Combining this, with (23), (24), and item (ii) of Lemma R3.5 yields that: +\begin{align} + &\left(\E\left[\left|U^0_n(t,x) - \E \left[U^0_n(t,x)\right]\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} \\ + &= \left(\E \left[\left|\frac{}{m^n}\left[\sum^{m^n}_{k=1}G_k(t,x)\right]- \E \left[G_1(t,x)\right]\right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} \\ + &\leq \frac{\mathbbm{1}(n)\mathfrak{m}}{m^{\frac{n}{2}}}\left(\E \left[\left| G_1(t,x)\right| ^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \\ + &= \frac{\mathbbm{1}(n)\mathfrak{m}}{m^{\frac{n}{2}}} \left[\left(\E \left[\left|g\left(x+W^1_{T-t}\right)\right|^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}}\right] +\end{align} +This and the fact that $W^0$ has independent increments ensure that for all $n\in \mathbb{N}_0$, $t\in [0,T]$, $x\in \mathbb{R}^d$ it holds that: +\begin{align} + \left(\E \left[\left| U^0_n \left(t,x+W^0_t\right) - \E \left[U^0_n \left(t,x+W^0_t\right)\right]\right|^\mathfrak{p}\right]\right)^{\frac{1}{\mathfrak{p}}} \leq \frac{\mathfrak{m}}{m^{\frac{n}{2}}} \left[\left(\E \left[\left| g \left(x+W^0_T\right)\right|^\p\right]\right)^{\frac{1}{\mathfrak{p}}} \right] +\end{align} +This completes the proof of Lemma R3.9 +$\hfill \square$ + +\medskip +\textbf{Lemma R3.10:} Assume setting R3.2. Then it holds for all $n\in \mathbb{N}_0$, $t\in [0,T]$, $x\in \mathbb{R}^d$ that: +\begin{align} + \left(\E \left[ \left| U^0_n \left(t,x+W^0_t\right) - u \left(t,x+W^0_t\right) \right|^\mathfrak{p} \right]\right)^\frac{1}{\mathfrak{p}} \leq \left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right) \left( \E \left[\left| g \left(x+W^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} +\end{align} + +Observe that from Lemma R3.5 item (ii) we have: +\begin{align} + \E\left[U^0_n(t,x)\right] = \E \left[ g \left(x+W^{(0,0,-1)}_{T-t}\right) \right] +\end{align} +This and ($4$) ensure that: +\begin{equation} + u(t,x) - \E \left[U^0_n(t,x) \right] = \begin{cases} + u(t,x) & :n=0 \\ + 0 & :n\in \mathbb{N} + \end{cases} +\end{equation} +This, and the fact that $W^0$ has independent increments, assure that for all $n\in \mathbb{N}_0$, $t\in [0,T]$, $x\in \mathbb{R}^d$, it holds that: +\begin{align} + \left(\E \left[\left| \E \left[U^0_n(t,x+W^0_t)\right] - u(t,x+W^0_t)\right| ^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} \leq \mathbbm{1}_{\{0\}}(n) \left(\E \left[|u(t,x+W^0_t)|^\p\right]\right) +\end{align} +This along with (4) ensure that: +\begin{align} + \left(\E \left[\left| \E \left[U^0_n(t,x+W^0_t)\right] - u(t,x+W^0_t)\right| ^\mathfrak{p}\right]\right)^\frac{1}{\mathfrak{p}} \leq \mathbbm{1}_{\{0\}}(n) \left( \E \left[\left| g \left(x+W^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} +\end{align} +Notice that the triangle inequality then gives us: +\begin{align} + \left(\E \left[ \left| U^0_n \left(t,x+W^0_t\right) - u \left(t,x+W^0_t\right) \right|^\mathfrak{p} \right]\right)^\frac{1}{\mathfrak{p}} \leq \left(\E \left[\left| U^0_n(t,x+W^0_t) - \E \left[U^0(t,x+W^0_t)\right]\right|^\p\right]\right)^{\frac{1}{\p}} \\ + +\left(\E \left[\left| \E \left[U^0_n(t,x+W^0_t)\right]-u(t,x+W^0_t)\right|^\p\right]\right)^{\frac{1}{\p}} +\end{align} +This, combined with (34) and (29) gives us: +\begin{align} + \left(\E \left[ \left| U^0_n \left(t,x+W^0_t\right) - u \left(t,x+W^0_t\right) \right|^\mathfrak{p} \right]\right)^\frac{1}{\mathfrak{p}} \leq \left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right) \left( \E \left[\left| g \left(x+W^0_T\right) \right|^\mathfrak{p} \right]\right)^{\frac{1}{\mathfrak{p}}} +\end{align} +This completes the proof of Lemma R3.10 +$\hfill \square$ + +\medskip + +\textbf{Lemma R3.14:} Assume setting R3.2 then it holds for all $n \in \mathbb{N}_0$, $t\in [0,T]$, $x\in \mathbb{R}^d$ that: +\begin{align} + \left( \E \left[ \left| U^0_n \left(t,x+W^0_t \right) - u \left( t, x+W^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \leq \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} + +\medskip + +\textit{Proof of Lemma R3.14}: Observe that Lemma R3.10 ensures that: +\begin{align} + \left( \E \left[ \left| U^0_n \left(t,x+W^0_t \right) - u \left( t, x+W^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \leq \left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\E \left[\left| g \left(x+W^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} +\end{align} +Observe next that (4) ensures that: +\begin{align} + \left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\E \left[\left| g \left(x+W^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} \leq \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\E \left[ \left( 1+ \left| \left| x+W^0_T \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +Which in turn yields that: +\begin{align} + \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\E \left[ \left( 1+ \left| \left| x+W^0_T \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} \leq \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +Combining (36),(37), and (38) yields that: +\begin{align} + \left( \E \left[ \left| U^0_n \left(t,x+W^0_t \right) - u \left( t, x+W^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \leq \left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\E \left[\left| g \left(x+W^0_T\right)\right|^\p\right]\right)^\frac{1}{\p} \leq \\ + \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s\in[0,T]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +This completes the proof of Lemma R3.14 +$\hfill \square$ +\medskip + +\textbf{Corollary R3.15:} Assume setting R3.2. Then it holds for all $n\in \mathbb{N}_0$, $t\in[0,T]$, $x\in \R^d$ that: +\begin{align} + \left( \E \left[ \left| U^0_n(t,x) -u(t,x) \right| ^\p \right] \right) ^{\frac{1}{\p}} \leq \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +\textit{Proof of Corollary R3.15:} Observe that for all $t \in [0,T-\mft]$ and $\mft \in [0,T]$, and the fact that $W^0$ has independent increments it is the case that: +\begin{align} + u(t+\mft,x) = \E \left[g \left(x+W^0_{T-(t+\mft)}\right)\right] = \E \left[g \left(x+W^0_{(T-\mft)-t)}\right)\right] +\end{align} +And it is also the case that: +\begin{align*} + U^\theta_n(t+\mft,x) = \frac{}{m^n} \left[\sum^{m^n}_{k=1} g \left(x+W^{(\theta,0,-k)}_{T-(t+\mft)}\right)\right] = \frac{}{m^n} \left[\sum^{m^n}_{k=1} g \left(x+W^{(\theta,0,-k)}_{(T-\mft)-t}\right)\right] +\end{align*} +\medskip +Then, applying \textbf{Lemma 3.14}, applied for all $\mft \in [0,T]$, with $\mathfrak{L} \curvearrowleft \mathfrak{L}$, $p \curvearrowleft p$, $\mathfrak{p} \curvearrowleft \mathfrak{p}$, $T \curvearrowleft (T-\mft)$ is such that for all $\mft \in [0,T]$, $t \in [0,T-\mft]$, $x \in \R^d$, and $n\in \mathbb{N}_0$ we have: +\begin{align} + \left( \E \left[ \left| U^0_n \left(t+\mft,x+W^0_t \right) - u \left( t+\mft, x+W^0_t \right) \right|^\p \right] \right)^{\frac{1}{\p}} \leq \\ + \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s \in [0,T-\mft]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} \\ + \leq \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +Thus we get for all $\mft \in [0,T]$, $x\in \R^d$, $n \in \mathbb{N}_0$: +\begin{align} + \left( \E \left[ \left| U^0_n \left(\mft,x \right) - u \left(\mft, x \right) \right|^\p \right] \right)^{\frac{1}{\p}} &= \left( \E \left[ \left| U^0_n \left(\mft,x+W^0_0 \right) - u \left(\mft, x+W^0_0 \right) \right|^\p \right] \right)^{\frac{1}{\p}} \\ + &\leq \mathfrak{L}\left( \frac{ \mathfrak{m}}{m^{\frac{n}{2}}}\right)\left(\sup_{s \in [0,T]}\E \left[ \left( 1+ \left| \left| x+W^0_s \right| \right|^p \right)^\p\right]\right)^\frac{1}{\p} +\end{align} +This completes the proof of Lemma R3.15 +$\hfill \square$ +\medskip + +\textbf{Proposition 4.4:} Let $T,L,p,q \in [0,\infty), m \in \mathbb{N}$, $\Theta = \bigcup_{n\in \mathbb{N}} \Z^n$, let $g_d\in C(\R^d,\R)$, and assume that $d\in \N$, $t \in [0,T]$, $x = (x_1,x_2,...,x_d)\in \R^d$, $v,w \in \R$ and that $\max \{ |g_d(x)|\} \leq Ld^p \left(1+\Sigma^d_{k=1}\left|x_k \right|\right)$, let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $W^{d,\theta}: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, $\theta \in \Theta$, be independent standard Brownian motions, assume for every $d\in \N$ that $\left(W^{d,\theta}\right)_{\theta \in \Theta}$ are independent, let $u_d \in C([0,T] \times \R^d,\R)$, $d \in \N$, satisfy for all $d\in \N$, $t\in [0,T]$, $x \in \R^d$ that $\E \left[g_x \left(x+W^{d,0}_{T-t} \right)\right] < \infty$ and: +\begin{align} + u_d\left(t,x\right) = \E \left[g_d \left(x + W^{d,0}_{T-t}\right)\right] +\end{align} +Let $U^{d,\theta}_{n}: [0,T] \times \R^d \times \Omega \rightarrow \R$, $d$, $n\in \Z$, $\theta \in \Theta$, satisfy for all $n \in \N_0$, $d\in \N$, $\theta \in \Theta$, $t\in [0,T]$, $x\in \R^d$ that: +\begin{align} + U^{d,\theta}_{n}(t,x) = \frac{1}{m} \left[\sum^{m}_{k=1} g_d \left(x + W^{d,(\theta, 0,-k)}_{T-t}\right)\right] +\end{align} +and let $\mathfrak{C}_{d,n} \in \R$, $d$, $n$, $ \in \N_0$, satisfy for all $d \in \N$, $n\in \N_0$, that: +\begin{align} + \mathfrak{C}_{d} \leq \mathbbm{1}_\N(n) \alpha d^{\mathfrak{d}}m +\end{align} +Then there exists $\mathscr{M}_\epsilon \in \mathbb{Z}$ such that it holds that: + +$\mathscr{N}: \N \times \R \rightarrow \N$ such that for all $d \in \N$, $\epsilon$, $\delta \in (0,\infty)$, $ \mathfrak{p} \in [2,\infty)$ with $\text{limsup}_{j \rightarrow \infty} \left[\frac{(m_j)^{\frac{p}{2}}}{j}\right] < \infty$ it holds that: +\begin{align} + (\n(d,\epsilon))^\beta \C_{d,\n(d,\epsilon),\n(d,\epsilon)} \leq \alpha d^{\mathfrak{d}}( \mathscr{N} (d,\epsilon))^\beta(1+ \sqrt{2})^{\n(d,\epsilon)} \\ + \leq \alpha c_{\p,\delta} d^{\mathfrak{d}+(p+q)(2+\delta)}(\min\{1,\epsilon\})^{-(2+\delta)} \leq \epsilon +\end{align} +and: +\begin{align} + \sup_{t\in[0,T]} \sup_{x \in [-L,L]^d} \left(\E \left[\left| u_d(t,x) - U^{d,0}_{\n(d,\epsilon),\n(d,\epsilon)}\right|^\p\right]\right)^\frac{1}{\p} \leq \epsilon +\end{align} +\textit{Proof of Proposition 4.4:} Throughout the proof let $\mathfrak{m}_\mathfrak{p} = \sqrt{\mathfrak{p} -1}$, $\mathfrak{p} \in [2,\infty)$, let $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfy for all $d \in \N$, $t\in [0,T]$ that: +\begin{align} +\mathbb{F}^d_t = \begin{cases} + \bigcap_{s\in[t,T]} \sigma \left(\sigma \left(W^{d,0}_r: r \in [0,s]\right) \cup \{A\in \mathcal{F}: \mathbb{P}(A)=0\}\right) & :t -- 29 +< -- 0 ++ -- 13 +- -- 10 +* -- 61 +:= -- 122 +add.period$ -- 7 +call.type$ -- 2 +change.case$ -- 11 +chr.to.int$ -- 0 +cite$ -- 2 +duplicate$ -- 18 +empty$ -- 57 +format.name$ -- 10 +if$ -- 141 +int.to.chr$ -- 0 +int.to.str$ -- 2 +missing$ -- 2 +newline$ -- 14 +num.names$ -- 4 +pop$ -- 6 +preamble$ -- 1 +purify$ -- 9 +quote$ -- 0 +skip$ -- 11 +stack$ -- 0 +substring$ -- 63 +swap$ -- 4 +text.length$ -- 0 +text.prefix$ -- 0 +top$ -- 0 +type$ -- 8 +warning$ -- 0 +while$ -- 7 +width$ -- 3 +write$ -- 24 diff --git a/Templates/Articles/Article_template.log b/Templates/Articles/Article_template.log new file mode 100644 index 0000000..9711133 --- /dev/null +++ b/Templates/Articles/Article_template.log @@ -0,0 +1,586 @@ +This is pdfTeX, 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[1][-]{section.1}{Introduction}{}% 1 +\BOOKMARK [1][-]{section.2}{Monte Carlo approximations}{}% 2 diff --git a/Templates/Articles/Article_template.pdf b/Templates/Articles/Article_template.pdf new file mode 100644 index 0000000..e174d1b Binary files /dev/null and b/Templates/Articles/Article_template.pdf differ diff --git a/Templates/Articles/Article_template.synctex.gz b/Templates/Articles/Article_template.synctex.gz new file mode 100644 index 0000000..9e4089b Binary files /dev/null and b/Templates/Articles/Article_template.synctex.gz differ diff --git a/Templates/Articles/Article_template.tex b/Templates/Articles/Article_template.tex new file mode 100644 index 0000000..648ca7b --- /dev/null +++ b/Templates/Articles/Article_template.tex @@ -0,0 +1,356 @@ +\documentclass[12pt]{article} + +\usepackage{amsmath, + mleftright, + amssymb, + amsthm, + nicefrac, + etoolbox, + xparse, + geometry, + enumitem, + mathtools, + } + +\mleftright + +\usepackage[colorlinks=true]{hyperref} + +\geometry{margin=1in} + +\usepackage[sort,capitalize]{cleveref} +\newcommand{\creflastconjunction}{, and\nobreakspace} +\crefformat{equation}{(#2#1#3)} +\crefname{enumi}{item}{items} +\crefname{equation}{}{} + +\theoremstyle{plain} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{example}[theorem]{Example} +\newtheorem{setting}[theorem]{Setting} +\newtheorem{conjecture}[theorem]{Conjecture} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} + +\numberwithin{equation}{section} + +%%% + +\input{commands.tex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +% +% +% + +\title{Article template} + +% +% +% + +\author{ +Joshua Lee Padgett$^{1,2}$ +\bigskip +\\ +\small{$^1$ Department of Mathematical Sciences, University of Arkansas,} +\vspace{-0.1cm}\\ +\small{Arkansas, USA, e-mail: \texttt{padgett@uark.edu}} +\smallskip +\\ +\small{$^2$ Center for Astrophysics, Space Physics, and Engineering Research,} +\vspace{-0.1cm}\\ +\small{Baylor University, Texas, USA, e-mail: \texttt{padgett@uark.edu}} +} + +% +% +% + +\date{\today} + +% +% +% + +\maketitle + +\begin{abstract} +% +Abstract goes here\dots +% +\end{abstract} + +% +% +% + +\tableofcontents + +% +% +% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Introduction} +\label{sec:intro} + +% +% +% + +Add an appropriate introduction\dots +% +% + +% +% +% + +\section{Monte Carlo approximations} +\label{sec:monte_carlo} + +% +% +% + +\begin{athm}{lemma}{Lp_monte_carlo} +% +Let +$p \in (2,\infty)$, +$n \in \N$, +let +$ ( \Omega, \cF, \P ) $ +be a probability space, +and let +$ X_i \colon \Omega \to \R $, $ i \in \{1, 2, \dots, n\} $, +be i.i.d.\ random variables with +$\E[ \abs{ X_1 } ] < \infty$. +% +% +Then it holds that +% +\begin{equation} +\pr[\Big]{ \E \br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[\big]{ \smallsum_{ i = 1 }^{ n } X_{ i } } }^p } }^{\!\!\nicefrac{1}{p}} +\le +\br[\big]{ \tfrac{p-1}{n} }^{\nicefrac{1}{2}} +\pr[\Big]{ \E \br[\pig]{ \abs[\big]{ X_1 - \E[ X_1 ] }^p } }^{\!\!\nicefrac{1}{p}} +\dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +First, \nobs that +\Enum{ +the hypothesis that for all $i\in\{1,2,\allowbreak\dots,\allowbreak n\}$ it holds that +$ X_i\colon \Omega\to \R $ +are i.i.d.\ random variables +} +that +% +\begin{equation} +\begin{split} +\E\br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr{ \smallsum_{ i = 1 }^{ n } X_{ i } } }^p } +& += +\E\br[\pig]{ \abs[\big]{ \tfrac{1}{n} \pr{ \smallsum_{ i = 1 }^{ n } (\E[X_1] - X_{ i }) } }^p } +\\ +& += +n^{-p} \, \E\br[\pig]{ \abs[\big]{ \smallsum_{ i = 1 }^{ n } (\E[X_i] - X_{ i }) }^p } +\dpp +\end{split} +\end{equation} +% +% +\Enum{ +This +;the fact that for all $i\in\{1,2,\dots,n\}$ it holds that +$ X_i\colon \Omega\to \R $ +are i.i.d.\ random variables +; +\eg Rio~\cite[Theorem 2.1]{rio2009moment} (applied with $p \with p$, $(S_i)_{i\in\{0,1,\dots,n\}} \with (\sum_{k=1}^i (\E[X_k] - X_k))_{i\in\{0,1,\dots,n\}}$, $(X_i)_{i \in\{1,2,\dots,n\}} \with (\E[X_i] - X_i)_{i\in\{1,2,\dots,n\}}$ in the notation of Rio~\cite[Theorem 2.1]{rio2009moment}) +} +that +% +\begin{equation} +\begin{split} +\pr[\Big]{ \E \br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[\big]{ \smallsum_{ i = 1 }^{ n } X_{ i } } }^p } }^{\!\!\nicefrac{2}{p}} +& += +\tfrac{1}{n^2} \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \smallsum_{ i = 1 }^{ n } (\E[X_i] - X_{ i }) }^p } }^{\!\!\nicefrac{2}{p}} +\\ +& +\le +\tfrac{(p-1)}{n^2} \br[\Big]{ \smallsum_{ i = 1 }^{ n } \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \E[X_i] - X_{ i } }^p } }^{\!\!\nicefrac{2}{p}} } +\\ +& += +\tfrac{(p-1)}{n^2} \br[\Big]{ n \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \E[X_1] - X_{ 1 } }^p } }^{\!\!\nicefrac{2}{p}} } +\\ +& += +\tfrac{(p-1)}{n} \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ \E[X_1] - X_{ 1 } }^p } }^{\!\!\nicefrac{2}{p}} +\dpp +\end{split} +\end{equation} +% +% +\end{aproof} + +% +% +% + +\begin{athm}{corollary}{prop:mlp} +% +Let $p \in [2,\infty)$, +$n \in \N$, +let +$ ( \Omega, \cF, \P ) $ +be a probability space, +and +let +$ X_i \colon \Omega \to \R $, +$ i \in \{1, 2, \dots, n\} $, +be i.i.d.\ random variables with +$\E\br[]{ \abs{ X_1 } } < \infty$. +% +% +Then it holds that +% +\begin{equation}\label{eq:2_9} +\pr[\Big]{\E\br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[]{ {\smallsum_{ i = 1 }^{ n }} X_{ i } } }^p } }^{\!\!\nicefrac{1}{p}} +\le +\br[\big]{ \tfrac{p-1}{n} }^{\nicefrac{1}{2}} +\pr[\Big]{ \E\br[\pig]{ \abs[\big]{ X_1 - \E[ X_1 ] }^p } }^{\!\!\nicefrac{1}{p}} +\dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +\Nobs that +\Enum{ +\eg Grohs et al.\ \cite[Lemma 2.3]{grohs2018proof} +; +\cref{Lp_monte_carlo} +} +\cref{eq:2_9}. +% +% +\end{aproof} + +% +% +% + +\begin{athm}{definition}{rand_const} +% +Let $p \in [2,\infty)$. +% +% +Then we denote by +$\firstConstant{p} \in \R$ +the real number given by +% +\begin{equation}\label{eq:rand_const} +\firstConstant{p} += +\inf\left\{ c \in \R \colon \left[ +\begin{aligned} +& \text{It holds for every probability space $(\Omega,\cF,\P)$ and every} \\ +& \text{random variable $X \colon \Omega \to \R$ with $\E[\abs{X}] < \infty$ that } \\ +& \pr[\big]{ \E\br[\big]{ \abs{ X - \E[X] }^p } }^{\!\nicefrac{1}{p}} \le c \pr[\big]{ \E\br[\big]{ \abs{X}^p } }^{\!\nicefrac{1}{p}} +\end{aligned} +\right] \right\} +\dpp +\end{equation} +% +% +\end{athm} + +% +% +% + +\begin{athm}{corollary}{cor:exp_bd} +% +Let +$p\in[2,\infty)$, +$n\in\N$, +let +$(\Omega,\cF,\P)$ +be a probability space, +and let +$X_i \colon \Omega \to \R$, $i\in\{1,2,\dots,n\}$, +be i.i.d.\ random variables with +$\E[\abs{X_1}] < \infty$. +% +% +Then +% +\begin{equation}\label{prob_bd1} +\pr[\Big]{\E\br[\pig]{ \abs[\big]{ \E[X_1] - \tfrac{1}{n} \pr[]{ {\smallsum_{ i = 1 }^{ n }} X_{ i } } }^p } }^{\!\!\nicefrac{1}{p}} +\le +\tfrac{\secondConstant{p}}{n^{\nicefrac{1}{2}}} \pr[\Big]{ \E\br[\pig]{ \abs[\big]{ X_1 }^p } }^{\!\!\nicefrac{1}{p}} +\end{equation} +% +(cf.\ \cref{rand_const}). +% +% +\end{athm} + +% +% +% + +\begin{aproof} +% +\Nobs +that +\Enum{ +\cref{rand_const} +; +\cref{prop:mlp} +} +that \cref{prob_bd1} holds. +% +\end{aproof} + +% +% +% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\bibliographystyle{acm} +\bibliography{bibfile} + +\end{document} \ No newline at end of file diff --git a/Templates/Articles/Article_template.toc b/Templates/Articles/Article_template.toc new file mode 100644 index 0000000..5c1f331 --- /dev/null +++ b/Templates/Articles/Article_template.toc @@ -0,0 +1,2 @@ +\contentsline {section}{\numberline {1}Introduction}{1}{section.1}% +\contentsline {section}{\numberline {2}Monte Carlo approximations}{1}{section.2}% diff --git a/Templates/Articles/bibfile.bib b/Templates/Articles/bibfile.bib new file mode 100644 index 0000000..671f37a --- /dev/null +++ b/Templates/Articles/bibfile.bib @@ -0,0 +1,21 @@ +% Encoding: UTF-8 + +@article{grohs2018proof, + title={A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of {B}lack-{S}choles partial differential equations}, + author={Grohs, Philipp and Hornung, Fabian and Jentzen, Arnulf and Von Wurstemberger, Philippe}, + journal={arXiv:1809.02362}, + note={\emph{To appear in Mem.\ Amer.\ Math.\ Soc.}}, + year={2018}, + pages={124 pages}, +} + +@article{rio2009moment, + title={Moment inequalities for sums of dependent random variables under projective conditions}, + author={Rio, Emmanuel}, + journal={J.\ Theor.\ Probab.}, + volume={22}, + number={1}, + pages={146--163}, + year={2009}, + publisher={Springer} +} \ No newline at end of file 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+\newcommand{\qandqShort}{\quad\text{and}\quad} +\newcommand{\andShort}{\text{ and }} +\newcommand{\forallDist}{\forall\,} +\newcommand{\identity}[1]{\mathrm{Id}_{#1}} +\newcommand{\smallsum}{\textstyle\sum} +\newcommand{\SmallSum}[2]{ {\textstyle\sum\limits_{#1}^{#2}}} +\newcommand{\idRelu}{\mathfrak I} +\newcommand{\Exists}{\exists\,} +\newcommand{\Forall}{\forall\,} +\newcommand{\mc}[1]{\mathcal{#1}} +\newcommand{\mf}[1]{\mathfrak{#1}} +\newcommand{\affineMap}{\mathbb{A}} +\newcommand{\idMatrix}{\operatorname{I} \cfadd{def:identityMatrix}} +\newcommand{\ANNs}{\mathbf{N} \cfadd{def:ANN}} +\newcommand{\MatrixANN}[1]{\mf N_{#1}} +\newcommand{\activation}{a} +\newcommand{\activationDim}[1]{\mathfrak{M}_{\activation,#1} \cfadd{multi}} +\newcommand{\functionANN}[1]{\mathcal{R}_{#1} \cfadd{def:ANNrealization}} +\newcommand{\paramANN}{\mathcal{P} \cfadd{def:ANN}} +\newcommand{\paramNotZeroANN}{\mathscr{P}} +\newcommand{\lengthANN}{\mathcal{L} \cfadd{def:ANN}} +\newcommand{\inDimANN}{\mathcal{I} \cfadd{def:ANN}} +\newcommand{\compANN}[2]{{#1 \bullet \allowbreak #2} \cfadd{def:ANNcomposition}} +\newcommand{\concANN}[2]{{#1 \odot #2}} +\newcommand{\concPsiANN}[2]{{#1 \odot_{\Psi} #2}} +\newcommand{\paraANN}[1]{\mathbf{P}_{#1}} +\newcommand{\paraLANN}[2]{\mathbf{P}_{#1,#2}} +\newcommand{\outDimANN}{\mathcal{O} \cfadd{def:ANN}} +\newcommand{\longerANN}[1]{\mathcal{E}_{#1} \cfadd{def:ANNenlargement}} +\newcommand{\idANN}[2]{\mathbb{I}_{#1,#2}} +\newcommand{\idANNshort}[1]{\mathbb{I}_{#1}} +\newcommand{\dims}{\mathcal{D} \cfadd{def:ANN}} +\newcommand{\hiddenLength}{\mathcal{H} \cfadd{def:ANN}} +\newcommand{\hiddenDimId}{\mathfrak{i}} +\newcommand{\qandq}{\qquad\text{and}\qquad} +\newcommand{\andq}{\text{and}\qquad} +\newcommand{\rect}{\mathfrak r \cfadd{rectifier}} +\newcommand{\Rect}{\mathfrak R} +\newcommand{\multdim}{\mathfrak M} +\DeclareMathOperator{\id}{id} +\newcommand{\matrixANN}[1]{\mathfrak{W}_{#1}} +\newcommand{\vectorANN}{\mathfrak{B}} +\newcommand{\sumANN}{\mathfrak{S} \cfadd{def:ANN:sum}} +\newcommand{\extensionANN}{\mathfrak{T} \cfadd{def:ANN:extension}} +\newcommand{\dimANNlevel}{\mathbb{D} \cfadd{def:ANN}} +\newcommand{\power}[2]{#1^{\bullet #2} \cfadd{def:iteratedANNcomposition}} +\newcommand{\scalar}[2]{ #1 \circledast #2 \cfadd{def:ANNscalar}} +\newcommand{\transpose}{* \cfadd{def:Transpose}} +\newcommand{\oSum}{\oplus \cfadd{def:ANNsum:same}} +\newcommand{\OSum}[2]{{\mathop\oplus\limits_{#1}^{#2} } \cfadd{def:ANNsum:same} } +\newcommand{\bSum}{{\mathop\boxplus} \cfadd{def:ANN:sum_diff}} +\newcommand{\BSum}[3]{{\mathop\boxplus\limits_{#1,#2}^{#3}} \cfadd{def:ANN:sum_diff}} +\newcommand{\modCont}{w \cfadd{mod_cont_def}} + +\allowdisplaybreaks + +\usepackage{scalerel,stackengine} +\newcommand\pig[1]{\scalerel*[5pt]{\big#1}{% + \ensurestackMath{\addstackgap[1.5pt]{\big#1}}}} +\newcommand\pigl[1]{\mathopen{\pig{#1}}} +\newcommand\pigr[1]{\mathclose{\pig{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Enum +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentCommand{\enum}{ O{;} m o } + { + \my_enum:nnn { #1 } { #2 } { #3 } + } + +\seq_new:N \l__my_enum_seq +\tl_new:N \l__my_enum_item_tl +\prop_new:N \l__verbs +\prop_put:Nnn \l__verbs {show} {shows} +\prop_put:Nnn \l__verbs {imply} {implies} +\prop_put:Nnn \l__verbs {demonstrate} {demonstrates} +\prop_put:Nnn \l__verbs {prove} {proves} +\prop_put:Nnn \l__verbs {establish} {establishes} +\prop_put:Nnn \l__verbs {ensure} {ensures} +\prop_put:Nnn \l__verbs {assure} {assures} +\prop_put:Nnn \l__verbs {yield} {yields} + +\int_new:N \l__number_of_args + +\cs_new_protected:Nn \my_enum:nnn + { + %\seq_clear:N \l__my_enum_seq + %\clist_map_variable:nNn { #1 } \l__my_enum_item_tl + % { + % \__my_enum_add:V \l__my_enum_item_tl + % } + \seq_set_split:Nnn \l__my_enum_seq { #1 } { #2 } + \seq_remove_all:Nn \l__my_enum_seq {} + \int_set_eq:NN \l__number_of_args { \seq_count:N \l__my_enum_seq } + %triangle + %\int_compare:nF { \seq_count:N \l__my_enum_seq < 2 } { s } + %\nobreakspace + \seq_use:Nnnn \l__my_enum_seq { ~and~ } { ,~ } { ,~and~ } + \IfNoValueTF{#3}{}{ + \space + \int_compare:nNnTF{ \l__number_of_args } < {2}{ \prop_item:Nn \l__verbs {#3} }{ #3 } + } + } + +\ExplSyntaxOff + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic cf-lists + +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\seq_new:N \g_cflist_loaded + +\seq_new:N \g_cflist_pending + +\NewDocumentCommand{\cfadd}{ m } + +{ + + \seq_if_in:NnF \g_cflist_loaded { #1 } { + + \seq_if_in:NnF \g_cflist_pending { #1 } { + + \seq_gput_right:Nn \g_cflist_pending { #1 } + + } + + } + +} + +\NewDocumentCommand{\cfload}{ o } + +{ + + \seq_if_empty:NTF \g_cflist_pending {\unskip} { + + (cf.\ \cref{\seq_use:Nn \g_cflist_pending {,}})\IfValueTF{#1}{#1~}{\unskip} + + \seq_gconcat:NNN \g_cflist_loaded \g_cflist_loaded \g_cflist_pending + + \seq_gclear:N \g_cflist_pending + + } + +} + +\NewDocumentCommand{\cfclear} {} { + + \seq_gclear:N \g_cflist_loaded + + \seq_gclear:N \g_cflist_pending + +} + +\NewDocumentCommand{\cfout}{ o } + +{ + + \seq_if_empty:NTF \g_cflist_pending {\unskip} { + + (cf.\ \cref{\seq_use:Nn \g_cflist_pending {,}})\IfValueTF{#1}{#1~}{\unskip} + + \seq_gclear:N \g_cflist_pending + + } + +} + +\NewDocumentCommand{\ifnocf} { m } { + + \seq_if_empty:NT \g_cflist_pending { #1 } + +} + +\NewDocumentCommand{\cfconsiderloaded}{ m }{ + + \seq_gput_right:Nn \g_cflist_loaded {#1} + +} + +\ExplSyntaxOff + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Automatic proof headers and endings, local labels +%%%%%%%%%%%%%%%%%%%%%%%% + +\NewDocumentEnvironment {athm} {m m} {% +%\newenvironment{athm}[2]{% +\begin{#1}\label{#2}\global\def\loc{#2}% +}{% +\end{#1}% +} + +\NewDocumentEnvironment{aproof} {} {% +\begin{proof}[Proof~of~\cref{\loc}]% +}{% +\finishproofthus +\end{proof}% +} + +\ExplSyntaxOff + +\newcommand{\eqqref}[1]{\cref{eq:\loc.#1}} +\newcommand{\eqlabel}[1]{\label{eq:\loc.#1}} + +\newcommand{\finishproofthus}{The proof of \cref{\loc} is thus complete.} +\newcommand{\finishproofthis}{This completes the proof of \cref{\loc}.} + +%%%%%%%%%%%%%%%%%%%%%%%% +%%% Flexible math displays +%%%%%%%%%%%%%%%%%%%%%%%% + +\ExplSyntaxOn + +\NewDocumentEnvironment{flexmath}{ m o }{ + \str_if_eq:noTF {a} {#1} { + \begin{equation} + \IfValueT{#2}{\label{eq:\loc.#2}} + \begin{aligned} + } { + \catcode`&=9 + \renewcommand{\\}{} + \str_if_eq:noTF {d} {#1} { + \begin{equation} + \IfValueT{#2}{\label{eq:\loc.#2}} + } { + \begin{math} + } + } +}{ + \str_if_eq:noTF {i} {#1} { + \end{math} + \catcode`&=4 + } { + \str_if_eq:noTF {d} {#1} { + \end{equation} + } { + \end{aligned} + \end{equation} + } + } +} + +\ExplSyntaxOff + +\ExplSyntaxOn + +\bool_new:N \g_noteobserve + +\NewDocumentCommand{\setnote}{}{ + \bool_gset_true:N \g_noteobserve +} + +\NewDocumentCommand{\setobserve}{}{ + \bool_gset_false:N \g_noteobserve +} + +\NewDocumentCommand{\nobs}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \str_if_eq:noTF {Note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \bool_gset_false:N \g_noteobserve + } + } + } + \bool_if:nTF { \g_noteobserve } { + \bool_gset_false:N \g_noteobserve + note + } { + \bool_gset_true:N \g_noteobserve + observe + } + \IfValueF{#1}{~} +} + +\NewDocumentCommand{\Nobs}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \str_if_eq:noTF {Note} {#1} { + \bool_gset_true:N \g_noteobserve + } { + \bool_gset_false:N \g_noteobserve + } + } + } + \bool_if:nTF { \g_noteobserve } { + \bool_gset_false:N \g_noteobserve + Note + } { + \bool_gset_true:N \g_noteobserve + Observe + } + \IfValueF{#1}{~} +} + + +\bool_new:N \g_hencetherefore + +\NewDocumentCommand{\hence}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \str_if_eq:noTF {Hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \bool_gset_false:N \g_hencetherefore + } + } + } + \bool_if:nTF { \g_hencetherefore } { + \bool_gset_false:N \g_hencetherefore + hence + } { + \bool_gset_true:N \g_hencetherefore + therefore + } + \IfValueF{#1}{~} +} + +\NewDocumentCommand{\Hence}{ o }{ + \IfValueT{#1}{ + \str_if_eq:noTF {hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \str_if_eq:noTF {Hence} {#1} { + \bool_gset_true:N \g_hencetherefore + } { + \bool_gset_false:N \g_hencetherefore + } + } + } + \bool_if:nTF { \g_hencetherefore } { + \bool_gset_false:N \g_hencetherefore + Hence + } { + \bool_gset_true:N \g_hencetherefore + Therefore + } + \IfValueF{#1}{~} +} + +\int_new:N \g_furthermore + +\NewDocumentCommand{\Moreover}{ o o }{ + \IfValueT{#1}{ + \str_case:nn {#1} { + {Next} {\int_gset:Nn {\g_furthermore} {0}} + {Furthermore} {\int_gset:Nn {\g_furthermore} {1}} + {Moreover} {\int_gset:Nn {\g_furthermore} {2}} + {In~addition} {\int_gset:Nn {\g_furthermore} {3}} + {note} {\bool_gset_true:N \g_noteobserve} + {observe} {\bool_gset_false:N \g_noteobserve} + } + \IfValueT{#2}{ + \str_case:nn {#2} { + {Next} {\int_gset:Nn {\g_furthermore} {0}} + {Furthermore} {\int_gset:Nn {\g_furthermore} {1}} + {Moreover} {\int_gset:Nn {\g_furthermore} {2}} + {In~addition} {\int_gset:Nn {\g_furthermore} {3}} + {note} {\bool_gset_true:N \g_noteobserve} + {observe} {\bool_gset_false:N \g_noteobserve} + } + } + } + \int_case:nn { \int_mod:nn {\g_furthermore} {4} } { + { 0 } { Next,~\nobs that} + { 1 } { Furthermore,~\nobs that} + { 2 } { Moreover,~\nobs that} + { 3 } { In~addition,~\nobs that} + } + \int_gincr:N \g_furthermore + \IfValueF{#1}{~} +} + +\ExplSyntaxOff + + + +\ExplSyntaxOn + +\int_new:N \g_enumit + +\NewDocumentCommand{\Enum}{ m o }{ + \IfValueT{#2}{ + \str_case:nn {#2} { + {ensure} {\int_gset:Nn {\g_enumit} {0}} + {ensures} {\int_gset:Nn {\g_enumit} {0}} + {assure} {\int_gset:Nn {\g_enumit} {1}} + {assures} {\int_gset:Nn {\g_enumit} {1}} + {prove} {\int_gset:Nn {\g_enumit} {2}} + {proves} {\int_gset:Nn {\g_enumit} {2}} + {show} {\int_gset:Nn {\g_enumit} {3}} + {shows} {\int_gset:Nn {\g_enumit} {3}} + {demonstrate} {\int_gset:Nn {\g_enumit} {4}} + {demonstrates} {\int_gset:Nn {\g_enumit} {4}} + {yield} {\int_gset:Nn {\g_enumit} {5}} + {yields} {\int_gset:Nn {\g_enumit} {5}} + {establish} {\int_gset:Nn {\g_enumit} {6}} + {establishes} {\int_gset:Nn {\g_enumit} {6}} + {imply} {\int_gset:Nn {\g_enumit} {7}} + {implies} {\int_gset:Nn {\g_enumit} {7}} + } + } + \int_case:nn { \int_mod:nn {\g_enumit} {8} } { + { 0 } { \enum{ #1 }[ensure]{ } } + { 1 } { \enum{ #1 }[assure] } + { 2 } { \enum{ #1 }[prove] } + { 3 } { \enum{ #1 }[show] } + { 4 } { \enum{ #1 }[demonstrate] } + { 5 } { \enum{ #1 }[yield] } + { 6 } { \enum{ #1 }[establish] } + { 7 } { \enum{ #1 }[imply] } + } + \int_gincr:N \g_enumit +} + +\ExplSyntaxOff \ No newline at end of file diff --git a/Templates/Beamer/.DS_Store b/Templates/Beamer/.DS_Store new file mode 100644 index 0000000..39a9b49 Binary files /dev/null and b/Templates/Beamer/.DS_Store differ diff --git 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number in head/foot}{fg=date in head/foot.bg} +\begin{frame}[noframenumbering] +{Table of Contents} + +{ +\hypersetup{linkcolor=blue} +\tableofcontents[hideothersubsections] +} +\end{frame} +}} + +\newcommand{\frameEND}{\section*{} + +{ +\setbeamercolor{page number in head/foot}{fg=date in head/foot.bg} +\begin{frame}[noframenumbering] +{$\vcenter{\hbox{{\Huge THANK YOU! +\vspace*{1.2mm}}}}$ +} + +\begin{figure} +\centerline{ +\includegraphics[width=42mm]{ +murphy} +\hspace{5mm} +\includegraphics[width=42mm]{ +memphis.jpeg} +} +\end{figure} + +\begin{center} +\Huge Questions? +\end{center} +\end{frame} +}} + +% End TOC set-up + +\usepackage{tikz} +\makeatletter +\newcommand\mathcircled[1]{% + \mathpalette\@mathcircled{#1}% +} +\newcommand\@mathcircled[2]{% + \tikz[baseline=(math.base)] \node[draw,circle,inner sep=1pt,red] (math) {$\m@th#1#2$};% +} +\makeatother + +\setbeamertemplate{headline}{} + +\setbeamertemplate{navigation symbols}{} + +\title[{\color{white}University of Arkansas}]{SOM-where in Chicago: What self-organizing maps reveal about ride-sharing in Chicago during the pandemic.} +\author[Shakil Rafi]{Shakil Rafi} +\institute[]{Ph.D. Candidate, Department of Mathematical Sciences + +University of Arkansas + +\medskip + +\includegraphics[scale=0.2]{UA_Logo_Horizontal} + +\bigskip + +OAK Fall 2023 +} + +\date{{October 26, 2023}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} +\maketitle +\begin{frame}{Contents} + What are SOMs \\~\\ + The Dataset\\~\\ + The Clusters\\~\\ + Future +\end{frame} +\begin{frame} +\frametitle{Self-organizing maps, or Kohonen maps} +Self-organizing maps are neural networks architectures used primarily for dimensionality reduction. \\~\\ +Proposed by Tuevo Kohonen in 1980s, hence also called Kohonen maps. +\\ +\end{frame} +\begin{frame} + \begin{center} + \includegraphics[scale=0.4]{wrap.png} + \end{center} +\end{frame} +\begin{frame}{The mathematical description} + \begin{block}{The algorithm} + initialize lattice nodes;\\ + initialize weight vectors; \\ + N $\leftarrow$ iteration count; \\ + \textbf{for} $i$ in $N$: \\ + \quad $x \leftarrow$ pick random point in dataset; \\ + \quad $c \leftarrow$ select lattice closest to $x$;\\ + \quad move weight vector of c closer to $x$;\\ + \quad move weight vectors of the neighbors of $c$ slightly closer to $x$ + \end{block} + +\end{frame} +\begin{frame}{Lattice Architecture} + Typically we initialize a lattice over the data as either square lattices or hexagonal lattices: + \begin{center} + \includegraphics[scale=0.3]{lattice-structure} + \end{center} + For rectangular lattice Kohonen suggests the $(x,y)$ should be the ratio of the two largest eigenvalues of the autocorrelation matrix. +\end{frame} +\begin{frame}{Weight initialization} + The most common ways of initialization is:\\~\\ + + \textbf{Random initialization} Slow to converge but because SOMs are fast this may not be an issue \\~\\ + \textbf{Random sampling initialization} Take samples from the dataset. +\end{frame} +\begin{frame}{Training the model: Step 1} + Each lattice point has two vectors: $w_i$ representing its weight and $l_j$ representing its position within the lattice. \\~\\ + We start by picking a random point $x \in \mathbb{R}^d$ of our data. We calculate the Euclidean distance of each lattice point from that data and select the lattice point with the smallest such distance: + \begin{align} + c_i = \arg \min_i \| x-w_i \| + \end{align} + Where $c_i$ is the index of the best matching unit. BMU. +\end{frame} +\begin{frame}{Training the model: Step 2} + We update the weights for all the lattice points for the $k+1$-th iteration: + \begin{align} + w^{(k+1)}_i = w_i^{(k)} + \alpha_k\cdot \eta_{c_i,i} \cdot \left\| x-w_i^{(k)} \right\| + \end{align} + Note that $\alpha_k$ is the learning rate s.t.: + \begin{align} + \alpha_{k+1} \leq \alpha_k \\ + \sum^\infty_{k=0} \alpha_k = \infty \\ + \sum^\infty_{k=0} \alpha_k^2 \leq \infty + \end{align} +\end{frame} +\begin{frame}{Training the model: Step 2 cont.} + Note that $\eta_{c_i,i}$ is a neighborhood function, designed to: + \\~\\ + Achieve a maximum when $\|l_i-l_j \| =0$ \\~\\ + $\eta_{c_i,i} \rightarrow 0$ as $\| l_i-l_j\| \rightarrow \infty$ + \\~\\ + The most common is Gaussian neighborhood: + \begin{align} + \eta_{c_i,i} = \exp \left[ -\frac{\|l_i - l_{c_i}\|}{2\sigma_k^2}\right] + \end{align} +\end{frame} +\begin{frame}{Error metrics} + Most common is quantization error: + Let $c = \arg\min_i \|x-w_i \|$. The quantization error is then: + \begin{align} + E_Q = \sum_i \| x_i -w_c\| + \end{align} + The topographic error preserves the underlying topology of the data defined as: + \begin{align} + E_T = \frac{1}{|D|} \sum_{x\in D} te(x) + \end{align} + where: + \begin{align} + te(x) = \begin{cases} + 1 & \text{$c_1$ and $c_2$ are neighbors}\\ + 0 & \text{else} + \end{cases} + \end{align} +\end{frame} +\begin{frame}{The dataset} + We take inspiration from Soria, Chen, and Stathoupolos, 2019.\\~\\ + We look at ride-share pickups (Uber and Lyft) for the city of Chicago in the year 2020. Our data came from the City of Chicago Data Portal\\~\\ + We collate the data by census tracts, and merge the data with demographics (median income) from the US Census. \\~\\ + And built-environment characteristics (percentage of zero car ownerships, distance to nearest transit) from the EPA smart locations dataset. +\end{frame} + +\begin{frame}{The variables} + Our variables are as follows: + \begin{center} + \includegraphics[scale=0.3]{variables.png} + \end{center} +\end{frame} + +\begin{frame}{Preliminary analysis} + Previous work by the author suggests that much can be gained by doing a segmentation analysis of the dataset. \\~\\ + A principal component analysis of the scaled data shows that there exists an eigenvalue in one direction wit both components accounting for $\approx 50\%$ of the variance + \begin{center} + \includegraphics[scale=0.45]{pca-w.png} + \end{center} +\end{frame} + +\begin{frame}{The elbow} + Taking a cue from previous work we do create a $2 \times 2$ lattice. Four clusters seem to be optimal from an elbow method perspective. \\~\\ + + \begin{center} + \includegraphics[scale=0.3]{elbow.png} + \end{center} +\end{frame} +\begin{frame}{Quantization errors} + Quantization errors across iteration counts shows $\sim 500$ iterations to be optimal + \begin{center} + \includegraphics[scale=0.45]{quantization-error-w.png} + \end{center} +\end{frame} +\begin{frame}{The Clustering} + We get a clustering as such: + \begin{center} + \includegraphics[scale=0.55]{cluster-pca.png} + \end{center} +\end{frame} +\begin{frame}{The Clusters I} + +\begin{table}[] +\begin{tabular}{@{}llllll@{}} + +Cluster & MedIncome & PopDensity & EmpDensity & LUDiversity & \% 0 Car\\ \cline{1-6} +0 & \$85.9k & 38.18 & 10.2 & 0.82 & 24\% \\ +1 & \$149k & 10.27 & 10.46 & 3.17 & 10\% \\ +2 & \$73k & 18.32 & 3.41 & 1.01 & 14\% \\ +3 & \$164k & 31.01 & 258.6 & 45.5 & 19\% \\ \cline{1-3} \cline{3-6} +\end{tabular} +\end{table} + +\end{frame} +\begin{frame}{The Clusters cont.} +\begin{table}[] +\begin{tabular}{llll} +\hline +\multicolumn{1}{l|}{Cluster} & \multicolumn{1}{l|}{Median Income} & Trip Seconds & Trip Miles \\ \hline +0 & \$85.9k & 1000.3 & 5.8 \\ +1 & \$149 & 1961 & 7.5 \\ +2 & \$73 & 1249 & 8.5 \\ +3 & \$164 & 951 & 6.14 \\ \cline{1-3} \cline{3-4} +\end{tabular} +\end{table} +\end{frame} +\begin{frame}{The map} + \begin{center} + \includegraphics[scale=0.34]{map.png} + \end{center} +\end{frame} +\begin{frame}{The Clusters cont.} + The full table: + \begin{center} + \includegraphics[scale=0.25]{full-table.png} + \end{center} +\end{frame} +\begin{frame}{The takeaways} + Key takeaways: + \\~\\ + People from richer neighborhoods take shorter Uber trips \\~\\ + Employment density is a better predictor than population density for the number of pickups \\~\\ + Percentage of zero car interacts with median income. + +\end{frame} +\begin{frame}{Future work} + Possible future work: + \\~\\ + Could we do a detailed SHAP analysis of the factors predicting pickups, i.e. is it the case that median income of tracts can be a good predictor?\\~\\ + Could we do a time series analysis and see a seasonality decrease around March 2020. +\end{frame} + +\begin{frame}{Bibliography} + City of Chicago Data Portal, 2021. Transportation Network Providers - Trips + https://data.cityofchicago.org \\~\\ + Ponmalai, Ravi, and Kamath, Chandrika. 2019. "Self-Organizing Maps and Their Applications to Data Analysis". United States. https://doi.org/10.2172/1566795. https://www.osti.gov/servlets/purl/1566795.\\~\\ + R., S.; Nithila, Arna Nishita (2022). Who rides Uber anyway? A census-tract level analysis and clustering of ride-shares for the city of Chicago during the era of the pandemic. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.21076042.v2\\~\\ + Soria, J., Chen, Y., Stathopoulos,A., 2020. K-Prototypes Segmentation Analysis on Large-Scale Ridesourcing Trip Data, Transportation Research Board 2020, DOI: 10.1177/0361198120929338 \\~\\ + Smart Location Database. https://www.epa.gov/smartgrowth/smart- location-mapping\\ + +\end{frame} +\begin{frame} + Check it out: + \begin{center} + \includegraphics[scale=0.7]{qr.png} + \end{center} +\end{frame} +\end{document} diff --git a/Templates/Beamer/JLP_SIAM_TX_LA_11_6_21.toc b/Templates/Beamer/JLP_SIAM_TX_LA_11_6_21.toc new file mode 100644 index 0000000..938ec0e --- /dev/null +++ b/Templates/Beamer/JLP_SIAM_TX_LA_11_6_21.toc @@ -0,0 +1,5 @@ +\beamer@sectionintoc {1}{Introduction}{3}{0}{1} +\beamer@sectionintoc {2}{Semidiscretized algorithm}{22}{0}{2} +\beamer@sectionintoc {3}{Properties of the fully-discretized algorithm}{30}{0}{3} +\beamer@sectionintoc {4}{Convergence of the fully-discretized algorithm}{37}{0}{4} +\beamer@sectionintoc {5}{Conclusions and future endeavors}{42}{0}{5} diff --git 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typesetting math +\usepackage{amssymb} % Adds new symbols to be used in math mode + +\usepackage{booktabs} % Top and bottom rules for tables +\usepackage{enumitem} % Used to reduce itemize/enumerate spacing +\usepackage{palatino} % Use the Palatino font +\usepackage[font=small,labelfont=bf]{caption} % Required for specifying captions to tables and figures + +\usepackage{multicol} % Required for multiple columns +\setlength{\columnsep}{1.5em} % Slightly increase the space between columns +\setlength{\columnseprule}{0mm} % No horizontal rule between columns + +\usepackage{tikz} % Required for flow chart +\usetikzlibrary{shapes,arrows} % Tikz libraries required for the flow chart in the template + +\newcommand{\compresslist}{ % Define a command to reduce spacing within itemize/enumerate environments, this is used right after \begin{itemize} or \begin{enumerate} +\setlength{\itemsep}{1pt} +\setlength{\parskip}{0pt} +\setlength{\parsep}{0pt} +} + +\definecolor{UofA}{RGB}{157,34,53} % Defines the color used for content box headers + +\begin{document} + +\begin{poster} +{ +headerborder=closed, % Adds a border around the header of content boxes +colspacing=1em, % Column spacing +bgColorOne=white, % Background color for the gradient on the left side of the poster +bgColorTwo=white, % Background color for the gradient on the right side of the poster +borderColor=UofA, % Border color +headerColorOne=UofA, % Background color for the header in the content boxes (left side) +headerColorTwo=UofA, % Background color for the header in the content boxes (right side) +headerFontColor=white, % Text color for the header text in the content boxes +boxColorOne=white, % Background color of the content boxes +textborder=roundedleft, % Format of the border around content boxes, can be: none, bars, coils, triangles, rectangle, rounded, roundedsmall, roundedright or faded +eyecatcher=false, % Set to false for ignoring the left logo in the title and move the title left +headershade=plain, +headerheight=0.1\textheight, % Height of the header +headershape=roundedright, % Specify the rounded corner in the content box headers, can be: rectangle, small-rounded, roundedright, roundedleft or rounded +headerfont=\Large\bf\textsc, % Large, bold and sans serif font in the headers of content boxes +%textfont={\setlength{\parindent}{1.5em}}, % Uncomment for paragraph indentation +linewidth=2pt % Width of the border lines around content boxes +} +%---------------------------------------------------------------------------------------- +% TITLE SECTION +%---------------------------------------------------------------------------------------- +% +{\includegraphics[height=4em]{UA_Logo_Horizontal.eps}} % First university/lab logo on the left +{\bf\textsc{A Census-Tract Based Analysis of Rideshare Usage by K- Means Chicago in the pandemic​}\vspace{0.5em}} % Poster title +{\textsc{Joshua Lee Padgett \\ +Department of Mathematical Sciences, University of Arkansas} +\textsc{Joshua Lee Padgett \\ +Department of Mathematical Sciences, University of Arkansas}} % Author names and institution +{\includegraphics[height=4em]{UA_Logo_Horizontal.eps}} % Second university/lab logo on the right + +%---------------------------------------------------------------------------------------- +% OBJECTIVES +%---------------------------------------------------------------------------------------- + +\headerbox{Objectives}{name=objectives,column=0,row=0}{ + +Donec non nisl a \textbf{arcu consequat} varius. Sed suscipit cursus luctus. Nulla sit amet elit augue. Curabitur scelerisque mollis dolor, quis blandit lorem condimentum at. Pellentesque sed nibh vel \textbf{dolor} sagittis semper. + +\begin{enumerate}\compresslist +\item Feugiat vitae elit +\item bibendum ante sed lacinia eros in +\item Curabitur scelerisque arcu consequat varius +\item Dapibus nulla id purus consectetur +\item Fringilla integer +\end{enumerate} + +\vspace{0.3em} % When there are two boxes, some whitespace may need to be added if the one on the right has more content +} + +%---------------------------------------------------------------------------------------- +% INTRODUCTION +%---------------------------------------------------------------------------------------- + +\headerbox{Introduction}{name=introduction,column=1,row=0,bottomaligned=objectives}{ + +Aliquam non lacus dolor, \textit{a aliquam quam}. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Nulla in nibh mauris. Donec vel ligula nisi, a lacinia arcu. Sed mi dui, malesuada vel consectetur et, egestas porta nisi. Sed eleifend pharetra dolor, et dapibus est vulputate eu. \textbf{Integer faucibus elementum felis vitae fringilla.} In hac habitasse platea dictumst. Duis tristique rutrum nisl, nec vulputate elit porta ut. Donec sodales sollicitudin turpis sed convallis. Etiam mauris ligula, blandit adipiscing condimentum eu, dapibus pellentesque risus. +} + +%---------------------------------------------------------------------------------------- +% RESULTS 1 +%---------------------------------------------------------------------------------------- + +\headerbox{Results 1}{name=results,column=2,span=2,row=0}{ + +\begin{multicols}{2} +\vspace{1em} +\begin{center} +\includegraphics[width=0.8\linewidth]{placeholder} +\captionof{figure}{Figure caption} +\end{center} + +Aliquam auctor, metus id ultrices porta, risus enim cursus sapien, quis iaculis sapien tortor sed odio. Mauris ante orci, euismod vitae tincidunt eu, porta ut neque. Aenean sapien est, viverra vel lacinia nec, venenatis eu nulla. Maecenas ut nunc nibh, et tempus libero. Aenean vitae risus ante. Pellentesque condimentum dui. Etiam sagittis purus non tellus tempor volutpat. Donec et dui non massa tristique adipiscing. +\end{multicols} + +%------------------------------------------------ + +\begin{multicols}{2} +\vspace{1em} +Sed fringilla tempus hendrerit. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; Etiam ut elit sit amet metus lobortis consequat sit amet in libero. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Phasellus vel sem magna. Nunc at convallis urna. isus ante. Pellentesque condimentum dui. Etiam sagittis purus non tellus tempor volutpat. Donec et dui non massa tristique adipiscing. Quisque vestibulum eros eu. + +\begin{center} +\includegraphics[width=0.8\linewidth]{placeholder} +\captionof{figure}{Figure caption} +\end{center} + +\end{multicols} +} + +%---------------------------------------------------------------------------------------- +% REFERENCES +%---------------------------------------------------------------------------------------- + +\headerbox{References}{name=references,column=0,above=bottom}{ + +\renewcommand{\section}[2]{\vskip 0.05em} % Get rid of the default "References" section title +\nocite{*} % Insert publications even if they are not cited in the poster +\small{ % Reduce the font size in this block +\bibliographystyle{unsrt} +\bibliography{sample} % Use sample.bib as the bibliography file +}} + +%---------------------------------------------------------------------------------------- +% FUTURE RESEARCH +%---------------------------------------------------------------------------------------- + +\headerbox{Future Research}{name=futureresearch,column=1,span=2,aligned=references,above=bottom}{ % This block is as tall as the references block + +\begin{multicols}{2} +Integer sed lectus vel mauris euismod suscipit. Praesent a est a est ultricies pellentesque. Donec tincidunt, nunc in feugiat varius, lectus lectus auctor lorem, egestas molestie risus erat ut nibh. + +Maecenas viverra ligula a risus blandit vel tincidunt est adipiscing. Suspendisse mollis iaculis sem, in \emph{imperdiet} orci porta vitae. Quisque id dui sed ante sollicitudin sagittis. +\end{multicols} +} + +%---------------------------------------------------------------------------------------- +% CONTACT INFORMATION +%---------------------------------------------------------------------------------------- + +\headerbox{Contact Information}{name=contact,column=3,aligned=references,above=bottom}{ % This block is as tall as the references block + +\begin{description}\compresslist +\item[Web] www.university.edu/smithlab +\item[Email] john@smith.com +\item[Phone] +1 (000) 111 1111 +\end{description} +} + +%---------------------------------------------------------------------------------------- +% CONCLUSION +%---------------------------------------------------------------------------------------- + +\headerbox{Conclusion}{name=conclusion,column=2,span=2,row=0,below=results,above=references}{ + +\begin{multicols}{2} + +\tikzstyle{decision} = [diamond, draw, fill=blue!20, text width=4.5em, text badly centered, node distance=2cm, inner sep=0pt] +\tikzstyle{block} = [rectangle, draw, fill=blue!20, text width=5em, text centered, rounded corners, minimum height=4em] +\tikzstyle{line} = [draw, -latex'] +\tikzstyle{cloud} = [draw, ellipse, fill=red!20, node distance=3cm, minimum height=2em] + +\begin{tikzpicture}[node distance = 2cm, auto] +\node [block] (init) {Initialize Model}; +\node [cloud, left of=init] (Start) {Start}; +\node [cloud, right of=init] (Start2) {Start Two}; +\node [block, below of=init] (init2) {Initialize Two}; +\node [decision, below of=init2] (End) {End}; +\path [line] (init) -- (init2); +\path [line] (init2) -- (End); +\path [line, dashed] (Start) -- (init); +\path [line, dashed] (Start2) -- (init); +\path [line, dashed] (Start2) |- (init2); +\end{tikzpicture} + +%------------------------------------------------ + +\begin{itemize}\compresslist +\item Pellentesque eget orci eros. Fusce ultricies, tellus et pellentesque fringilla, ante massa luctus libero, quis tristique purus urna nec nibh. Phasellus fermentum rutrum elementum. Nam quis justo lectus. +\item Vestibulum sem ante, hendrerit a gravida ac, blandit quis magna. +\item Donec sem metus, facilisis at condimentum eget, vehicula ut massa. Morbi consequat, diam sed convallis tincidunt, arcu nunc. +\item Nunc at convallis urna. isus ante. Pellentesque condimentum dui. Etiam sagittis purus non tellus tempor volutpat. Donec et dui non massa tristique adipiscing. +\end{itemize} + +\end{multicols} +} + +%---------------------------------------------------------------------------------------- +% MATERIALS AND METHODS +%---------------------------------------------------------------------------------------- + +\headerbox{Materials \& Methods}{name=method,column=0,below=objectives,bottomaligned=conclusion}{ % This block's bottom aligns with the bottom of the conclusion block + +The following materials were required to complete the research: + +\begin{itemize}\compresslist +\item Curabitur pellentesque dignissim +\item Eu facilisis est tempus quis +\item Duis porta consequat lorem +\item Eu facilisis est tempus quis +\end{itemize} + +The following equations were used for statistical analysis: + +\begin{equation} +\cos^3 \theta =\frac{1}{4}\cos\theta+\frac{3}{4}\cos 3\theta +\label{eq:refname} +\end{equation}\ + +\begin{equation} +E = mc^{2} +\label{eqn:Einstein} +\end{equation} + +Phasellus imperdiet, tortor vitae congue bibendum, felis enim sagittis lorem, et volutpat ante orci sagittis mi. Morbi rutrum laoreet semper. Morbi accumsan enim nec tortor consectetur non commodo nisi sollicitudin. Proin sollicitudin. Pellentesque eget orci eros. Fusce ultricies, tellus et pellentesque fringilla, ante massa luctus libero, quis tristique purus urna nec nibh. +} + +%---------------------------------------------------------------------------------------- +% RESULTS 2 +%---------------------------------------------------------------------------------------- + +\headerbox{Results 2}{name=results2,column=1,below=objectives,bottomaligned=conclusion}{ % This block's bottom aligns with the bottom of the conclusion block + +Donec faucibus purus at tortor egestas eu fermentum dolor facilisis. Maecenas tempor dui eu neque fringilla rutrum. Mauris \emph{lobortis} nisl accumsan. + +\begin{center} +\begin{tabular}{l l l} +\toprule +\textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2}\\ +\midrule +Treatment 1 & 0.0003262 & 0.562 \\ +Treatment 2 & 0.0015681 & 0.910 \\ +Treatment 3 & 0.0009271 & 0.296 \\ +\bottomrule +\end{tabular} +\captionof{table}{Table caption} +\end{center} + +Nulla ut porttitor enim. Suspendisse venenatis dui eget eros gravida tempor. Mauris feugiat elit et augue placerat ultrices. Morbi accumsan enim nec tortor consectetur non commodo. + +\begin{center} +\begin{tabular}{l l l} +\toprule +\textbf{Treatments} & \textbf{Response 1} & \textbf{Response 2}\\ +\midrule +Treatment 1 & 0.0003262 & 0.562 \\ +Treatment 2 & 0.0015681 & 0.910 \\ +Treatment 3 & 0.0009271 & 0.296 \\ +\bottomrule +\end{tabular} +\captionof{table}{Table caption} +\end{center} +} + +%---------------------------------------------------------------------------------------- + +\end{poster} + +\end{document} \ No newline at end of file diff --git a/Templates/Poster/baposter.cls b/Templates/Poster/baposter.cls new file mode 100644 index 0000000..bcc67da --- /dev/null +++ b/Templates/Poster/baposter.cls @@ -0,0 +1,1094 @@ +%% +%% This is file `baposter.cls' +%% +%% A relatively comfortable latex class to produce posters with a grid based +%% layout. It comes with a number of combinable styles and is (maybe only for +%% the writer) easy to extend, as all the graphics is based on pgf. +%% +%% It is meant to be used with pdftex, but might also work with pslatex if you +%% are not interested in things like transparency. +%% +%% Copyright (C) 2007-2011 Brian Amberg +%% Copyright (C) 2011 Reinhold Kainhofer +%% +%% 29. September 2011: +%% - Finally fixed confusion with paper size handling and landscape. This required seperate handling of papersizes +%% known to the geometry package and other packages. +%% 26. September 2011: +%% - Reverted drawing of faded borders to manual method, as the current result does not work with evince, +%% and produced spurious colored boxes with okular and acroread. +%% - Added one more example due to popular request +%% 16. September 2011: +%% - Removed nag package, such that it compiles on older systems +%% - Added more examples +%% 15. September 2011: +%% - Merged fork by (rk) back into mainline, essentially taking over all the great improvements that Reinhold Kainhofer coded. +%% We call this version 2, as it breaks the interface in some small points. Essentially watch out for this: +%% - no/yes keys are now false/true +%% - the shade-lr etc. keys have the hypen removed, and are now called shadelr etc. +% - Added more examples, which are all adapted to work with the current version +%% 27. August 2011 (rk): +%% - Completely factored out all drawing commands to handler functions, assigned via choice keys +%% - Added cornerradius, boxheaderheight, boxpadding options +%% - Added missing roundedright +%% 26. August 2011 (rk): +%% - Turned headerbox cmd into posterbox environment (allows verbatim) +%% - pass several options to packages +%% - added debug cls option to reduce default output +%% - rewrote several loops and ifthenelse +%% - Use boolean and choice keys for all options +%% - Changed all choice options to set a global function to +%% \baposter@OPTION@OPTIONVALUE, which are a functions that execute the +%% option's setting (drawing routine, color setting for tikz, etc.) +%% - Add a5paper +%% 07. April 2011: Fixed paper size handling +%% - Fixed Paper Size handling, you can now specify all page sizes using paperwidth=, paperheight= class options. +%% - Added rudimentary documentation of the available keys. +%% 11. December 2010: Fixed "Empty Second Page" bug +%% - A second empty page was introduced when a document ended with +%% \end{poster} +%% \end{document} +%% This was fixed thanks to a suggestion by Martin Aulbach +%% - Added +%% \thispagestyle{empty} +%% to suppress page numbers, which became visible when the margins are large. +%% The underlying problem of page-size handling has not yet been solved. +%% 30. September 2010: Border Bugfixes +%% - Correct background color with textborder=rounded-small, Thanks to Ke Chen for the bugreport and patch +%% - Correctly draw rectangular text boxes, Thanks to abenkst for the bugreport and fix +%% 10. June 2010: Added option to set the number of columns +%% - added a class option to set the number of columns +%% - columns=5: sets the number of columns to 5, possible values are 1..6, default is 3 in portrait and 4 in landscape format +%% 29. April 2009: Incorporated Patches by Arne Henningsen +%% - added some class options +%% - a4shrink: shrink the paper to A4 size (for printing drafts or handouts) +%% - movebody=Xpt: move the text/poster body Xpt to the right +%% (or to the left if Xpt is negative), +%% e.g. for manually centering the poster on the page +%% - showframe: use the "showframe" option of the "geometry" package +%% - a0paper (default): set paper size to A0 +%% - archE: set paper size to Arch E +%% - setting "background" can be "none" now (otherwise the "showframe" +%% option has no effect) +%% - the page number has been removed (it was mostly not visible before) +%% - the "margin=" option works now +%% 04. December 2008 +%% - Mainly an update to the landscape example +%% 14. November 2008 +%% - Actually center the title when eyecatcher is used. +%% 04. November 2008 +%% - Fixed bug with eyecatcher not working. +%% 26. June 2008 +%% - Fixed bug with plain background mode. +%% 14. June 2008 +%% - Support for portrait/landscape switching. +%% - Some smaller bugfixes. +%% 01. June 2007 +%% - First version released. +%% +%% Use this class with pdflatex +%% +%% I have confirmed that this package works with +%% - texlive 2007 and +%% - miktex 2.7 +%% +%% It does not seem to work with +%% - miktex 2.2 +%% - some old versions of tetex +%% +%% +%% TODO: +%% -) Rename backgrounds back to shaded-tb shade-lr +%% -) Rename textborder back to rounded-small (consistent naming needed!) +%% -) Rename headershade back to shade-lr, shade-tb, shade-tb-inverse +%% -) Rename headershape back to small-rounded +%% -) Option value consistency (small-rounded vs. rounded-small; missing ones) +%% -) Rename \baposterHeaderSetShade, \baposterHeaderDrawText to include @ and verb +%% +%% +%% Licence: GPL +\ProvidesClass{baposter}[2011/11/26 v2.0 baposter class] +\NeedsTeXFormat{LaTeX2e}[1995/06/01] +\LoadClass{article} +\typeout{baposter: Brian Amberg, 2007, 2008, 2009, 2010, 2011 | http://www.brian-amberg.de/uni/poster/} +\typeout{baposter: Reinhold Kainhofer, 2011 | http://reinhold.kainhofer.com/} + +%% Define lengths only once on inclusion, such that we can make multiple posters +\newlength{\baposter@basepaperwidth} +\newlength{\baposter@basepaperheight} +\newlength{\baposter@basemargin} +\newlength{\baposter@finalpaperwidth} +\newlength{\baposter@finalpaperheight} +\newlength{\baposter@finalmargin} +\newlength{\headerheight}% +\newlength{\colwidth}% +\newlength{\colheight}% +\newlength{\baposter@@colspacing}% +\newlength{\baposter@box@@cornerradius}% +\newlength{\baposter@box@@boxheaderheight}% +\newlength{\baposter@box@@boxpadding}% +\newlength{\boxstartx}% +\newlength{\boxstarty}% +\newlength{\boxwidth}% +\newlength{\boxheight}% +\newlength{\baposter@titleimage@left@width}% +\newlength{\baposter@titleimage@right@width}% +\newlength{\baposter@titleimage@textwidth}% +\newbox\baposter@box@content% +\newbox\baposter@titleimage@left% +\newbox\baposter@titleimage@title% +\newbox\baposter@titleimage@right% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Packages +%------------------------------------------------------------------------------- +% The only ``weird'' dependency of this package is pgf. All the rest should be +% installed on any decent system. +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\typeout{Use Packages} +\RequirePackage{xkeyval} +\RequirePackage{calc} +\RequirePackage{xcolor} +\RequirePackage{tikz} +\RequirePackage{pgf} +\RequirePackage{ifthen} +\RequirePackage[T1]{fontenc} +%\RequirePackage[l2tabu, orthodox]{nag} +\usetikzlibrary{decorations} +\usetikzlibrary{fadings} +%%%\usetikzlibrary{snakes} +\usetikzlibrary{calc} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Settings +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Choose a smaller value for larger fonts +\newcommand{\baposter@fontscale}{0.292} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Zoom +%------------------------------------------------------------------------------- +% We scale the page from fontscale * papersize up to papersize +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%% Paper sizes +\newif\if@landscape +\newif\if@geometryKnowsThisSize +\DeclareOptionX{landscape}{\@landscapetrue} +\DeclareOptionX{portrait}{} + +\newcommand{\baposter@setfinalpapersize}[2]{% +\if@geometryKnowsThisSize + \setlength{\baposter@finalpaperwidth}{#1}% + \setlength{\baposter@finalpaperheight}{#2}% +\else +\if@landscape +% Transpose length, if geometry does not handle the papersize based on the key + \setlength{\baposter@finalpaperwidth}{#2}% + \setlength{\baposter@finalpaperheight}{#1}% +\else + \setlength{\baposter@finalpaperwidth}{#1}% + \setlength{\baposter@finalpaperheight}{#2}% +\fi +\fi +} + +% Default paperwidth and paperheight = a0paper +\DeclareOptionX{paperwidth}[841mm]{\setlength{\baposter@finalpaperwidth}{#1}} +\DeclareOptionX{paperheight}[1189mm]{\setlength{\baposter@finalpaperheight}{#1}} +\DeclareOptionX{archA} { \baposter@setfinalpapersize{9in}{12in}}% +\DeclareOptionX{archB} { \baposter@setfinalpapersize{12in}{18in}}% +\DeclareOptionX{archC} { \baposter@setfinalpapersize{18in}{24in}}% +\DeclareOptionX{archD} { \baposter@setfinalpapersize{24in}{36in}}% +\DeclareOptionX{archE} { \baposter@setfinalpapersize{36in}{48in}}% +\DeclareOptionX{archE1} { \baposter@setfinalpapersize{30in}{42in}}% +\DeclareOptionX{archE2} { \baposter@setfinalpapersize{26in}{38in}}% +\DeclareOptionX{archE3} { \baposter@setfinalpapersize{27in}{39in}}% +\DeclareOptionX{a0paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{841mm}{1189mm}}%g +\DeclareOptionX{a1paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{594mm}{841mm}}%g +\DeclareOptionX{a2paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{420mm}{594mm}}%g +\DeclareOptionX{a3paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{297mm}{420mm}}%g +\DeclareOptionX{a4paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{210mm}{297mm}}%g +\DeclareOptionX{a5paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{148mm}{210mm}}%g +\DeclareOptionX{a6paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{105mm}{148mm}}%g +\DeclareOptionX{b0paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{1000mm}{1414mm}}%g +\DeclareOptionX{b1paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{707mm}{1000mm}}%g +\DeclareOptionX{b2paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{500mm}{707mm}}%g +\DeclareOptionX{b3paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{353mm}{500mm}}%g +\DeclareOptionX{b4paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{250mm}{353mm}}%g +\DeclareOptionX{b5paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{176mm}{250mm}}%g +\DeclareOptionX{b6paper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{125mm}{176mm}}%g +\DeclareOptionX{ansiapaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{8.5in}{11in}}% +\DeclareOptionX{ansibpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{11in}{17in}}% +\DeclareOptionX{ansicpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{17in}{22in}}% +\DeclareOptionX{ansidpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{22in}{34in}}% +\DeclareOptionX{ansiepaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{34in}{44in}}% +\DeclareOptionX{letterpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{8.5in}{11in}}% +\DeclareOptionX{legalpaper} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{8.5in}{14in}}% +\DeclareOptionX{executivepaper}{\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{7.25in}{10.5in}}% +\DeclareOptionX{screen} {\@geometryKnowsThisSizetrue\baposter@setfinalpapersize{225mm}{180mm}}%g + +% Margin +\setlength{\baposter@finalmargin}{1.5cm} +\DeclareOptionX{fontscale}[0.292]{\renewcommand{\baposter@fontscale}{#1}} +\DeclareOptionX{margin} [1.5cm]{\setlength{\baposter@finalmargin}{#1}} + +% move text/poster body to the right (or to the left if negative) +\newlength{\baposter@movebody} +\setlength{\baposter@movebody}{0cm} +\DeclareOptionX{movebody}[0cm]{\setlength{\baposter@movebody}{#1}} + + +\newif\if@debug +\DeclareOptionX{debug}{\@debugtrue} +%% Will be passed on to other packages (xcolor and geometry), still we don't want unused warnings +\DeclareOptionX{table}{} +\DeclareOptionX{showframe}{} + +\ProcessOptionsX + +\if@debug +\newcommand{\debug}[1]{\typeout{#1}} +\else +\newcommand{\debug}[1]{} +\fi + + + +\setlength{\baposter@basepaperwidth} {\baposter@fontscale\baposter@finalpaperwidth } +\setlength{\baposter@basepaperheight}{\baposter@fontscale\baposter@finalpaperheight} +\setlength{\baposter@basemargin} {\baposter@fontscale\baposter@finalmargin} +\newlength{\baposter@basemarginright} +\setlength{\baposter@basemarginright}{\baposter@basemargin} +\addtolength{\baposter@basemarginright}{-\baposter@fontscale\baposter@movebody} +\newlength{\baposter@basemarginleft} +\setlength{\baposter@basemarginleft}{\baposter@basemargin} +\addtolength{\baposter@basemarginleft}{\baposter@fontscale\baposter@movebody} + +\typeout{Paperwidth=\the\baposter@finalpaperwidth} +\typeout{Paperheight=\the\baposter@finalpaperheight} +\typeout{BasePaperwidth=\the\baposter@basepaperwidth} +\typeout{BasePaperheight=\the\baposter@basepaperheight} +\usepackage[ + paperwidth=\baposter@basepaperwidth, + paperheight=\baposter@basepaperheight, + tmargin=\baposter@basemargin, + bmargin=\baposter@basemargin, + lmargin=\baposter@basemarginleft, + rmargin=\baposter@basemarginright, + ]{geometry} + +\usepackage{pgfpages} +\if@landscape +\if@geometryKnowsThisSize +\pgfpagesuselayout{resize to}[physical paper width=\baposter@finalpaperheight,physical paper height=\baposter@finalpaperwidth] +\else +\pgfpagesuselayout{resize to}[physical paper width=\baposter@finalpaperwidth,physical paper height=\baposter@finalpaperheight] +\fi +\else +\pgfpagesuselayout{resize to}[physical paper width=\baposter@finalpaperwidth,physical paper height=\baposter@finalpaperheight] +\fi + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Default functions for borders/backgrounds +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% These functions will all be redefined from the actual option values. In +%% particular, they will be set to \baposter@optionname@optionvalue, which +%% should do the actual work / setting for that particular optionvalue. + +\newcommand{\baposterPosterDrawBackground}[2]{} % Draw poster background + +\newcommand{\baposterBoxGetShape}{} % Returns path of text box shape +\newcommand{\baposterBoxDrawBackground}[2]{} % Draw bg of boxes +\newcommand{\baposterBoxDrawBorder}[1]{} % Draw border of individual boxes + +\newcommand{\baposterHeaderGetShape}{} % Returns path of text box shape +\newcommand{\baposterHeaderSetShade}[3]{} % Set bg style for box headers +\newcommand{\baposterHeaderDrawBackground}[3]{} % Draw background of box header +\newcommand{\baposterHeaderDrawBorder}[1]{} % Draw border of box header +\newcommand{\baposterHeaderDrawText}[1]{} % Draw text inside box header + +\newcommand{\@@previousbox}{notset} % stores the previously processed box for below=auto + +% Function to set a user-defined background +\newcommand{\baposter@backgroundCmd}{\error{No background command defined. Use \background{...} to define background}} +\newcommand{\background}[1]{\renewcommand{\baposter@backgroundCmd}{#1}} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Handle poster and box options +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\debug{Handling keys} + +%% +%% POSTER OPTIONS +%% +%% Store all poster options in variables of the form \baposter@option +%% choose-keys also store the index in \baposter@optionnr +%% choose-keys typically also assign a function +\definecolor{baposter@silver}{cmyk}{0,0,0,0.7} +\define@boolkey[ba]{poster}[baposter@]{grid} [false] {} +\define@boolkey[ba]{poster}[baposter@]{eyecatcher} [true] {} +\define@cmdkey [ba]{poster}[baposter@]{headerheight} [0.1\textheight]{} +\define@cmdkey [ba]{poster}[baposter@]{columns} [{}] {} +\define@cmdkey [ba]{poster}[baposter@]{colspacing} [1em] {} +\define@cmdkey [ba]{poster}[baposter@]{bgColorOne} [baposter@silver]{} +\define@cmdkey [ba]{poster}[baposter@]{bgColorTwo} [green] {} + +% background can be one of: shadeLR, shadeTB, plain, user, none +\define@choicekey*+[ba]{poster}{background}% + [\baposter@background\baposter@backgroundnr]% + {shadeLR, shadeTB, plain, user, none} [plain] {% + \debug{Poster background: \baposter@background} + \renewcommand{\baposterPosterDrawBackground}[2]{ + \csname baposter@background@\baposter@background\endcsname{##1}{##2}} +}{ + \PackageWarning{baposter}{Unknown background `\baposter@background' (use + shadeLR, shadeTB, plain, none, or user). If user is used, you also + have to define \background{...}.} + \renewcommand{\baposterPosterDrawBackground}[2]{\baposter@background@none{##1}{##2}} +} + + +%% +%% BOX OPTIONS +%% +\define@cmdkey[ba]{posterbox}[baposter@box@]{cornerradius} [1em] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxheaderheight} [2em] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxpadding} [0.5em] {} + + +% textborder can be one of: none, bars, coils, triangles, rectangle, rounded, +% roundedleft, roundedsmall, faded; UNIMPLEMENTED: roundedright +\edef\baposter@box@textborder@validvalues{none,bars,coils,triangles,rectangle,rounded,roundedleft,roundedsmall,faded} +\define@choicekey*+[ba]{posterbox}{textborder}% + [\baposter@box@textborder\baposter@box@textbordernr]% + {none,bars,coils,triangles,rectangle,rounded,roundedleft,roundedright,roundedsmall,faded} [rectangle] {% + \debug{Text border: \baposter@box@textborder} + \renewcommand{\baposterBoxGetShape}{ + \csname baposter@box@boxshape@\baposter@box@textborder\endcsname} + \renewcommand{\baposterBoxDrawBorder}[1]{ + \csname baposter@box@drawborder@\baposter@box@textborder\endcsname{##1}} +}{ + \PackageWarning{baposter}{Unknown text-border style `\baposter@box@textborder'. + Edit your file to choose a valid option (\baposter@box@textborder@validvalues).} + \renewcommand{\baposterBoxGetShape}{\baposter@boxshape@rectangle} + \renewcommand{\baposterBoxDrawBorder}[1]{\baposter@drawborder@rectangle{##1}} +} + +% boxshade can be one of: shadeLR, shadeTB, plain, none +\define@choicekey*+[ba]{posterbox}{boxshade}% + [\baposter@box@boxshade\baposter@box@boxshadenr]% + {shadelr,shadetb,plain,none} [none] {% + \debug{Box shade: \baposter@box@boxshade} + \renewcommand{\baposterBoxDrawBackground}[2]{ + \csname baposter@box@drawbackground@\baposter@box@boxshade\endcsname{##1}{##2}} +}{ + \PackageWarning{baposter}{Unknown boxshade style `\baposter@boxshade'. + Edit your file to choose a valid option.} + \renewcommand{\baposterBoxDrawBackground}[2]{\baposter@box@drawbackground@none{##1}{##2}} +} + +% headershade can be one of: shade-lr, shade-tb, shade-tb-inverse, plain +\define@choicekey*+[ba]{posterbox}{headershade}% + [\baposter@box@headershade\baposter@box@headershadenr]% + {shadelr, shadetb, shadetbinverse, plain} [shadelr] {% + \debug{Header shade: \baposter@box@headershade} + \renewcommand{\baposterHeaderSetShade}[3]{ + \csname baposter@box@headershade@\baposter@box@headershade\endcsname{##1}{##2}{##3}} +}{ + \PackageWarning{baposter}{Unknown headershade style `\baposter@box@headershade'. + Edit your file to choose a valid option.} + \renewcommand{\baposterHeaderSetShade}[3]{\baposter@box@headershade@none{##1}{##2}{##3}} +} + +% headershape can be one of: rectangle, rounded, smallrounded, roundedleft, roundedright +\define@choicekey*+[ba]{posterbox}{headershape}% + [\baposter@box@headershape\baposter@box@headershapenr]% + {rectangle,rounded,smallrounded,roundedleft,roundedright} [roundedright] {% + \debug{Header shape: \baposter@box@headershape} + \renewcommand{\baposterHeaderGetShape}{ + \csname baposter@box@headershape@\baposter@box@headershape\endcsname} + \renewcommand{\baposterHeaderDrawText}[1]{ + \csname baposter@box@headerdrawtext@\baposter@box@headershape\endcsname{##1}} + \renewcommand{\baposterHeaderDrawBorder}[1]{ + \csname baposter@box@headerdrawborder@\baposter@box@headershape\endcsname{##1}} +}{ + \PackageWarning{baposter}{Unknown headershape style `\baposter@headershape'. + Edit your file to choose a valid option.} + \renewcommand{\baposterHeaderGetShape}{\baposter@box@headershape@rectangle} + \renewcommand{\baposterHeaderDrawText}[1]{\baposter@box@headerdrawtext@rectangle{##1}} + \renewcommand{\baposterHeaderDrawBorder}[1]{\baposter@box@headerdrawborder@rectangle{##1}} +} + +% headerborder can be one of: open, closed, none +\define@choicekey*+[ba]{posterbox}{headerborder}% + [\baposter@box@headerborder\baposter@box@headerbordernr]% + {open,closed,none} [open] {% + \debug{Header border: \baposter@box@headerborder} +% \renewcommand{\baposterHeaderBorder}{ +% \csname baposter@headerborder@\baposter@box@headerborder\endcsname} +}{ + \PackageWarning{baposter}{Unknown headerborder style `\baposter@headerborder'. + Edit your file to choose a valid option.} +% \renewcommand{\baposterHeaderBorder}{\baposter@box@headerborder@rectangle} +} + + +\define@cmdkey[ba]{posterbox}[baposter@box@]{borderColor} [yellow] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerColorOne} [red] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerColorTwo} [brown] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerFontColor} [black] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxColorOne} [magenta] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{boxColorTwo} [cyan] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{headerfont} [\scshape\Large] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{textfont} [{}] {} + +\define@cmdkey[ba]{posterbox}[baposter@box@]{linewidth} [2pt] {} + +\define@cmdkey[ba]{posterbox}[baposter@box@]{below} [notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{above} [notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{aligned}[notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{bottomaligned}[notset]{} +\define@cmdkey[ba]{posterbox}[baposter@box@]{column} [0] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{row} [0] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{span} [1] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{height} [auto] {} +\define@cmdkey[ba]{posterbox}[baposter@box@]{name} [noname]{} + +% Set some default values, the poster and posterbox environments can override: +\setkeys[ba]{poster}{ + % Debug grid + grid=false, + % Is there an eyecatcher image + eyecatcher=true, + columns={}, + % Colours + bgColorOne=baposter@silver, + bgColorTwo=green, + % + colspacing=1em, + headerheight=0.1\textheight, + background=shadeLR, +}{} +\setkeys[ba]{posterbox}{ + % Position + column=0,row=0,span=1, + below=notset,above=notset, + bottomaligned=notset, + aligned=notset, + height=auto, + % Name + name=noname, + % Box design: border: + linewidth=2pt, + borderColor=yellow, + cornerradius=1em, + % text box: + textfont={}, + boxshade=plain, + boxColorOne=magenta, + boxColorTwo=cyan, + textborder=faded, + boxpadding=0.5em, + % header + headerfont=\scshape\Large,% or headerfont=\color{white}\textsf\textbf + headerFontColor=black, + headerColorOne=red, + headerColorTwo=brown, + headershape=rectangle, + headershade=shadeLR, + headerborder=none, + boxheaderheight=2em, +}{} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Background options and functions (one function for each possible value) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\baposter@background@shadelr}[2]{ + \debug{BAPOSTER: Using shade left right background.} + \begin{tikzpicture}[remember picture,overlay]% + \shade [shading=axis,left color=#1,right color=#2] (current page.north west) + rectangle(current page.south east); + \end{tikzpicture}% +} +\newcommand{\baposter@background@shadetb}[2]{ + \debug{BAPOSTER: Using shade top to bottom background.} + \begin{tikzpicture}[remember picture,overlay]% + \shade [shading=axis,top color=#1,bottom color=#2] (current page.north west) + rectangle(current page.south east); + \end{tikzpicture}% +} +\newcommand{\baposter@background@plain}[2]{ + \debug{BAPOSTER: Using plain background.} + \begin{tikzpicture}[remember picture,overlay]% + \fill [fill=#1] (current page.north west) rectangle(current page.south east); + \end{tikzpicture}% +} +\newcommand{\baposter@background@user}[2]{ + \debug{BAPOSTER: Using user background.} + \baposter@backgroundCmd% +} +\newcommand{\baposter@background@none}[2]{ + \debug{BAPOSTER: Using no background.} +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Return shape path of text box (depending on the box shape) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\baposter@box@boxshape@none}{\baposter@box@boxshape@rectangle} +\newcommand{\baposter@box@boxshape@bars}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) % + (\baposter@box@name se) -- (\baposter@box@name tne) +} +\newcommand{\baposter@box@boxshape@coils}{\baposter@box@boxshape@bars} +\newcommand{\baposter@box@boxshape@triangles}{\baposter@box@boxshape@bars} +\newcommand{\baposter@box@boxshape@rectangle}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) -- % + (\baposter@box@name se) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@boxshape@faded}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) % + (\baposter@box@name tne) -- (\baposter@box@name se) + } +\newcommand{\baposter@box@boxshape@rounded}{ + [rc] \baposter@box@boxshape@rectangle% +} +\newcommand{\baposter@box@boxshape@roundedsmall}{ + [src] \baposter@box@boxshape@rectangle +} +\newcommand{\baposter@box@boxshape@roundedleft}{ + (\baposter@box@name tnw) {[rc]-- (\baposter@box@name sw)} -- % + (\baposter@box@name se) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@boxshape@roundedright}{ + (\baposter@box@name tnw) -- (\baposter@box@name sw) {[rc]-- % + (\baposter@box@name se)} -- (\baposter@box@name tne)% +} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Draw box background (one function for each possible value) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take no arguments +\newcommand{\baposter@box@drawbackground@none}[2]{ + \tikzstyle{box colors}=[] +} +\newcommand{\baposter@box@drawbackground@plain}[2]{ + \tikzstyle{box colors}=[fill=#1] + \fill[box colors] \baposterBoxGetShape; +} +\newcommand{\baposter@box@drawbackground@shadelr}[2]{ + \tikzstyle{box colors}=[shading=axis, left color=#1, right color=#2]% + \fill[box colors] \baposterBoxGetShape; +} +\newcommand{\baposter@box@drawbackground@shadetb}[2]{ + \tikzstyle{box colors}=[shading=axis, top color=#1, bottom color=#2]% + \fill[box colors] \baposterBoxGetShape; +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Draw box border +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take two arguments: borderColor +\newcommand{\baposter@box@drawborder@none}[1]{} +\newcommand{\baposter@box@drawborder@bars}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@coils}[1]{ + \draw[color=#1,segment amplitude=0.35em,segment length=0.4em,snake=coil] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@triangles}[1]{ + \draw[color=#1,segment amplitude=0.2em,segment length=0.4em,snake=triangles] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@rectangle}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@rounded}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@roundedleft}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@roundedright}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} +\newcommand{\baposter@box@drawborder@faded}[1]{ + % This is the right way to do it, but it does not work with evince, and has problems during printing, so instead we do + %\draw[color=#1,path fading=south] \baposterBoxGetShape;% + % this + \foreach \x in {0,1,...,90} \draw[color=#1!\x] ($(\baposter@box@name tnw)!{(100-\x)/100}!(\baposter@box@name sw)$) -- ($(\baposter@box@name tnw)!{(100-(\x+10))/100}!(\baposter@box@name sw)$);% + \foreach \x in {0,1,...,90} \draw[color=#1!\x] ($(\baposter@box@name tne)!{(100-\x)/100}!(\baposter@box@name se)$) -- ($(\baposter@box@name tne)!{(100-(\x+10))/100}!(\baposter@box@name se)$);% +} +\newcommand{\baposter@box@drawborder@roundedsmall}[1]{ + \draw[color=#1] \baposterBoxGetShape;% +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Return shape path of text box (depending on the box shape) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take no arguments +% TODO: For headerborder==none, use (\baposter@box@name outer tnw) instead! +\newcommand{\baposter@box@headershape@rectangle}{% + (\baposter@box@name tnw) -- (\baposter@box@name nw) -- % + (\baposter@box@name ne) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@smallrounded}{% + (\baposter@box@name tnw) {[src] -- (\baposter@box@name nw) -- % + (\baposter@box@name ne)} -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@roundedright}{% + (\baposter@box@name tnw) -- (\baposter@box@name nw) {[rc] -- % + (\baposter@box@name ne)} -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@roundedleft}{% + (\baposter@box@name tnw) {[rc]-- (\baposter@box@name nw)} -- % + (\baposter@box@name ne) -- (\baposter@box@name tne)% +} +\newcommand{\baposter@box@headershape@rounded}{% + (\baposter@box@name tnw) {[rc] -- (\baposter@box@name nw) -- % + (\baposter@box@name ne) } -- (\baposter@box@name tne)% +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Header text drawing (one function for each possible value of headershape) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take one argument: the header text +\newcommand{\baposter@box@headerdrawtext@rectangle}[1]{ + \path (\baposter@box@name nw) +(0em,-0.5\baposter@box@@boxheaderheight) node[anchor=west,inner sep=0.4em] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@smallrounded}[1]{ + \path (\baposter@box@name nw) +(0.5\boxwidth,-0.5\baposter@box@@boxheaderheight) node[anchor=center] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@roundedright}[1]{ + \path (\baposter@box@name nw) +(0em,-0.5\baposter@box@@boxheaderheight)% + node[anchor=west,inner sep=0.4em,text depth=0.0em] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@roundedleft}[1]{ + \path (\baposter@box@name nw) +(0em,-0.5\baposter@box@@boxheaderheight)% + node[anchor=west,inner sep=0.4em] {#1};% +} +\newcommand{\baposter@box@headerdrawtext@rounded}[1]{ + \path (\baposter@box@name nw) +(0.5\boxwidth,-0.5\baposter@box@@boxheaderheight) node[anchor=center] {#1};% +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% Header shade options and functions (one function for each possible value) +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% These functions take two arguments: headerColorOne, headerColorTwo and borderColor +\newcommand{\baposter@box@headershade@shadelr}[3]{ + \debug{Header-Shade: Shade Left - Right} + \tikzstyle{header colors}=[% + color=#3,% + shading=axis,% + left color=#1,% + right color=#2% + ]% +} +\newcommand{\baposter@box@headershade@shadetb}[3]{ + \debug{Header-Shade: Shade Top - Bottom} + \tikzstyle{header colors}=[% + color=#3,% + shading=axis,% + top color=#1,% + bottom color=#2% + ]% +} +\newcommand{\baposter@box@headershade@shadetbinverse}[3]{ + \tikzstyle{header colors}=[% + top color=#1!75!#2,% + bottom color=#2!100!#1,% + shading angle=20% + ]% + \colorlet{baposterHeaderFontColor}{white}% +} +\newcommand{\baposter@box@headershade@plain}[3]{ + \debug{Header-Shade: Plain} + \tikzstyle{header colors}=[% + color=#3,% + fill=#1% + ]% +} +\newcommand{\baposter@box@headershade@none}[3]{ + \debug{Header-Shade: none} + \tikzstyle{header colors}=[] +} + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% The main poster environment +%%% \begin{baposter}{settings}{Eye Catcher}{Title}{Author}{University Logo} +%%%----------------------------------------------------------------------------- +%%% The settings are +%%% - grid=true,[false]:Show grid to help with alignment +%%% - colspacing=0.7em: Column spacing +%%% - columns=4: number of columns (default 4 in landscape and 3 in portrait format) (maximum number is 6) +%%% - color=[orange]: xcolor color definition used as the main color of the poster +%%% - colortwo=[white]: The other color for gradient based layouts +%%% - textborder=none,bars,coils,triangles,rectangle,rounded,roundedsmall,roundedleft,roundedright,[faded] +%%% The style of the box around the text area +%%% - headerborder=none,closed,open +%%% No extra border around box header, full border around box header or border that is open below. +%%% - headershape=rectangle,rounded,roundedleft,roundedright +%%% Shape of the box-header region +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newenvironment{poster}[5]{% + \thispagestyle{empty}% Suppress Page Number + \debug{Poster Starts}% + % This setkeys call parses all provided options and depending on the option + % value, assigns different handler functions to the \baposter(Box|Header)* + % functions. Once that is done, we don't have to care about particular + % values for border, shading, etc. All we have to do is call the + % handler functions and let them do their job. + % This also allows the user to override the poster-wide defaults on a per-box + % basis. + \setkeys[ba]{posterbox,poster}{#1}% +% + % TODO: Move all those assignments to the key macros! + % Parse Keys% + \colorlet{bgColorOne}{\baposter@bgColorOne} + \colorlet{bgColorTwo}{\baposter@bgColorTwo} +% + %% Boxes% + \setlength{\headerheight}{\baposter@headerheight}% + \setlength{\colheight}{\textheight-\baposter@headerheight}% + \renewcommand{\@@previousbox}{notset} + + \debug{Format}% + % Set default for columns if unset (4 for landscape, 3 for portrait) + \ifthenelse{\equal{\baposter@columns}{}}{% + \renewcommand{\baposter@columns}{\if@landscape4\else3\fi}% + }{} +% + \debug{Columns: \baposter@columns}% + \setlength{\baposter@@colspacing}{\baposter@colspacing}% + \setlength{\colwidth}{\textwidth}% + \addtolength{\colwidth}{\baposter@@colspacing*(1-\baposter@columns)}% + \ifcase\baposter@columns\relax + \error{You need to have at least one column!} + \or % 1 + \setlength{\colwidth}{\colwidth}% + \or % 2 + \setlength{\colwidth}{0.5\colwidth}% + \or % 3 + \setlength{\colwidth}{0.3333333333333\colwidth}% + \or % 4 + \setlength{\colwidth}{0.25\colwidth}% + \or % 5 + \setlength{\colwidth}{0.2\colwidth}% + \or % 6 + \setlength{\colwidth}{0.16666666666\colwidth}% + \else % >6 + \error{You do not want so many columns} + \fi +% + \newcommand{\baposter@reference}{north west}% +% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % A box with a header and some content. The basic unit of the poster% + %---------------------------------------------------------------------------% + % Each box has a name and can be placed absolutely or relatively.% + % The only inconvenience is that you can only specify a relative position % + % towards an already declared box. So if you have a box attached to the % + % bottom, one to the top and a third one which should be inbetween, you % + % have to specify the top and bottom boxes before you specify the middle % + % box.% + %% + % below= name of other node% + % above= name of other node% + % aligned=name of other node% + % bottomaligned=name of other node% + % column= [0] % + % row= [0] % + % span= [1] % + % height= ,[auto]% + % name= [noname]% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % Backward-compatibility definition (\headerbox command uses posterbox env): + \newcommand{\headerbox}[3]{% + \begin{posterbox}[##2]{##1} + ##3 + \end{posterbox} + } + + \newenvironment{posterbox}[2][]{% + \debug{Poster box options: ##1}% + % Override the poster-wide defaults on a per-box basis + \setkeys[ba]{posterbox}{##1}% +% + \def\baposter@box@title{##2} +% + \colorlet{borderColor}{\baposter@box@borderColor} + \colorlet{headerColorOne}{\baposter@box@headerColorOne} + \colorlet{headerColorTwo}{\baposter@box@headerColorTwo} + \colorlet{headerFontColor}{\baposter@box@headerFontColor} + \colorlet{boxColorOne}{\baposter@box@boxColorOne} + \colorlet{boxColorTwo}{\baposter@box@boxColorTwo} +% + \setlength{\baposter@box@@cornerradius}{\baposter@box@cornerradius}% + \setlength{\baposter@box@@boxheaderheight}{\baposter@box@boxheaderheight}% + \setlength{\baposter@box@@boxpadding}{\baposter@box@boxpadding}% + + + %% The columns is always given absolute + % boxstartx = column * colwidth + column * colspacing + \setlength{\boxstartx}{(\colwidth+\baposter@@colspacing)*\baposter@box@column}% +% + %% The width is gvien absolute + % Box Width = span * colwidth + (span-1) * colspacing + \setlength{\boxwidth}{\baposter@box@span\colwidth} % + \addtolength{\boxwidth}{\baposter@@colspacing*(\baposter@box@span-1)}% +% + %% Measure the content of the box% + \setbox\baposter@box@content=\hbox\bgroup% + \begin{pgfinterruptpicture}% + \begin{minipage}[t]{\boxwidth-\baposter@box@@boxpadding*2}% + \baposter@box@textfont\bgroup% + }% End of posterbox preamble + %%% HERE COME THE ACTUAL CONTENTS OF THE HEADERBOX ENVIRONMENT + {% posterbox handling after contents (i.e. drawing everything) + \egroup% + \end{minipage}% + \end{pgfinterruptpicture}% + \egroup% + \setlength{\boxheight}{\ht\baposter@box@content}% + \addtolength{\boxheight}{\dp\baposter@box@content}% + \addtolength{\boxheight}{\baposter@box@@boxheaderheight} % Header% + \addtolength{\boxheight}{2\baposter@box@@boxpadding} % Inner Sep +% + \ifthenelse{\equal{\baposter@box@height}{bottom}}{% + }{\ifthenelse{\equal{\baposter@box@height}{auto}}{% + }{ % Neither auto nor bottom% + \setlength{\boxheight}{\baposter@box@height\colheight}% + }}% +% + %% Determine the box position% + \debug{Setting Coordinates}% + \debug{Upper Right}% + \debug{\baposter@box@name}% +% + %%% Upper Right Corner% + % if below=auto, set it to the previous box + % TODO: We should generalize this to the previous box of the used column, + % currently we use the previous box, which might be in a different column + \ifthenelse{\equal{\baposter@box@below}{auto}}{% + \edef\baposter@box@below{\@@previousbox} + \debug{Box \baposter@box@name has below=auto, placing it below box \baposter@box@below.} + }{} + \xdef\@@previousbox{\baposter@box@name} + + \ifthenelse{\not\equal{\baposter@box@below}{notset} }{% + %% Below% + \debug{Below}% + \path[shape=coordinate] (\boxstartx,0pt |- \baposter@box@below se) ++(0pt,-\baposter@@colspacing) coordinate(\baposter@box@name nw);% + }{% + \ifthenelse{\not\equal{\baposter@box@aligned}{notset} }{% + %% Aligned% + \debug{Aligned: \baposter@box@aligned}% + \path[shape=coordinate] (\boxstartx,0pt |- \baposter@box@aligned nw) coordinate(\baposter@box@name nw);% + }{% + %% Fixed% + \debug{Fixed}% + \setlength{\boxstarty}{\baposter@box@row\colheight}% + \path[shape=coordinate] (\boxstartx,\colheight-\boxstarty) coordinate(\baposter@box@name nw);% + }}% +% + %% Lower Left Corner% + \debug{Lower Left}% + \ifthenelse{\equal{\baposter@box@above}{bottom}}{% + %% Above = Bottom% + \debug{Above bottom}% + \ifthenelse{\equal{\baposter@box@below}{notset} \and \equal{\baposter@box@aligned}{notset}}{% + \path[shape=coordinate] (\boxstartx,\boxheight) coordinate(\baposter@box@name nw);% + }{}% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt) coordinate(\baposter@box@name se);% + }{\ifthenelse{\not \equal{\baposter@box@bottomaligned}{notset}}{% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt |- \baposter@box@bottomaligned se) coordinate(\baposter@box@name se);% + }{{\ifthenelse{\not \equal{\baposter@box@above}{notset}}{% + %% Above = Node% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt |- \baposter@box@above nw) +(0pt,\baposter@@colspacing) coordinate(\baposter@box@name se);% + }{% + %% Above = notset% + \debug{Above=not set}% + \ifthenelse{\equal{\baposter@box@height}{bottom}}{% + %% height=bottom% + \debug{height=bottom}% + \path[shape=coordinate] (\boxstartx+\boxwidth,0pt) coordinate(\baposter@box@name se);% + }{ %% height=auto or fixed% + \debug{height=auto or fixed}% + \path[shape=coordinate] (\baposter@box@name nw) ++(\boxwidth,-\boxheight) coordinate(\baposter@box@name se);% + }}}}}% +% + % + % Set coordinates relative to nw,se% + \debug{Fixing Coordinates}% + \path[shape=coordinate]% + (\baposter@box@name nw) +(0pt,-\baposter@box@@boxheaderheight) coordinate(\baposter@box@name tnw)% + (\baposter@box@name nw |- \baposter@box@name se) coordinate(\baposter@box@name sw)% + (\baposter@box@name se |- \baposter@box@name nw) coordinate(\baposter@box@name ne)% + (\baposter@box@name ne) +(0pt,-\baposter@box@@boxheaderheight) coordinate(\baposter@box@name tne)% +% + (\baposter@box@name nw) +(-0.025em,0pt) coordinate(\baposter@box@name outer nw)% + (\baposter@box@name tnw) +(-0.025em,0pt) coordinate(\baposter@box@name outer tnw)% + (\baposter@box@name sw) +(-0.025em,0pt) coordinate(\baposter@box@name outer sw)% +% + (\baposter@box@name ne) +( 0.025em,0pt) coordinate(\baposter@box@name outer ne)% + (\baposter@box@name tne) +( 0.025em,0pt) coordinate(\baposter@box@name outer tne)% + (\baposter@box@name se) +( 0.025em,0pt) coordinate(\baposter@box@name outer se);% +% + %% Setting the bg colors of the box header + \baposterHeaderSetShade{headerColorOne}{headerColorTwo}{borderColor} +% + \tikzstyle{rc}=[rounded corners=\baposter@box@@cornerradius];% + \tikzstyle{src}=[rounded corners=0.5em];% +% + + %% Now that everything is set up, draw the actual box, with bg and header + \begin{scope}[line width=\baposter@box@linewidth] + %% Header% + \debug{Header}% + \debug{Header-Shape: \baposter@box@headershape, header-border: \baposter@box@headerborder (\baposter@box@headerbordernr)}% + % TODO: Also turn this last ifcase construct into a handler function + % We only need to determine (fill|shade)(draw|)... +% \baposterHeaderDrawBackground{bgColorOne}{bgColorTwo}{borderColor} +% \baposterHeaderDrawBorder{borderColor} + \ifcase\baposter@box@headerbordernr\relax% + % open + \ifthenelse{\equal{\baposter@box@headershade}{plain}}{ + \filldraw [style=header colors] \baposterHeaderGetShape;% + }{ + \shadedraw [style=header colors] \baposterHeaderGetShape;% + } + \or + % closed + \ifthenelse{\equal{\baposter@box@headershade}{plain}}{ + \filldraw [style=header colors] \baposterHeaderGetShape -- cycle;% + }{ + \shadedraw [style=header colors] \baposterHeaderGetShape -- cycle;% + } + \or + % none + \ifthenelse{\equal{\baposter@box@headershade}{plain}}{ + \fill [style=header colors] \baposterHeaderGetShape;% + }{ + \shade [style=header colors] \baposterHeaderGetShape;% + } + \fi + % + %% Draw the text inside the box header: + \baposterHeaderDrawText{\color{headerFontColor}\baposter@box@headerfont{\baposter@box@title}};% + % + %% Text borders (border around boxes) + \debug{Poster boxes}% + % First set box shade + \baposterBoxDrawBackground{boxColorOne}{boxColorTwo} + \baposterBoxDrawBorder{borderColor} + %% + %% Text Box% + \debug{Drawing Text}% + \path (\baposter@box@name tnw) node(text) [anchor=north west, + outer sep=-0.000em,text width=\boxwidth-2\baposter@box@@boxpadding,inner sep=\baposter@box@@boxpadding, + text justified] {\usebox{\baposter@box@content}};% + \end{scope} + % + % Finally store the box name as the previous box for the next call +% \xdef\@@previousbox{\baposter@box@name}% + }% END of posterbox definition +% + %% Poster Background% + \baposterPosterDrawBackground{bgColorOne}{bgColorTwo}% + %% Poster header/title + \hspace{-1.5em}% + \begin{tikzpicture}[inner sep=0pt,outer sep=0pt,line width=0.05em]% + \useasboundingbox (0em,0em) rectangle(\textwidth,\textheight);% + \path[shape=coordinate]% + (0pt,\colheight) coordinate(north west) (\textwidth,\colheight) coordinate(north east)% + (0pt,0pt) coordinate(south west) (\textwidth,0pt) coordinate(south east);% +% + \ifbaposter@eyecatcher% Has eye catcher + \debug{Eyecatcher found!} + \setbox\baposter@titleimage@left=\hbox{#2}% + \else% Has no eye catcher% + \setbox\baposter@titleimage@left=\hbox{}% + \fi% + \setlength{\baposter@titleimage@left@width}{\wd\baposter@titleimage@left}% + \setbox\baposter@titleimage@right=\hbox{#5}% + \setlength{\baposter@titleimage@right@width}{\wd\baposter@titleimage@right}% + \setlength{\baposter@titleimage@textwidth}{\textwidth}% + \addtolength{\baposter@titleimage@textwidth}{-\baposter@titleimage@left@width}% + \addtolength{\baposter@titleimage@textwidth}{-\baposter@titleimage@right@width}% + + \debug{#3} + % + % + % % Draw Header% + \draw (north west) +(0em,1em+0.5\headerheight) node(image)[anchor=west] { {\usebox{\baposter@titleimage@left }} };% + \draw (north east) +(0em,1em+0.5\headerheight) node(logo) [anchor=east] { {\usebox{\baposter@titleimage@right}} };% + % + \ifbaposter@eyecatcher% Has eye catcher% + \draw (image.east) node(title)[anchor=west,text width=\baposter@titleimage@textwidth]{% + \begin{minipage}{\baposter@titleimage@textwidth}% + \begin{center}% + \textbf{\Huge #3}\\% + {\Large #4}% + \end{center}% + \end{minipage} + };% + \else% Has no eye catcher + \draw (image.east) node(title)[anchor=west] { {\begin{minipage}{\baposter@titleimage@textwidth}{\bfseries\Huge #3}\\{\Large #4}\end{minipage}} };% + \fi + }% END poster begin +% The body + {% BEGIN poster end + % The end, draw gridlines if neccesary + \ifbaposter@grid + \newdimen{\gridpos} + \pgfmathsetmacro{\z}{\baposter@columns-1} + \foreach \y in {0,...,\z} + { + \setlength{\gridpos}{\y\colwidth+\y\baposter@@colspacing} + \draw[draw=green,draw opacity=0.7] (\gridpos,0pt) -- (\gridpos,\colheight) + (\gridpos+\colwidth,0pt) -- (\gridpos+\colwidth,\colheight);% + } + % Horizontal lines, every 0.1: + %% Explicitly list all percentages, because with {0.0, 0.1, ..., 1.0} we + %% get rounding errors in the displayed numbers! + \foreach \y in {0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0} + \draw[draw=green,draw opacity=0.7] (0pt,\colheight-\y\colheight) -- + (\textwidth,\colheight-\y\colheight) node[anchor=west] {\y};% + \fi% + \end{tikzpicture}% + % \xkvview{} + \par + }% END poster end diff --git a/Templates/Poster/figures/UA_Logo_Horizontal-eps-converted-to.pdf b/Templates/Poster/figures/UA_Logo_Horizontal-eps-converted-to.pdf new file mode 100644 index 0000000..629a51e Binary files /dev/null and b/Templates/Poster/figures/UA_Logo_Horizontal-eps-converted-to.pdf differ diff --git a/Templates/Poster/figures/UA_Logo_Horizontal.eps b/Templates/Poster/figures/UA_Logo_Horizontal.eps new file mode 100644 index 0000000..ebf65f3 Binary files /dev/null and b/Templates/Poster/figures/UA_Logo_Horizontal.eps differ diff --git a/Templates/Poster/figures/logo.png b/Templates/Poster/figures/logo.png new file mode 100644 index 0000000..df92477 Binary files /dev/null and b/Templates/Poster/figures/logo.png differ diff --git a/Templates/Poster/figures/placeholder.jpg b/Templates/Poster/figures/placeholder.jpg new file mode 100644 index 0000000..d5c08e0 Binary files /dev/null and b/Templates/Poster/figures/placeholder.jpg differ diff --git a/Templates/Poster/sample.bib b/Templates/Poster/sample.bib new file mode 100755 index 0000000..1443ef4 --- /dev/null +++ b/Templates/Poster/sample.bib @@ -0,0 +1,19 @@ +@BOOK{Smith:2012qr, + title = {{B}ook {T}itle}, + publisher = {Publisher}, + author = {Smith, J.~M. and Jones, A.~B.}, + year = {2012}, + edition = {7th}, +} + +@ARTICLE{Smith:2013jd, + author = {Jones, A.~B. and Smith, J.~M.}, + title = {{A}rticle {T}itle}, + journal = {Journal title}, + year = {2013}, + volume = {13}, + pages = {123-456}, + number = {52}, + month = {March}, + publisher = {Publisher} +} \ No newline at end of file diff --git a/Writing Style/Writing_Style.aux b/Writing Style/Writing_Style.aux new file mode 100644 index 0000000..e44b882 --- /dev/null +++ b/Writing Style/Writing_Style.aux @@ -0,0 +1,84 @@ +\relax 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Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +You're in trouble here. Try typing to proceed. +If that doesn't work, type X to quit. + + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: LaTeX Error: \expandafterallowed only in math mode. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.1800 \mathfrak{Y} + +(That makes 100 errors; please try again.) +Here is how much of TeX's memory you used: + 34945 strings out of 478994 + 797124 string characters out of 5858184 + 1197992 words of memory out of 5000000 + 51646 multiletter control sequences out of 15000+600000 + 414936 words of font info for 72 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 113i,21n,404p,756b,609s stack positions out of 5000i,500n,10000p,200000b,80000s + +/Users/shakilrafi/Padgett_Rafi/Writing Style/Writing_Style.tex:1800: ==> Fatal error occurred, no output PDF file produced! diff --git a/Writing Style/Writing_Style.out b/Writing Style/Writing_Style.out new file mode 100644 index 0000000..72680cb --- /dev/null +++ b/Writing Style/Writing_Style.out @@ -0,0 +1,19 @@ +\BOOKMARK [1][-]{section.1}{\376\377\000P\000r\000e\000f\000a\000c\000e}{}% 1 +\BOOKMARK [1][-]{section.2}{\376\377\000G\000e\000n\000e\000r\000a\000l\000\040\000c\000o\000m\000m\000e\000n\000t\000s}{}% 2 +\BOOKMARK [2][-]{subsection.2.1}{\376\377\000L\000o\000g\000i\000c\000a\000l\000\040\000p\000r\000e\000c\000i\000s\000i\000o\000n}{section.2}% 3 +\BOOKMARK [2][-]{subsection.2.2}{\376\377\000W\000r\000i\000t\000i\000n\000g\000\040\000s\000t\000y\000l\000e\000\040\000a\000n\000d\000\040\000s\000c\000i\000e\000n\000t\000i\000f\000i\000c\000\040\000E\000n\000g\000l\000i\000s\000h}{section.2}% 4 +\BOOKMARK [1][-]{section.3}{\376\377\000P\000r\000e\000c\000i\000s\000e\000\040\000s\000p\000e\000c\000i\000f\000i\000c\000a\000t\000i\000o\000n\000s}{}% 5 +\BOOKMARK [2][-]{subsection.3.1}{\376\377\000S\000t\000r\000u\000c\000t\000u\000r\000e\000\040\000o\000f\000\040\000t\000h\000e\000\040\000d\000o\000c\000u\000m\000e\000n\000t}{section.3}% 6 +\BOOKMARK [2][-]{subsection.3.2}{\376\377\000L\000a\000T\000e\000X\000\040\000h\000e\000a\000d\000e\000r}{section.3}% 7 +\BOOKMARK [2][-]{subsection.3.3}{\376\377\000M\000a\000i\000n\000\040\000t\000e\000x\000t\000\040\000o\000f\000\040\000t\000h\000e\000\040\000d\000o\000c\000u\000m\000e\000n\000t}{section.3}% 8 +\BOOKMARK [2][-]{subsection.3.4}{\376\377\000L\000a\000T\000e\000X\000\040\000e\000n\000v\000i\000r\000o\000n\000m\000e\000n\000t\000s}{section.3}% 9 +\BOOKMARK [3][-]{subsubsection.3.4.1}{\376\377\000R\000e\000m\000a\000r\000k\000s}{subsection.3.4}% 10 +\BOOKMARK [3][-]{subsubsection.3.4.2}{\376\377\000S\000e\000t\000t\000i\000n\000g\000s\000\040\000a\000n\000d\000\040\000f\000r\000a\000m\000e\000w\000o\000r\000k\000s}{subsection.3.4}% 11 +\BOOKMARK [3][-]{subsubsection.3.4.3}{\376\377\000D\000e\000f\000i\000n\000i\000t\000i\000o\000n\000s}{subsection.3.4}% 12 +\BOOKMARK [3][-]{subsubsection.3.4.4}{\376\377\000R\000e\000s\000u\000l\000t\000s}{subsection.3.4}% 13 +\BOOKMARK [3][-]{subsubsection.3.4.5}{\376\377\000P\000r\000o\000o\000f\000s}{subsection.3.4}% 14 +\BOOKMARK [1][-]{section.4}{\376\377\000A\000n\000\040\000i\000l\000l\000u\000s\000t\000r\000a\000t\000i\000v\000e\000\040\000e\000x\000a\000m\000p\000l\000e}{}% 15 +\BOOKMARK [2][-]{subsection.4.1}{\376\377\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{section.4}% 16 +\BOOKMARK [3][-]{subsubsection.4.1.1}{\376\377\000N\000o\000t\000a\000t\000i\000o\000n}{subsection.4.1}% 17 +\BOOKMARK [3][-]{subsubsection.4.1.2}{\376\377\000S\000e\000t\000t\000i\000n\000g}{subsection.4.1}% 18 +\BOOKMARK [2][-]{subsection.4.2}{\376\377\000R\000e\000s\000u\000l\000t\000s}{section.4}% 19 diff --git a/Writing Style/Writing_Style.pdf b/Writing Style/Writing_Style.pdf new file mode 100644 index 0000000..1d1f8ed Binary files /dev/null and b/Writing Style/Writing_Style.pdf differ diff --git a/Writing Style/Writing_Style.tex b/Writing Style/Writing_Style.tex new file mode 100644 index 0000000..9b9a0e8 --- /dev/null +++ b/Writing Style/Writing_Style.tex @@ -0,0 +1,1805 @@ +.%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Preamble of document %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Sets up the standard "article" document class +\documentclass[12pt]{article} + +% Loads packages +% Note that I typically load all packages at once (as below). +% Sometimes you must load them in a certain order... +\usepackage{amsmath, + amsfonts, + amssymb, + amsthm, + enumerate, + enumitem, + geometry, + mleftright, + nicefrac, + mathtools, + xparse, + ifthen, + tikz, + pgfplots, + floatrow, + multicol, + caption, + bbm, + xurl} + +% Set up PGF plots (used for graphing) +\pgfplotsset{compat=newest} + +% Fixes spacing caused by using "\left" or "\right" +\mleftright + +% Sets the margins of the document. +% One can do this using plain TeX, but why? +\geometry{margin=1in} + +% Lines 45-48 set up my use of the cleveref package. +% This provides a "smarter" cross referencing style +\usepackage[colorlinks=true]{hyperref} +\usepackage[nosort,capitalise]{cleveref} +\newcommand{\creflastconjunction}{, and\nobreakspace} +\crefname{enumi}{item}{items} + +% My personal command macros that I use. +% Note that "\displaySpace" is use to create the correct spacing for displayed math puctuation (and can be easily updated throughout the entire document by changing line 50). +\newcommand{\R}{\mathbb{R}} +\newcommand{\C}{\mathbb{C}} +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\E}{\mathbb{E}} +\renewcommand{\P}{\mathbb{P}} + +% Creates my paired math delimiters for easy use. +\DeclarePairedDelimiter{\pr}{(}{)} +\DeclarePairedDelimiter{\br}{[}{]} +\DeclarePairedDelimiter{\cu}{\{}{\}} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclarePairedDelimiter{\norm}{\lVert}{\rVert} +\DeclarePairedDelimiter{\vt}{\langle}{\rangle} + +% Creates desired header and footer for document. +% Note that it uses the adaptively created date macro. +\setlength{\headheight}{15pt} +\usepackage{fancyhdr} +\pagestyle{fancy} +\fancyhf{} +\fancyhead[R]{} +\cfoot{} +\lfoot{\small{\textsc{Dissemination prohibited. \today}}} +\rfoot{\thepage} + +% Updates equation and figure numbering. +\counterwithin{equation}{section} +\counterwithin{figure}{section} + +% Defines theorem-like environments +\theoremstyle{definition} +\newtheorem{definition}{Definition}[section] +\newtheorem{theorem}[definition]{Theorem} +\newtheorem{problem}[definition]{Problem} +\newtheorem{remark}[definition]{Remark} +\newtheorem{assumption}[definition]{Assumption} +\newtheorem{setting}[definition]{Setting} +\newtheorem{corollary}[definition]{Corollary} +\newtheorem{proposition}[definition]{Proposition} +\newtheorem{lemma}[definition]{Lemma} +\renewcommand{\thetheorem}{\theenumi\arabic{theorem}} + +% Helps create better arrows in figures +\usetikzlibrary{arrows.meta} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\title{Specifications for writing\\ a thesis or a scientific article} + +\author{Joshua Lee Padgett} + +\maketitle + +\tableofcontents + +\section{Preface} + +% + +The following document provides detailed specifications +regarding the way in which a thesis or a scientific article should be written. +In principle, all specifications are up for discussion. +If you intend to deviate from one of the specifications below, then please contact me +\emph{in advance} before deviating from the proposed specifications +to discuss with me whether I agree to the deviations that you propose. +If you do not contact me regarding the proposed specifications, +then I assume that you fully agree with the proposed specifications. +Please let me know if you have any questions regarding this document. +Please also study carefully the \LaTeX\ code used to create this document! + +\section{General comments} + + +\subsection{Logical precision} + +\begin{enumerate} +\item +\emph{Each predicate} should be quantified by the use of a quantifier (often in written text instead of with a symbol) +\emph{in front} of the predicate. The most common quantifiers are the \emph{universal} quantifier (text: ``for all,'' symbol: $ \forall $) +and the \emph{existential} quantifier +(text: ``there exist(s),'' symbol: $ \exists $). +The word ``For'' is \underline{not} a quantifier. + +For example, please write: +note that for all $ x \in \R $ there exists a real number $ y \in \R $ +such that for all $ t \in [0,\infty) $ +it holds that +\begin{equation} + e^x + \leq + y \, e^t +\end{equation} +instead of, for example: +% +% +% +\begin{itemize} + + + \item +note that for all $ x \in \R $ there exists a real number $ y \in \R $ such that +\begin{equation} +\label{eq:pr1} + e^x + \leq + y \, e^t +\end{equation} +for all $ t \in [0,\infty) $ +(Problem: it is not entirely clear whether \cref{eq:pr1} means +``$ \forall \, x \in \R \colon \exists \, y \in \R \colon \big( e^x \leq y \, e^t \ \forall \, t \in [0,\infty) \big) $'' +or whether it means +``$ \forall \, x \in \R \colon \big( \big( \exists \, y \in \R \colon e^x \leq y \, e^t \big) \, \forall \, t \in [0,\infty) \big)$''. +Putting the quantifiers in front of the predicate avoids such issues.), + + +\item +note that for all $ x \in \R $ there exists a real number $ y \in \R $ such that +for $ t \in [0,\infty) \colon $ +\begin{equation} + e^x + \leq + y \, e^t +\end{equation} +(Problem: it is not clear what ``for $ t \in [0,\infty) $'' means. Does it mean +``for all $ t \in [0,\infty) $ it holds that'' or does it mean +``for some $ t \in [0,\infty) $ it holds that''? +``For some $ t \in [0,\infty) $ it holds that,'' in turn, is another way of saying that +``there exists a $ t \in [0,\infty) $ such that.''), +or + +\item +note that for all $ x \in \R $ there exists a real number $ y \in \R $ such that +\begin{equation} +\label{eq:pr3} + e^x + \leq + y \, e^t +\end{equation} +(Problem: $ t $ is not quantified in \cref{eq:pr3}; that is, \cref{eq:pr3} is not a statement +but an $ 1 $-ary predicate. So, it does not make sense to say that we should note +that \cref{eq:pr3} holds.). + +\end{itemize} + +Wikipedia (German) +\url{http://de.wikipedia.org/wiki/Pr%C3%A4dikatenlogik_erster_Stufe} +(access date: October 8, 2021): + +``The first-order logic is a branch of mathematical logic. +It is concerned with the structure of certain mathematical expressions and the logical inference that goes from such expressions to others. +In doing so, it is possible to define both the language and the inference in a purely syntactic way, i.e., without reference to mathematical meanings. +The interplay of purely syntactic considerations on the one hand and semantic considerations on the other hand leads to important findings that are important for all of mathematics, because this can be formulated using the Zermelo-Fraenkel set theory in the first-level predicate logic.'' + + +\item +Please also use +``Let $ n \in [0,\infty) $'' +or +``Let $ n \in \{ 0, 1, 2, \dots \} $'' +(depending on what you mean) +instead of +``Let $ n \geq 0 $'' +as it is not entirely clear whether +``Let $ n \geq 0 $'' +means +``Let $ n \in \{ 0, 1, 2, \dots \} $'' +or +``Let $ n \in [0,\infty) $.'' + + +\item +We never use the word ``define'' and we never use the expression ``:='' in the article\slash thesis. +Instead we write, for example, +\begin{enumerate}[label=(\roman*)] + \item let $ A $ be the set given by $ A = \{ 2, 4, 6, 8, \dots \} $, + \item let $ A_r $, $ r \in \R $, be the sets which satisfy for all $ r \in \R $ + that $ A_r = [r,\infty) $, + or + \item +let $ f_r \colon \R \to \R $, $ r \in \R \backslash \{ 0 \} $, +be the functions which satisfy for all $ r \in \R \backslash \{ 0 \} $, $ x \in \R $ +that +\begin{equation} + f_r( x ) = + \frac{ 1 }{ \sqrt{ 2 \pi r^2 } } + \exp\left( + - \frac{ x^2 }{ 2 r^2 } + \right) + \text{.} +\end{equation} +\end{enumerate} + +\end{enumerate} + + +\subsection{Writing style and scientific English} + +\begin{enumerate}[label=(\roman*)] +\item +The \LaTeX\ package \texttt{hyperref} must be included into the document. + +\item +Every time after the symbol $ \forall $ or the symbol $ \exists $ has appeared +there must appear the expression \verb+\,+ +in the \LaTeX\ code. + +\item +The following +\LaTeX\ phrases may not appear in your document: +\begin{enumerate} +\item +``\verb+cf. +'' +\item +``\verb+i.i.d. +'' +\end{enumerate} +Instead the following +\LaTeX\ phrases may appear in your document: +\begin{enumerate} +\item +``\verb+cf.\ +'' +\item +``\verb+i.i.d.\ +'' +\end{enumerate} +Analogous comments apply to other abbreviations. + +\item +The following +\LaTeX\ phrases may not appear in your document: +\begin{enumerate} +\item +``\verb+$ d, m \in \N, x \in \R $+'' +\item +``\verb+$ d_1, d_2, \dots, d_k \in \N, x \in \R $+'' +\end{enumerate} +Instead the following +\LaTeX\ phrases may appear in your document: +\begin{enumerate} +\item +``\verb+$ d, m \in \N $, $ x \in \R $+'' +\item +``\verb+$ d_1, d_2, \dots, d_k \in \N $, $ x \in \R $+'' +\end{enumerate} +Analogous comments apply to similar expressions. + +\item +Immediately after the phrase ``e.g.,'' +immediately after the phrase ``for example,'' +and immediately after the phrase ``for instance'' +must follow a comma. + +\item +One should never write that a function ``is increasing.'' +Instead one should write that a function ``is strictly increasing'' or ``non-decreasing'' (depending on the correct content). +The analogous comment applies to the phrase that a function ``is decreasing.'' + +\item +The expressions +$ \subset $, +$ \R^{ + } $, +$ \R_{ \geq 0 } $, +or +$ \R^{ * } $ +may nowhere appear in the document. +Instead the expressions +$ \subseteq $, $ \subsetneq $, +$ (0,\infty) $, +$ [0,\infty) $, +or +$ \R \backslash \{ 0 \} $ +can be used in the document. + +\item +Immediately after every +result (by which we mean a theorem, a lemma, a proposition, or a corollary) +of the document (by which we mean a scientific article or a thesis) +\begin{enumerate}[label=(\roman*)] +\item +we add a proof which starts with +``Proof of $ \dots $'' and which ends, e.g., with +``The proof of $ \dots $ is thus complete.'' or +\item +we add a reference to another article\slash thesis +for the proof of the result in front of the result. +\end{enumerate} +See the proof of \cref{prop:prop1} for an example. + + \item + Typically, every sentence, + except for the last one (and sometimes also the first one) + in the proof of a theorem, a lemma, + a proposition, or a corollary + starts with something like + \begin{enumerate} + \item This implies/shows/proves/ensures/assures/yields/demonstrates that $ \dots $ + \item This, equation~(...), inequality~(...), estimate~(...), Theorem~..., Corollary~..., + Lemma~..., and Lemma~... imply/show/prove/ensure/assure/yield/demonstrate that $ \dots $ + \item Combining $ \dots $ (with $ \dots $) (hence/therefore) implies/shows/$ \dots $ that $ \dots $ + \item Theorem~..., Corollary~..., Corollary~..., and Proposition~... (hence/therefore) imply/show/$ \dots $ + that $ \dots $ + \end{enumerate} + Typically, the last sentence in the proof reads + as ``The proof of $ \dots $ is thus complete.''. + Throughout the entire article/thesis the words ``thus'' typically appears + only in the last sentence of a proof. + A proof starts often with + ``Throughout this proof let $ \dots $, + let $ \dots $, + $ \dots $, + and let $ \dots $. Next observe/note that $ \dots $''. + +\item +Just before every display (\LaTeX\: \verb+\begin{equation} ... \end{equation}+) +in the document should appear the phrase ``that'', the phrase ``and'', the phrase ``Then'', or the phrase ``be the set given by'' +without any punctuation mark. + + +For example, we often write + +This implies for all $ \dots $ that \\ +\verb+\begin{equation} ... \end{equation}+ + +or + +This implies that for all $ \dots $ it holds that\\ +\verb+\begin{equation} ... \end{equation}+ + +in a proof. + + + +\item +\label{item:and_that} +The phrase ``and that'' may not appear in any result +nor in any proof. + + +\item +Typically, we do not write a long text between the results. +One sentence relating the result with other results/notions/concepts in the document +as well as with other results/notions/concepts in the literature is often sufficient. +We may add more explanatory sentences (written text) in the introduction and in the beginning of the +section/chapter. + +\item +The document should start with an abstract and, thereafter, a table of contents. + + +\item + Typically, all assumptions in a setting are presented in exactly one sentence. + In particular, we typically formulate a setting as follows: + + Throughout this article the following setting is (frequently\footnote{depending upon whether the setting + is always or only sometimes used throughout the document}) used. Assume/let $ \dots $, assume/let $ \dots $, + assume/let $ \dots $, $ \dots $, and assume/let $ \dots $. + +\item +Every display has exactly one number on the right hand side. In particular, we never use +\verb+\begin{equation*} ... \end{equation*}+ +or +\verb+$$ ... $$+.\\ +We always use\\ +\verb+\begin{equation} ... \end{equation}+ +\\ +or\\ +\verb+\begin{equation} \begin{split} ... \end{split} \end{equation}+ +\\ +instead; see \cref{eq:young}--\cref{eq:Lyapunov_end} for a few examples. + +\item +We never use +\begin{assumption} +Throughout this article/thesis assume that $ T p < 1 $, +that for all $ x \in \R $ it holds that +\begin{equation} + f( x ) \leq 2 x \text{,} +\end{equation} +and that for all $ y \in [0,\infty) $ it holds that +\begin{equation} + g( y ) \leq 2 y \text{.} +\end{equation} +\end{assumption} +but we may use one or more settings +(see \cref{item:and_that} above and see \cref{sec:setting} +and \cref{prop:prop3} for an example). + + + \item + Often the structure of a thesis/research article is as follows: + \begin{enumerate} + \item Introduction + \begin{enumerate} + \item[1.1] Notation + \item[1.2] Setting + \end{enumerate} + \item First section for results $ \dots $ + \item Second section for results $ \dots $ + \item $ \dots $ + \end{enumerate} + + + + +\item +Serial comma (also called Oxford comma): +We write +``$ A $, $ B $, and $ C $'' +instead of ``$ A $, $ B $ and $ C $''. +Similarly we write +``$ A $, $ B $, or $ C $'' +instead of +``$ A $, $ B $ or $ C $.'' +\begin{enumerate}[label=(\alph*)] + \item +However, please note that the phrase +``let $ f \colon \R \to \R $, and $ g \colon \R \to \R $ be functions which satisfy'' +may not appear in your document. +\item +Instead the phrase +``let $ f \colon \R \to \R $ and $ g \colon \R \to \R $ be functions which satisfy'' +may appear in your document. +\end{enumerate} + + +\item +There should be no significant \LaTeX\ badboxes in the document. + +\item +You are welcome to use the axiom schema of specification +to specify a set. More specifically, if $ S $ is a set and $ P $ +is a predicate with one single argument, then you can use +the set +\begin{equation} +\label{eq:set_with_colon_command} + \left\{ x \in S \colon P(x) \right\} +\end{equation} +according to the axiom schema of specification in your document. +In contrast, the expression +\begin{equation} + \left\{ x \in S \, | \, P(x) \right\} +\end{equation} +may not appear in your document. Moreover, the expression +\begin{equation} +\label{eq:set_without_colon_command} + \left\{ x \in S : P(x) \right\} +\end{equation} +may also not appear in your document. +As usual please check the employed \LaTeX\ code for \cref{eq:set_with_colon_command} +and \cref{eq:set_without_colon_command}, respectively. + + +\item +\label{item:cases} +You are welcome to employ to the +\LaTeX\ +command \verb+\begin{cases} ... \end{cases}+ +to create a distinction of cases within a display. +If you employ the +\LaTeX\ +command \verb+\begin{cases} ... \end{cases}+, +then it must employ the following structure: +\begin{verbatim} +\begin{cases} +Value in the first case +& +\colon +First case +\\ +Value in the second case +& +\colon +Second case +\\ +Value in the third case +& +\colon +Third case +\\ +... +\\ +Value in the n-th case +& +\colon +n-th case +\end{cases} +\end{verbatim} +For example, you may use the following phrase +``let $ a \in \R $, +let $ \operatorname{sgn} \colon \R \to \R $ be the function which +satisfies for all $ x \in \R $ that +\begin{equation} +\label{eq:sign_with_colon} + \operatorname{sgn}(x) + = + \begin{cases} + 1 + & + \colon x \geq 0 + \\ + - 1 + & + \colon x < 0 + \text{,} + \end{cases} +\end{equation} +and let $ \dots $'' +to introduce a real number $ a \in \R $ +and to specify the sign function (a sign function). +The phrase +``let $ \operatorname{sgn} \colon \R \to \R $ be the function that satisfies'' +may not appear in your document. + + +\item +The phrase ``let $ a \in \R $, +let $ \operatorname{sgn} : \R \to \R $ be the function which +satisfies'' +may not appear in your document. +The phrase ``let $ a \in \R $, +let $ sgn \colon \R \to \R $ be the function which +satisfies'' +may also not appear in your document. +Please note the differences +between +$ \operatorname{sgn} \colon \R \to \R $ +(cf.\ \cref{eq:sign_with_colon} in \cref{item:cases} above), +$ \operatorname{sgn} : \R \to \R $ (see above), +and +$ sgn \colon \R \to \R $ (see above) +and as usual please also study the \LaTeX\ codes used to create the above phrases. + + +\item +The phrase ``(c.f.\ \cref{eq:sign_with_colon})'' may not appear in your document. +The phrase ``(cf. \cref{eq:sign_with_colon}'' may also not appear in your document. +In contrast, the phrase ``(cf.\ \cref{eq:sign_with_colon})'' +and the phrase ``(see \cref{eq:sign_with_colon})'' +may appear in your document. +As usual please also study the \LaTeX\ codes used to create the above phrases. + + +\item +The phrase +``let $ \left\lVert \cdot \right\lVert_{ \R^d } \colon \R^d \to [0,\infty) $ be the Euclidean norm on $ \R^d $'' +may appear in your document. +\begin{enumerate}[label=(\alph*)] +\item +The phrase +``let $ \left\lVert \cdot \right\rVert_{ \R^d } \colon \R^d \to [0,\infty) $ be the euclidean norm on $ \R^d $'' +may not appear in your document. +Analogously, one writes ``is H\"{o}lder continuous'' instead of is ``h\"{o}lder continuous.'' + +\item +The phrase +``let $ \| \cdot \|_{ \R^d } \colon \R^d \to [0,\infty) $ be the Euclidean norm on $ \R^d $'' +may also not appear in your document. + +\end{enumerate} +As usual please also study the \LaTeX\ codes used to create the above phrases. + + +\item +The phrase ``function'' and the phrase ``functions'' may appear in your document. +In contrast, each of the following phrases may not appear in your document: +\begin{enumerate}[label=(\alph*)] +\item +mapping +\item +Mapping +\item +map +\item +Map +\item +Function +\item +Functions +\end{enumerate} +The following phrases may appear in your document: +\begin{enumerate}[label=(\alph*)] + +\item +let $ f \colon \R \to \R $ be the function which satisfies for all + +\item +let $ f \colon \R \to \R $ satisfy for all + +\item +let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, be the functions which satisfy for all +$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $ + +\item +let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, satisfy for all +$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $ + +\item +let $ f \colon \R^2 \to \R $ and $ g \colon \R \to \R $ be functions +\item +let $ f = ( f(x,y) )_{ (x,y) \in \R^2 } \colon \R^2 \to \R $ and $ g \colon \R \to \R $ be functions +\item +let $ f = ( f_x(y) )_{ (x,y) \in \R^2 } \colon \R^2 \to \R $ and $ g \colon \R \to \R $ be functions +\item +let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $, satisfy +\item +let $ a_n \in \R $, $ n \in \N $, satisfy for all $ n \in \N $ that +$ a_n = n $ +\item +let $ ( a_n )_{ n \in \N } \subseteq \R $ satisfy for all $ n \in \N $ that +$ a_n = n $ +\end{enumerate} +In contrast, the following phrases may not appear in your document: +\begin{enumerate}[label=(\alph*)] +\item +let $ f : \R \to \R $ be the function which satisfies for all +\item +let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $ be the functions which satisfy for all +$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $ +\item +let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, be the functions which satisfy for all +$ x \in \R $ that $ g_r( x ) = x + r $ +\item +let $ f \colon \R^2 \to \R $, $ g \colon \R \to \R $ be functions +\item +let $ f \colon \R^2 \to \R $, $ g \colon \R \to \R $, be functions + +\item +let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $ satisfy + +\item +Let $ g_r \in C( \R, \R ) $ satisfy + +\item +Let $ a \in \R $ and let $ g_r \in C( \R, \R ) $ satisfy + +\item +let $ g_r \in C( \R, \R ), r \in (0,\infty) $, satisfy + +\item +let $ g_r \in C( \R, \R ), r \in (0,\infty), $ satisfy + +\item +let $ a_n \in \R $, $ n \in \N $, satisfy that +$ a_n = n $ + +\item +let $ ( a_n )_{ n \in \N } \subseteq \R $ satisfy that +$ a_n = n $ + +\item +let $ ( a_n )_{ n \in \N } \subseteq \R $, $ n \in \N $, satisfy for all $ n \in \N $ that +$ a_n = n $ + + +\item +let $ ( a_n )_{ n \in \N } \subseteq \R $, $ n \in \N $ satisfy for all $ n \in \N $ that +$ a_n = n $ + +\end{enumerate} +As usual please also study the \LaTeX\ codes used to create the above phrases. +\begin{enumerate}[label=(\Alph*)] +\item +\label{item:with_functions} +In your document you may use the following phrases: +% +% +\begin{enumerate}[label=(\alph*)] +% +\item +let $ f \colon \R \to \R $ be the function which satisfies for all + +\item +let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, be the functions which satisfy for all +$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $ + +\item +let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $, satisfy + +\end{enumerate} +% +% +\item +\label{item:without_functions} +Instead of the phrases in +\cref{item:with_functions} +you may also use the following phrases: +\begin{enumerate}[label=(\alph*)] +\item +let $ f \colon \R \to \R $ satisfy for all + +\item +let $ g_r \colon \R \to \R $, $ r \in (0,\infty) $, satisfy for all +$ r \in (0,\infty) $, $ x \in \R $ that $ g_r( x ) = x + r $ + +\item +let $ g_r \in C( \R, \R ) $, $ r \in (0,\infty) $, satisfy +\end{enumerate} +\end{enumerate} +You may, however, not mix between +\cref{item:with_functions} +and +\cref{item:without_functions}. +Please use either the phrases in +\cref{item:with_functions} +or the phrases in +\cref{item:without_functions}. + + +\item +Except in the introduction and the text between the presented results, +the phrase ``satisfying'' +% , the phrase ``and for'', +and the phrase ``fulfilling'' may not appear in your document. + +\end{enumerate} + + +\section{Precise specifications} + + +\subsection{Structure of the document} + +A document (by which we mean a scientific article or a thesis) +consists of exactly the following parts (in this order): +\begin{enumerate} +\item +Title of the document (Title) +\item +Name of the authors +\item +Affiliations of the authors +\item +Abstract of the document (Abstract) +\item +Table of contents of the document +\item +Main text of the document (Main text) +\item +List of references. +\end{enumerate} +The main text contains of the following \LaTeX\ sections: +\begin{enumerate} +\item +Introduction +\item +Name of the 2nd section +\item +Name of the 3rd section +\item +\dots +\end{enumerate} + +\subsection{\LaTeX\ header} + +The \LaTeX\ header should not import packages which are not used in the document. +The following \LaTeX\ code provides an illustrative example. +\begin{verbatim} +\documentclass[a4paper,12pt]{article} +\usepackage{amsthm, + amsmath, + amssymb, + bbm, + enumerate, + geometry, + nicefrac, + hyperref + } + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{remark}{Remark} +\newtheorem{prop}{Proposition} +\newtheorem{cor}{Corollary} + +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\E}{\mathbb{E}} +\renewcommand{\P}{\mathbb{P}} +\newcommand{\tr}{\operatorname{trace}} + +\title{Title of the document} + +\author{First author$^1$, Second author$^2$, and Third Author$^{3}$ +\bigskip +\\ +\small{$^1$Department of Mathematics, Some University, +USA, e-mail: $ \dots @ \dots $} +\smallskip +\\ +\small{$^2$Department of Mathematics, ETH Zurich, +Switzerland, e-mail: $ \dots @ \dots $} +\smallskip +\\ +\small{$^3$Faculty of Mathematics, Bielefeld University, +Germany, e-mail: $ \dots @ \dots $}} + +\begin{document} + +\maketitle + +\begin{abstract} +Abstract ... +\end{abstract} + +\tableofcontents + +\section{Introduction} + +\bibliographystyle{acm} +\bibliography{bibfile} +\end{document} +\end{verbatim} + + +\subsection{Main text of the document} +% +% +% +The main text consists of environments (\LaTeX\ environments) +with free text in between. +The free text does not follow precise specifications. +In Section~1 there is typically a lot of free text. +In all other sections there is typically only very little free text. +The purpose of the free text in all other sections beside +Section~1 is typically to link the presented environment +to other environments in the document as well as +to link the presented environment to other findings in the +scientific literature. +% +% +% +% +\subsection{\LaTeX\ environments} +% +% +% +An environment is +\begin{enumerate} +\item a remark, +\item a setting, +\item a framework, +\item a definition, +\item a result (by which we mean a theorem, a lemma, a proposition, or a corollary), +or +\item the proof of a result. +\end{enumerate} +In \LaTeX, an environment is formulated by means of the +commands: +\begin{verbatim} +\begin{type_of_the_environment} +\label{...} +... +\end{type_of_the_environment} +\end{verbatim} +It is not allowed to use the word +\emph{where}, the word \emph{from}, the word \emph{since}, the word \emph{as}, the word \emph{by}, the word \emph{because}, +or the word \emph{using} +in a \LaTeX\ environment. + +% +% +\subsubsection{Remarks} +A remark does not necessarily need to follow precise specifications. All other environments follow +precise specifications. +% +% +% +\subsubsection{Settings and frameworks} +% +% +% +A setting and a framework consists of exactly one sentence. +This sentence has the following structure (illustrated here in the case of +a setting environment): +\begin{setting} +Assume/let $ \dots $, assume/let $ \dots $, +assume/let $ \dots $, +$ \dots $, +assume/let $ \dots $, +and assume/let $ \dots $. +\end{setting} +A framework environment is sometimes useful to describe algorithms. +In most cases the document under consideration +does not contain any framework but there might be several settings. +% +% +% +\subsubsection{Definitions} +% +% +Every definition either consists +of exactly one sentence +or consists of exactly two sentences. +% +% +\paragraph{Definitions consisting of exactly one sentence} +% +% +If a definition consists of exactly one sentence, then +the definition must follow the following structure: +% +% +\begin{definition} +We denote by $ \dots $. +\end{definition} +% +% +\paragraph{Definitions consisting of exactly two sentences} +% +% +If a definition consists of exactly two sentences, then +the definition must follow the following structure: +% +% +\begin{definition} +Let $ \dots $, let $ \dots $, $ \dots $, and let $ \dots $ +Then we denote by $ \dots $. +\end{definition} +% +% +\subsubsection{Results} +% +% +If a result consists of more than one sentence, then the second sentence of the result must start with the word ``Then.'' +% +\paragraph{Results consisting of exactly one sentence} +% A result consists of at least one sentence and at most two sentences. +If a result consists of exactly one sentence, +then the result must follow one of the following two structures (illustrated here in the case of +a theorem environment): +\begin{enumerate} +\item +First option for a result (theorem, lemma, proposition, or corollary) +with exactly one sentence (illustrated here in the case of +a theorem environment): +\begin{theorem} +It holds for all $ \dots $ that $ \dots $. +\end{theorem} +\item +Second option for a result (theorem, lemma, proposition, or corollary) +with exactly one sentence (illustrated here in the case of +a theorem environment): +\begin{theorem} +There exist(s) $ \dots $ that $ \dots $. +\end{theorem} +\end{enumerate} +% +% +\paragraph{Results consisting of exactly two sentences} +% +% +In most cases a result consists of exactly two sentences. +In this case the result (theorem, lemma, proposition, or corollary) +must follow the following structure (illustrated here in the case of +a theorem environment): +\begin{theorem} +\label{thm:example} +Assume/let $ \dots $, +assume/let $ \dots $, +assume/let $ \dots $, +$ \dots $, +assume/let $ \dots $, +and assume/let $ \dots $. +Then $ \dots $. +\end{theorem} +The first sentence in \cref{thm:example} formulates +the employed hypotheses. If a setting is employed as one hypothesis, +then one writes (where $ n \in \N = \{ 1 , 2 , 3, \dots \} $ is a natural number): +\emph{Assume Setting~??} (1st hypothesis), +\emph{assume/let $ \dots $} (2nd hypothesis), +\emph{assume/let $ \dots $} (3rd hypothesis), +\emph{$ \dots $}, +\emph{assume/let $ \dots $} ($ ( n - 1 ) $-th hypothesis), +\emph{and +assume/let $ \dots $} ($ n $-th hypothesis). +% The second sentence of a result must start with the word ``Then''. +In most cases the second sentence of a result starts +\begin{enumerate}[label=(\roman*)] +\item +with ``Then it holds for all $ \dots $'' or +\item +with ``Then there exist(s) $ \dots $.'' +\end{enumerate} +% +% +% +% +\paragraph{Results consisting of more than two sentences} +% +% +If a result (theorem, lemma, proposition, or corollary) consists of more than two sentences, +then the result must follow the following structure (illustrated here in the case of +a theorem environment): +\begin{theorem} +Assume/let $ \dots $, +assume/let $ \dots $, +assume/let $ \dots $, +$ \dots $, +assume/let $ \dots $, +and assume/let $ \dots $. +Then the following two/three/four/five/$ \dots $ statements are equivalent: +\begin{enumerate}[label=(\roman*)] +\item It holds $ \dots $/There exist(s) $ \dots $. +\item It holds $ \dots $/There exist(s) $ \dots $. +\item It holds $ \dots $/There exist(s) $ \dots $. +\item $ \dots $ +\item $ \dots $. +\end{enumerate} + +\end{theorem} +% +% +% +\subsubsection{Proofs} +% +% +% +A proof of a result should consist of \emph{the introductory paragraph of the proof} +(see \cref{sec:beginning} below), +of a finite possibly empty sequence of \emph{argumentation sentences} +(see \cref{sec:argumentation} below), +and of \emph{the closing sentence} (see \cref{sec:closing} below). +% +% +% +\paragraph{The introductory paragraph of the proof} +\label{sec:beginning} +Every proof of a result consists of at least two sentences. +The beginning of the proof starts +\begin{enumerate}[label=(\roman*)] +\item +with +``Throughout this proof assume w.l.o.g.\ that $ \dots $ (1st hypothesis), +assume w.l.o.g.\ that $ \dots $ (2nd hypothesis), +$ \dots $, +assume w.l.o.g.\ that $ \dots $ ($ k $-th hypothesis), +let $ \dots $ (1st set introduction), +let $ \dots $ (2nd set introduction), +$ \dots $, +let $ \dots $ ($ l - 1 ) $-th set introduction), +and +let $ \dots $ ($ l $-th set introduction). Note/observe that $ \dots $'' +(where $ k, l \in \N_0 = \N \cup \{ 0 \} = \{ 0, 1, 2, \dots \} $), +\item +with +``Note/observe that $ \dots $,'' +or +\item +with +``First, note/observe that $ \dots $.'' +\end{enumerate} +% +% +% +\paragraph{Argumentation sentences} +\label{sec:argumentation} +% +% +% +% Subsequent to these formulations, the proof consists +% of \emph{argumentation sentences} +% and one \emph{closing sentence} after these argumentation sentences +% (see Subsection~\ref{sec:closing} below). +In each argumentation sentence we distinguish between +\begin{enumerate}[label=\arabic*.] +\item +a \emph{Type A1 argumentation sentence}, +\item +a \emph{Type A2 argumentation sentence}, +\item +a \emph{Type B1 argumentation sentence}, and +\item +a \emph{Type B2 argumentation sentence}. +\end{enumerate} +% +% +An argumentation sentence which employs +the previous argumentation sentence is +a Type A1 argumentation sentence or +a Type A2 argumentation sentence. +An argumentation sentence which does not employ the previous +argumentation sentence +is a Type B1 argumentation sentence +or a Type B2 argumentation sentence. +Each argumentation sentence may only use the following verbs: +\begin{enumerate} +\item +note +\item +observe +\item +imply/implies +\item +assure/assures +\item +ensure/ensures +\item +demonstrate/demonstrates +\item +prove/proves +\item +establish/establishes +\item +show/shows +\item +yield/yields +\item +hold/holds +\item +combine +\item +obtain +\item +satisfy/satisfies +\item +exist/exists +\item +be +\end{enumerate} +Besides the above verbs, no other verb may be used in an argumentation sentence. +If an argumentation sentence contains the phrase ``combine,'' then +this argumentation sentence must contain exactly one of +the following phrases: +\begin{enumerate} +\item +``In the next step we combine $ \dots $ to obtain that $ \dots $.'' +\item +``Next, we combine $ \dots $ to obtain that $ \dots $.'' +\end{enumerate} +The phrase ``combine'' may only appear in a Type B1 argumentation sentence. +% +% +% +% +\paragraph{Type A1 argumentation sentences} +A Type A1 argumentation sentence +must contain exactly one of the following phrases: +\begin{enumerate} +\item ``therefore'' +\item ``Therefore, we obtain that $ \dots $'' +\item ``hence'' +\item ``Hence, we obtain that $ \dots $'' +\item ``this'' +\item ``This $ \dots $'' +\end{enumerate} +% +% +% +Each Type A1 argumentation sentence must use exactly one of the following structures: +\begin{enumerate} + + +\item +Hence, we obtain that $ \dots $ +(followed by possible quantification of a predicate) + + +\item +Therefore, we obtain that $ \dots $ +(followed by possible quantification of a predicate) + + +\item +Combining $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ +(where $ n \in \{ 2, 3, 4, \dots \} $) +hence\slash +therefore +implies\slash +ensures\slash +assures\slash +demonstrates\slash +proves\slash +establishes\slash +yields\slash +shows +(followed by possible\linebreak quantification of a predicate) + + +\item +Combining +$ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ +(where $ n \in \N = \{ 1, 2, 3, \dots \} $) +with +$ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ +(where $ k \in \N = \{ 1, 2, 3, \dots \} $) +hence\slash +therefore +implies\slash +ensures\slash +assures\slash +demonstrates\slash +proves\slash +establishes\slash +yields\slash +shows +(followed by possible quantification of a predicate) + + +\item +Combining this, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ +(where $ n \in \N $) +implies\slash +ensures\slash +assures\slash +demon\-strates\slash +proves\slash +establishes\slash +yields\slash +shows +(followed by possible quantification of a predicate) + + +\item +Combining +this, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ +(where $ n \in \N_0 = \N \cup \{0\} $) +with +$ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ +(where $ k \in \N$) +implies\slash +ensures\slash +assures\slash +demonstrates\slash +proves\slash +establishes\slash +yields\slash +shows +(followed by possible quantification of a predicate) + + +% \item +% Combining +% this, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ +% (where $ n \in \N_0 $) +% with +% $ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ +% (where $ k \in \N$) allows us to choose $ \dots $ + + +\item +This +implies\slash +ensures\slash +assures\slash +demonstrates\slash +proves\slash +establishes\slash +yields\slash +shows +that +(followed by possible quantification of a predicate) + + +\item +This, $ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ +(where $ n \in \N $) +imply\slash +ensure\slash +assure\slash +demonstrate\slash +prove\slash +establish\slash +yield\slash +show +% that +(followed by possible quantification of a predicate) + + +% \item +% $ S_1 $, $ S_2 $, $ \dots $, $ S_n $ (where $ n \in \N $), +% this, $ T_1 $, $ T_2 $, $ \dots $, and $ T_k $ (where $ k \in \N_0 $) +% imply/ensure/assure/demonstrate/prove/establish +% that +% (possible quantification of a predicate) + + +\item +$ S_1 $, $ S_2 $, $ \dots $, and $ S_n $ (where $ n \in \{ 2, 3, 4, \ldots \} $) +hence\slash +therefore +imply\slash +ensure\slash +assure\slash +demonstrate\slash +prove\slash +establish\slash +yield\slash +show +(followed by possible quantification of a predicate) +(For example: Lemma~1 and Theorem~3 hence prove that for all $ x \in \R $ it holds that $ f(x) = y $.) + + +\item +$ S_1 $ hence\slash therefore +implies\slash +ensures\slash +assures\slash +demonstrates\slash +proves\slash +establishes\slash +yields\slash\linebreak +shows +(followed by possible quantification of a predicate) +(For example: Lemma~1 hence proves that for all $ x \in \R $ it holds that $ f(x) = y $.) + + +\end{enumerate} +% +% +The phrase ``This'' may only appear in a Type A1 argumentation sentence. +The phrase ``this'' may only appear in a Type A1 argumentation sentence. +% +% +\paragraph{Type A2 argumentation sentences} +A Type A2 argumentation sentence +must follow the following structure: +``Observe/Note that $ \dots $ +imply\slash +implies\slash +ensure\slash +ensures\slash +demonstrate\slash +demonstrates\slash +show\slash +shows\slash +prove\slash +proves\slash +yield\slash +yields +% that +$ \dots $.'' +% +% +% +\paragraph{Type B1 argumentation sentences} +A Type B1 argumentation sentence +must contain exactly one of the following phrases: +\begin{enumerate} +\item ``Moreover, note that $ \dots $'' +\item ``Moreover, observe that $ \dots $'' +\item ``Furthermore, note that $ \dots $'' +\item ``Furthermore, observe that $ \dots $'' +\item ``In addition, note that $ \dots $'' +\item ``In addition, observe that $ \dots $'' +\item ``Next, note that $ \dots $'' +\item ``Next, observe that $ \dots $'' +\item ``In the next step, we note that $ \dots $'' +\item ``In the next step, we observe that $ \dots $'' +\item ``In the next step, we combine $ \dots $ to obtain that $ \dots $'' +\item ``Next, we combine $ \dots $ to obtain that $ \dots $'' +\end{enumerate} +% +% +\paragraph{Type B2 argumentation sentences} +% +% +A Type B2 argumentation sentence +must follow the following structure: +``In the next step/Next, let $ \dots $ +which satisfy/satisfies $ \dots $.'' +% +% +\paragraph{The closing sentence of the proof} +\label{sec:closing} +% +% +The last sentence of the proof reads +\begin{enumerate} +\item +as ``The proof of $ \dots $ is thus complete.'' or +\item +as ``This completes the proof of $ \dots $.'' + +\end{enumerate} + +%\begin{comment} + +\newpage + +\section{An illustrative example} + +We now provide an illustrative example of how these writing styles may be used in a scientific article/thesis. +Note that in an actual article/thesis, we would utilize \verb+\section{...}+ rather than \verb+\subsection{...}+. + +\subsection{Introduction} + +Here should be an introduction. + +\subsubsection{Notation} + +Throughout this article/thesis we use the following notation. +For sets $ A $ and $ B $ we denote by +$ \mathbb{M}( A, B) $ +the set of all functions from $ A $ to $ B $. +Moreover, for measurable spaces +$ (A, \mathcal{A} ) $ +and +$ (B, \mathcal{B} ) $ +we denote by +$ + \mathcal{M}( \mathcal{A}, \mathcal{B} ) +$ +the set of all +$ \mathcal{A} $/$ \mathcal{B} $-measurable spaces. + + +\subsubsection{Setting} +\label{sec:setting} + + +Throughout this article/thesis the following setting is frequently used. +Let $ T \in (0,\infty) $, $m \in \N$, +let +$ ( \Omega , \mathcal{F}, \P ) $ +be a probability space with a normal filtration +$ ( \mathcal{F}_t )_{ t \in [0,T] } $, +and let +$ + W \colon [0,T] \times \Omega \to \R^m +$ +be a standard $ ( \mathcal{F}_t )_{ t \in [0,T] } $-Brownian motion. + + + +\subsection{Results} + + +\begin{proposition}[Young's inequality] +\label{prop:prop1} +Let $ n \in \N $, $ a_1, \dots, a_n \in \R $, +$ p_1, \dots, p_n \in (1,\infty) $ +satisfy +$ + \frac{ 1 }{ p_1 } + \dots + \frac{ 1 }{ p_n } = 1 +$. +Then it holds that +\begin{equation} +\label{eq:young} + a_1 \cdot a_2 \cdot \ldots \cdot a_n +\leq + \frac{ + \left\lvert a_1 \right\rvert^{ p_1 } + }{ p_1 } + + + \ldots + + + \frac{ + \left\lvert a_n \right\rvert^{ p_n } + }{ p_n } + \text{.} +\end{equation} +\end{proposition} + + +\begin{proof}[Proof of \cref{prop:prop1}] +Throughout this proof without loss of generality assume that $ a_1 \cdot a_2 \cdot \ldots \cdot a_n \neq 0 $ +and let +$ x_1, \dots, x_n \in (0,\infty) $ +satisfy +$ + x_1 + = + \ln\left( + \left\lvert a_1 \right\rvert^{ p_1 } + \right) +$, +$ + x_2 + = + \ln\left( + \left\lvert a_2 \right\rvert^{ p_2 } + \right) +$, +$ \ldots $, +$ + x_n + = + \ln\left( + \left\lvert a_n \right\rvert^{ p_n } + \right) +$. +This ensures that for all $ k \in \{ 1, 2, \dots, n \} $ +it holds that +$ a_k \neq 0 $. +Next, note that the convexity of the function +$ + ( 0, \infty ) \ni x \mapsto e^x \in (0,\infty) +$ +and the assumption that +$ + \frac{ 1 }{ p_1 } + \dots + \frac{ 1 }{ p_n } = 1 +$ +ensure that +\begin{equation} +\begin{split} +a_1 \cdot a_2 \cdot \ldots \cdot a_n +& \leq + \left\lvert + a_1 \cdot a_2 \cdot \ldots \cdot a_n + \right\rvert +\\ & = + \exp\left( + \frac{ + \ln\left( + \left\lvert a_1 \right\rvert^{ p_1 } + \right) + }{ p_1 } + \right) + \cdot + \exp\left( + \frac{ + \ln\left( + \left\lvert a_2 \right\rvert^{ p_2 } + \right) + }{ p_2 } + \right) + \cdot + \ldots + \cdot + \exp\left( + \frac{ + \ln\left( + \left\lvert a_n \right\rvert^{ p_n } + \right) + }{ p_n } + \right) +\\ & = + \exp\left( + \frac{ + x_1 + }{ p_1 } + \right) + \cdot + \exp\left( + \frac{ + x_2 + }{ p_2 } + \right) + \cdot + \ldots + \cdot + \exp\left( + \frac{ + x_n + }{ p_n } + \right) += + \exp\left( + \frac{ x_1 }{ p_1 } + + + \frac{ x_2 }{ p_2 } + + + \ldots + + + \frac{ x_n }{ p_n } + \right) +\\ & \leq + \frac{ \exp( x_1 ) }{ p_1 } + + + \frac{ \exp( x_1 ) }{ p_1 } + + + \ldots + + + \frac{ \exp( x_1 ) }{ p_1 } + = + \frac{ + \left\vert a_1 \right\rvert^{ p_1 } + }{ p_1 } + + + \ldots + + + \frac{ + \left\lvert a_n \right\rvert^{ p_n } + }{ p_n } + \text{.} +\end{split} +\end{equation} +The proof of \cref{prop:prop1} +is thus complete. +\end{proof} + + +More results on stochastic differential equations can, e.g., be found +in Kloeden and Platen \cite{kp92}. + +\begin{proposition}[Lyapunov-type functions] +\label{prop:prop2} +Let $ c, T \in [0,\infty) $, +$ d, m \in \N $, +let $ D \subseteq \R^d $ be an open set, +let +$ + \mu \in + \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^d ) ) +$, +$ + \sigma \in + \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^{ d \times m } ) ) +$, +$ V \in C^2( D, [0,\infty) $ +satisfy that for all $ x \in D $ it holds that +\begin{equation} + V'(x) \mu( x ) + + + \sum_{ k = 1 }^m + V''( x )\left( + \sigma_k( x ) + , + \sigma_k( x ) + \right) +\leq + c \, V( x ) + \text{,} +\end{equation} +let $ ( \Omega, \mathcal{F}, \P ) $ be a probability space +with a normal filtration $ ( \mathcal{F}_t )_{ t \in [0,T] } $, +let +$ + W \colon [0,T] \times \Omega \to \R^m +$ +be a standard $ ( \mathcal{F}_t )_{ t \in [0,T] } $-Brownian motion, +and let $ X \colon [0,T] \times \Omega \to D $ be an +$ ( \mathcal{F}_t )_{ t \in [0,T] } $-adapted stochastic process +with continuous sample paths satisfying that for all +$ t \in [0,T] $ it holds $ \P $-a.s.\ that +\begin{equation} + X_t = + X_0 + + + \int_0^t + \mu( X_s ) \, {\rm d}s + + + \int_0^t + \sigma( X_s ) \, {\rm d}W_s + \text{.} +\end{equation} +Then for all $ t \in [0,T] $ +it holds that +\begin{equation} + \E\big[ V( X_t ) \big] + \leq + e^{ c t } + \, + \E\big[ V( X_0 ) \big] + \text{.} +\end{equation} +\end{proposition} + +\begin{proposition}[Lyapunov-type functions revisited] +\label{prop:prop3} +Assume the setting outlined in \cref{sec:setting}, +let $ c \in [0,\infty) $, +$ d\in \N $, +let $ D \subseteq \R^d $ be an open set, +let +$ + \mu \in + \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^d ) ) +$, +$ + \sigma \in + \mathcal{M}( \mathcal{B}( D ) , \mathcal{B}( \R^{ d \times m } ) ) +$, +$ V \in C^2( D, [0,\infty) ) $ +satisfy that for all $ x \in D $ it holds that +\begin{equation} + V'(x) \mu( x ) + + + \sum_{ k = 1 }^m + V''( x )\left( + \sigma_k( x ) + , + \sigma_k( x ) + \right) +\leq + c \, V( x ) + \text{,} +\end{equation} +and let $ X \colon [0,T] \times \Omega \to D $ be an +$ ( \mathcal{F}_t )_{ t \in [0,T] } $-adapted stochastic process +with continuous sample paths satisfying that for all +$ t \in [0,T] $ it holds $ \P $-a.s.\ that +\begin{equation} + X_t = + X_0 + + + \int_0^t + \mu( X_s ) \, {\rm d}s + + + \int_0^t + \sigma( X_s ) \, {\rm d}W_s + \text{.} +\end{equation} +Then it holds for all $ t \in [0,T] $ that +\begin{equation} +\label{eq:Lyapunov_end} + \E\big[ V( X_t ) \big] + \leq + e^{ c t } + \, + \E\big[ V( X_0 ) \big] + \text{.} +\end{equation} +\end{proposition} + + + +\begin{theorem} +\label{thm:sum} +Assume the setting outlined in \cref{sec:setting} +and let $ \alpha \in \R $. +Then +\begin{equation} + \lim_{ N \to \infty } + \left( + N^{ \alpha } + \, + \E\big[ \lvert W_{ T / N } \rvert \big] + \right) + = + \begin{cases} + \infty + & + \colon + \alpha \geq \nicefrac{ 1 }{ 2 } + \\ + \E\big[ \lvert W_{ T } \rvert \big] + & + \colon + \alpha = \nicefrac{ 1 }{ 2 } + \\ + 0 + & + \colon + \alpha < \nicefrac{ 1 }{ 2 } + \end{cases} + \text{.} +\end{equation} +\end{theorem} + +\bibliographystyle{acm} +\bibliography{bibfile} + +%\end{comment} + +\mathfrak{Y} + + +\end{document} + + \ No newline at end of file diff --git a/Writing Style/Writing_Style.toc b/Writing Style/Writing_Style.toc new file mode 100644 index 0000000..e69de29 diff --git a/Writing Style/bibfile.bib b/Writing Style/bibfile.bib new file mode 100644 index 0000000..50cf430 --- /dev/null +++ b/Writing Style/bibfile.bib @@ -0,0 +1,20774 @@ +% Encoding: UTF-8 + +@Preamble{ +"\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth +\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}" +} +} + +@article {MR4265903, + title={{A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems}}, + author={Reisinger, Christoph and Stockinger, Wolfgang and Zhang, Yufei}, + journal={arXiv:2108.06740}, + year={2021}, + pages={26 pages} +} +} + +@article {MR4265654, + AUTHOR = {Gu, Yiqi and Yang, Haizhao and Zhou, Chao}, + TITLE = {Select{N}et: self-paced learning for high-dimensional partial + differential equations}, + JOURNAL = {J. Comput. Phys.}, + FJOURNAL = {Journal of Computational Physics}, + VOLUME = {441}, + YEAR = {2021}, + PAGES = {Paper No. 110444, 18}, + ISSN = {0021-9991}, + MRCLASS = {65N99 (65M12 65M99 65N12)}, + MRNUMBER = {4265654}, + DOI = {10.1016/j.jcp.2021.110444}, + URL = {https://doi.org/10.1016/j.jcp.2021.110444}, +} + +@article{nusken2021solving, + title={{Solving high-dimensional Hamilton--Jacobi--Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space}}, + author={N{\"u}sken, Nikolas and Richter, Lorenz}, + journal={Partial Differ. Equ. Appl.}, + volume={2}, + number={4}, + pages={1--48}, + year={2021}, + publisher={Springer} +} + +@article{kapllani2020deep, + title={Deep Learning algorithms for solving high dimensional nonlinear Backward Stochastic Differential Equations}, + author={Kapllani, Lorenc and Teng, Long}, + journal={arXiv:2010.01319}, + year={2020}, + pages={21 pages}, +} + +@article{chassagneux2021learning, + title={{A learning scheme by sparse grids and Picard approximations for semilinear parabolic PDEs}}, + author={Chassagneux, Jean-Fran{\c{c}}ois and Chen, Junchao and Frikha, Noufel and Zhou, Chao}, + journal={arXiv:2102.12051}, + year={2021}, + pages={56 pages}, +} + +@article{beck2020overview, + title={An overview on deep learning-based approximation methods for partial differential equations}, + author={Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf and Kuckuck, Benno}, + journal={arXiv:2012.12348}, + year={2020}, + pages={23 pages}, + note={\emph{Revisions requested from Discrete Contin. Dyn. Syst. Ser. B}}, +} + +@article{hutzenthaler2021multilevel, + title={{Multilevel Picard approximations for McKean-Vlasov stochastic differential equations}}, + author={Hutzenthaler, Martin and Kruse, Thomas and Nguyen, Tuan Anh}, + journal={arXiv:2103.10870}, + year={2021}, + pages={14 pages}, +} + +@article{kloeden2007pathwise, + title={The pathwise convergence of approximation schemes for stochastic differential equations}, + author={Kloeden, Peter E and Neuenkirch, Andreas}, + journal={LMS J.\ Comput.\ Math.}, + volume={10}, + pages={235--253}, + year={2007}, + publisher={Cambridge University Press} +} + +@article{rio2009moment, + title={Moment inequalities for sums of dependent random variables under projective conditions}, + author={Rio, Emmanuel}, + journal={J.\ Theor.\ Probab.}, + volume={22}, + number={1}, + pages={146--163}, + year={2009}, + publisher={Springer} +} + +@article{FlandoliLuoRicci2020, +title = "A numerical approach to Kolmogorov equation in high dimension based on Gaussian analysis", +journal = "Journal of Mathematical Analysis and Applications", +pages = "124505", +year = "2020", +issn = "0022-247X", +doi = "https://doi.org/10.1016/j.jmaa.2020.124505", +url = "http://www.sciencedirect.com/science/article/pii/S0022247X20306673", +author = "Franco Flandoli and Dejun Luo and Cristiano Ricci", +keywords = "Kolmogorov equation, Numerical solution, Iteration schema, Gaussian process", +abstract = "For Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic Partial Differential Equations (SPDE) and thus contains an underlying Gaussian process which is the key of the algorithm. A series development of the solution in terms of iterated integrals of the Gaussian process is given, it is proved to converge - also in the infinite dimensional limit - and it is numerically tested in a number of examples." +} + +@Techreport{GononSchwab2020, + author = {L. Gonon and Ch. Schwab}, + title = {Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models}, + institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich}, + number = {2020-52}, + address = {Switzerland}, + url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-52.pdf }, + year = {2020} +} + +@misc{SalvadorOosterleeVanDerMeer2020, + title={{Financial option valuation by unsupervised learning with artificial neural networks}}, + author={Beatriz Salvador and Cornelis W. Oosterlee and Remco van der Meer}, + year={2020}, + journal={arXiv:2005.12059}, + primaryClass={q-fin.CP}, + pages={23 pages} +} + +@article{Essis-BrettonGaillardetz2020, + title={{Fast Lower and Upper Estimates for the Price of Constrained Multiple Exercise American Options by Single Pass Lookahead Search and Nearest-Neighbor Martingale}}, + author={Nicolas Essis-Breton and Patrice Gaillardetz}, + year={2020}, + journal={arXiv:2002.11258}, + primaryClass={q-fin.CP}, + pages={64 pages} +} + + +@article{BeckerCheriditoJentzen2020, +title={Pricing and Hedging American-Style Options with Deep Learning}, +volume={13}, +ISSN={1911-8074}, +url={http://dx.doi.org/10.3390/jrfm13070158}, +DOI={10.3390/jrfm13070158}, +number={7}, +journal={Journal of Risk and Financial Management}, +publisher={MDPI AG}, +author={Becker, Sebastian and Cheridito, Patrick and Jentzen, Arnulf}, +year={2020}, +month={Jul}, +pages={158} +} + +@ARTICLE {HutzenthalerJentzenKruse2019, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas and + Nguyen, Tuan Anh}, + TITLE = {A proof that rectified deep neural networks overcome the curse + of dimensionality in the numerical approximation of semilinear + heat equations}, + JOURNAL = {Partial Differ. Equ. Appl.}, + FJOURNAL = {Partial Differential Equations and Applications}, + VOLUME = {1}, + YEAR = {2020}, + NUMBER = {2}, + PAGES = {Paper No. 10}, + ISSN = {2662-2963}, + MRCLASS = {65C99}, + MRNUMBER = {4292849}, + DOI = {10.1007/s42985-019-0006-9}, + URL = {https://doi.org/10.1007/s42985-019-0006-9}, +} +} + +@article {MR3767958, + AUTHOR = {E, Weinan and Yu, Bing}, + TITLE = {The deep {R}itz method: a deep learning-based numerical + algorithm for solving variational problems}, + JOURNAL = {Commun. Math. Stat.}, + FJOURNAL = {Communications in Mathematics and Statistics}, + VOLUME = {6}, + YEAR = {2018}, + NUMBER = {1}, + PAGES = {1--12}, + ISSN = {2194-6701}, + MRCLASS = {35J20 (35J91 65N30)}, + MRNUMBER = {3767958}, + DOI = {10.1007/s40304-018-0127-z}, + URL = {https://doi.org/10.1007/s40304-018-0127-z}, +} + +@book{jones2001lebesgue, + title={Lebesgue integration on {E}uclidean space}, + author={Jones, Frank}, + year={2001}, + publisher={Jones \& Bartlett Learning} +} + +@article{bercher2020weak, + title={Weak error analysis for stochastic gradient descent optimization algorithms}, + author={Bercher, Aritz and Gonon, Lukas and Jentzen, Arnulf and Salimova, Diyora}, + journal={arXiv:2007.02723}, + year={2020}, + pages={123 pages} +} + +@article{grohs2018proof, + title={A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of {B}lack-{S}choles partial differential equations}, + author={Grohs, Philipp and Hornung, Fabian and Jentzen, Arnulf and Von Wurstemberger, Philippe}, + journal={arXiv:1809.02362}, + note={\emph{To appear in Mem.\ Amer.\ Math.\ Soc.}}, + year={2018}, + pages={124 pages}, +} + +@article{cox2016convergence, + title={{Convergence in H{\"o}lder norms with applications to Monte Carlo methods in infinite dimensions}}, + author={Cox, Sonja and Hutzenthaler, Martin and Jentzen, Arnulf and van Neerven, Jan and Welti, Timo}, + journal={IMA Journal of Numerical Analysis} +} + +@article{CheriditoEtAl2020, + title={Non-convergence of stochastic gradient descent in the training of deep neural networks}, + author={Patrick Cheridito and Arnulf Jentzen and Florian Rossmannek}, + journal={arXiv:2006.07075}, + year={2020}, + PAGES = {12 pages}, +} + +@article{beck2020nonlinear, + title={{On nonlinear Feynman--Kac formulas for viscosity solutions of semilinear parabolic partial differential equations}}, + author={Beck, Christian and Hutzenthaler, Martin and Jentzen, Arnulf}, + JOURNAL = {Stoch. Dyn.}, + FJOURNAL = {Stochastics and Dynamics}, + year={2021}, + publisher={World Scientific}, + note={doi:10.1142/S0219493721500489}, +} + + +@article{SankararamanEtAl2019, + title={{The Impact of Neural Network Overparameterization on Gradient Confusion and Stochastic Gradient Descent}}, + author={Karthik A. Sankararaman and Soham De, Zheng Xu and W. Ronny Huang and Tom Goldstein}, + journal={arXiv:1904.06963}, + year={2019}, + PAGES = {37 pages}, +} + +@article{SchaulZhangLeCun12, + author = {Schaul, Tom and Zhang, Sixin and LeCun, Yann}, + title = {No More Pesky Learning Rates}, + journal = {Proceedings of the 30th + International Conference on Machine Learning (ICML)}, + year = {2013}, + volume={28}, + number={3}, + pages = {343--351}, +} + + + +@article{HutzenthalerJentzenKruse201912, + title={{Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities}}, + author={Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse}, + journal={Found.\ Comput.\ Math.}, + year={2021}, + pages={1--62} +} + + +@article{NueskenRichter2020, + title={{Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space}}, + author={Nikolas N{\"u}sken and Lorenz Richter}, + journal={arXiv:2005.05409}, + year={2020}, + pages={40 pages} +} + +@article {BeckerBraunwarthEtAl2020, + AUTHOR = {Becker, Sebastian and Braunwarth, Ramon and Hutzenthaler, + Martin and Jentzen, Arnulf and von Wurstemberger, Philippe}, + TITLE = {Numerical simulations for full history recursive multilevel + {P}icard approximations for systems of high-dimensional + partial differential equations}, + JOURNAL = {Commun. Comput. Phys.}, + FJOURNAL = {Communications in Computational Physics}, + VOLUME = {28}, + YEAR = {2020}, + NUMBER = {5}, + PAGES = {2109--2138}, + ISSN = {1815-2406}, + MRCLASS = {65M75 (35R60 60H15)}, + MRNUMBER = {4188531}, + DOI = {10.4208/cicp.oa-2020-0130}, + URL = {https://doi.org/10.4208/cicp.oa-2020-0130}, +} + +@article{beck2020nested, + title={Nonlinear {M}onte {C}arlo methods with polynomial runtime for high-dimensional iterated nested expectations}, + author={Beck, Christian and Jentzen, Arnulf and Kruse, Thomas}, + journal={arXiv:2009.13989}, + year={2020} +} + +@article{BeckGononJentzen2020, + title={{Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations}}, + author={Christian Beck and Lukas Gonon and Arnulf Jentzen}, + journal={arXiv:2003.00596}, + year={2020}, + pages={50 pages}, +} + + +@article{JentzenMazzonettoSalimova2018, + title={Existence and uniqueness properties for solutions of a class of {B}anach space valued evolution equations}, + author={Jentzen, Arnulf and Mazzonetto, Sara and Salimova, Diyora}, + journal={arXiv:1812.06859}, + year={2018}, + pages={28 pages} +} + +@article{MazzonettoSalimova2020, + title={{Existence, uniqueness, and numerical approximations for stochastic Burgers equations}}, + author={Sara Mazzonetto and Diyora Salimova}, + journal={Stoch. 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Stat.}, + FJOURNAL = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et + Statistiques}, + VOLUME = {55}, + YEAR = {2019}, + NUMBER = {1}, + PAGES = {184--210}, + ISSN = {0246-0203}, + MRCLASS = {60J85 (35K58 35R25 60H07 65C05)}, + MRNUMBER = {3901645}, + DOI = {10.1214/17-aihp880}, + URL = {https://doi.org/10.1214/17-aihp880}, +} + + +@article{GilesJentzenWelti2019, + title={{Generalised multilevel Picard approximations}}, + author={Michael B. Giles and Arnulf Jentzen and Timo Welti}, + journal={arXiv:1911.03188}, + year={2019}, + pages={61 pages}, +} + +@article {DereichMuellerGronbach2019, + AUTHOR = {Dereich, Steffen and M\"{u}ller-Gronbach, Thomas}, + TITLE = {General multilevel adaptations for stochastic approximation + algorithms of {R}obbins-{M}onro and {P}olyak-{R}uppert type}, + JOURNAL = {Numer. 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Oden}, + DOI = {10.1007/978-1-4757-4338-8}, + URL = {https://doi.org/10.1007/978-1-4757-4338-8}, +} + + +@book {ChorinMarsden1993, + AUTHOR = {Chorin, Alexandre J. and Marsden, Jerrold E.}, + TITLE = {A mathematical introduction to fluid mechanics}, + SERIES = {Texts in Applied Mathematics}, + VOLUME = {4}, + EDITION = {Third}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1993}, + PAGES = {xii+169}, + ISBN = {0-387-97918-2}, + MRCLASS = {76-02 (35Qxx)}, + MRNUMBER = {1218879}, + DOI = {10.1007/978-1-4612-0883-9}, + URL = {https://doi.org/10.1007/978-1-4612-0883-9}, +} + + +@article {BeckHornungEtAl2019, + AUTHOR = {Beck, Christian and Hornung, Fabian and Hutzenthaler, Martin + and Jentzen, Arnulf and Kruse, Thomas}, + TITLE = {Overcoming the curse of dimensionality in the numerical + approximation of {A}llen-{C}ahn partial differential equations + via truncated full-history recursive multilevel {P}icard + approximations}, + JOURNAL = {J. Numer. Math.}, + FJOURNAL = {Journal of Numerical Mathematics}, + VOLUME = {28}, + YEAR = {2020}, + NUMBER = {4}, + PAGES = {197--222}, + ISSN = {1570-2820}, + MRCLASS = {60H30 (65C05 65M75)}, + MRNUMBER = {4191733}, + DOI = {10.1515/jnma-2019-0074}, + URL = {https://doi.org/10.1515/jnma-2019-0074}, +} + + +@article {HutzenthalerPricing2019, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and von + Wurstemberger, Philippe}, + TITLE = {Overcoming the curse of dimensionality in the approximative + pricing of financial derivatives with default risks}, + JOURNAL = {Electron. J. Probab.}, + FJOURNAL = {Electronic Journal of Probability}, + VOLUME = {25}, + YEAR = {2020}, + PAGES = {Paper No. 101, 73}, + MRCLASS = {60H35 (35K58 60H10 60H30 91G20)}, + MRNUMBER = {4144883}, + DOI = {10.1007/s13253-019-00378-y}, + URL = {https://doi.org/10.1007/s13253-019-00378-y}, +} + + +@article{Wang2018, + title={{An efficient explicit full discrete scheme for strong approximation of stochastic Allen-Cahn equation}}, + author={Wang, Xiaojie}, + journal={arXiv:1802.09413}, + year={2018}, + pages={25 pages} +} + + +@article{ChenGanWang2018arxiv, + title={{Mean-square approximations of L{\'e}vy noise driven SDEs with super-linearly growing diffusion and jump coefficients}}, + author= {Chen, Ziheng and Gan, Siqing and Wang, Xiaojie}, + journal={arXiv:1812.03069}, + year={2018}, + pages={19 pages} +} + + +@article {LiYang2018, + AUTHOR = {Li, Xiaocui and Yang, Xiaoyuan}, + TITLE = {Error estimates of finite element methods for fractional + stochastic {N}avier-{S}tokes equations}, + JOURNAL = {J. Inequal. Appl.}, + FJOURNAL = {Journal of Inequalities and Applications}, + YEAR = {2018}, + PAGES = {Paper No. 284, 15}, + ISSN = {1029-242X}, + MRCLASS = {65M60 (26A33 35Q30 35R60 60H15 65M15 65M75 76D06)}, + MRNUMBER = {3866691}, + DOI = {10.1186/s13660-018-1880-y}, + URL = {https://doi.org/10.1186/s13660-018-1880-y}, +} + + +@article {DuanYang2013, + AUTHOR = {Duan, Yuanyuan and Yang, Xiaoyuan}, + TITLE = {The finite element method of a {E}uler scheme for stochastic + {N}avier-{S}tokes equations involving the turbulent component}, + JOURNAL = {Int. J. Numer. Anal. Model.}, + FJOURNAL = {International Journal of Numerical Analysis and Modeling}, + VOLUME = {10}, + YEAR = {2013}, + NUMBER = {3}, + PAGES = {727--744}, + ISSN = {1705-5105}, + MRCLASS = {76D06 (35Q35 35R60 76F55 76M10)}, + MRNUMBER = {3064276}, +MRREVIEWER = {Azzouz Dermoune}, +} + + + +@article {KossiorisZouraris2013, + AUTHOR = {Kossioris, Georgios T. and Zouraris, Georgios E.}, + TITLE = {Finite element approximations for a linear + {C}ahn-{H}illiard-{C}ook equation driven by the space + derivative of a space-time white noise}, + JOURNAL = {Discrete Contin. Dyn. Syst. Ser. B}, + FJOURNAL = {Discrete and Continuous Dynamical Systems. Series B. A Journal + Bridging Mathematics and Sciences}, + VOLUME = {18}, + YEAR = {2013}, + NUMBER = {7}, + PAGES = {1845--1872}, + ISSN = {1531-3492}, + MRCLASS = {65M60 (65C20 65M15 65M75)}, + MRNUMBER = {3066325}, +MRREVIEWER = {B\"{u}lent Karas\"{o}zen}, + DOI = {10.3934/dcdsb.2013.18.1845}, + URL = {https://doi.org/10.3934/dcdsb.2013.18.1845}, +} + + + +@article {Gazeau2014, + AUTHOR = {Gazeau, Maxime}, + TITLE = {Probability and pathwise order of convergence of a + semidiscrete scheme for the stochastic {M}anakov equation}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {52}, + YEAR = {2014}, + NUMBER = {1}, + PAGES = {533--553}, + ISSN = {0036-1429}, + MRCLASS = {65N06 (35Q55 35R60 60H15 65N75)}, + MRNUMBER = {3166967}, +MRREVIEWER = {G. B. Loghmani}, + DOI = {10.1137/13090924X}, + URL = {https://doi.org/10.1137/13090924X}, +} + + +@article {GlattTemamWang2017, + AUTHOR = {Glatt-Holtz, Nathan and Temam, Roger and Wang, Chuntian}, + TITLE = {Time discrete approximation of weak solutions to stochastic + equations of geophysical fluid dynamics and applications}, + JOURNAL = {Chin. Ann. Math. Ser. B}, + FJOURNAL = {Chinese Annals of Mathematics. Series B}, + VOLUME = {38}, + YEAR = {2017}, + NUMBER = {2}, + PAGES = {425--472}, + ISSN = {0252-9599}, + MRCLASS = {35Q86 (35D30 35Q35 35R60 60H15)}, + MRNUMBER = {3615498}, + DOI = {10.1007/s11401-017-1077-6}, + URL = {https://doi.org/10.1007/s11401-017-1077-6}, +} + +@article {KamraniEtAl2018, + AUTHOR = {Kamrani, Minoo and Hosseini, S. Mohammad and Hausenblas, + Erika}, + TITLE = {Implicit {E}uler method for numerical solution of nonlinear + stochastic partial differential equations with multiplicative + trace class noise}, + JOURNAL = {Math. Methods Appl. Sci.}, + FJOURNAL = {Mathematical Methods in the Applied Sciences}, + VOLUME = {41}, + YEAR = {2018}, + NUMBER = {13}, + PAGES = {4986--5002}, + ISSN = {0170-4214}, + MRCLASS = {65M75 (35R60 60H15)}, + MRNUMBER = {3843574}, + DOI = {10.1002/mma.4946}, + URL = {https://doi.org/10.1002/mma.4946}, +} + +@article {BessaihEtAl2018, + AUTHOR = {Bessaih, Hakima and Hausenblas, Erika and Randrianasolo, Tsiry + Avisoa and Razafimandimby, Paul Andr\'{e}}, + TITLE = {Numerical approximation of stochastic evolution equations: + convergence in scale of {H}ilbert spaces}, + JOURNAL = {J. Comput. Appl. Math.}, + FJOURNAL = {Journal of Computational and Applied Mathematics}, + VOLUME = {343}, + YEAR = {2018}, + PAGES = {250--274}, + ISSN = {0377-0427}, + MRCLASS = {60H15 (35R60 65M60 65M75)}, + MRNUMBER = {3813549}, + DOI = {10.1016/j.cam.2018.04.067}, + URL = {https://doi.org/10.1016/j.cam.2018.04.067}, +} + +@article {CarelliProhl2012, + AUTHOR = {Carelli, Erich and Prohl, Andreas}, + TITLE = {Rates of convergence for discretizations of the stochastic + incompressible {N}avier-{S}tokes equations}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {50}, + YEAR = {2012}, + NUMBER = {5}, + PAGES = {2467--2496}, + ISSN = {0036-1429}, + MRCLASS = {65M75 (60H15 65M15 76D06)}, + MRNUMBER = {3022227}, +MRREVIEWER = {B\"{u}lent Karas\"{o}zen}, + DOI = {10.1137/110845008}, + URL = {https://doi.org/10.1137/110845008}, +} + +@article{BessaihMillet2018, + title={On strong {$L^2$} convergence of time numerical schemes for the stochastic 2{D} {N}avier-{S}tokes equations}, + author={Bessaih, Hakima and Millet, Annie}, + journal={arXiv:1801.03548}, + year={2018}, + pages={28 pages} +} + +@article{LiuQiao2018, + author = {Liu, Zhihui and Qiao, Zhonghua}, + title = {Strong Approximation of Stochastic {A}llen-{C}ahn Equation with White Noise}, +journal = {arXiv:1801.09348}, + year = {2018}, + pages = {17 pages} +} + + + + + + +@article {Kamrani2016, + AUTHOR = {Kamrani, Minoo}, + TITLE = {Numerical solution of stochastic partial differential + equations using a collocation method}, + JOURNAL = {ZAMM Z. Angew. Math. Mech.}, + FJOURNAL = {ZAMM. Zeitschrift f\"{u}r Angewandte Mathematik und Mechanik. + Journal of Applied Mathematics and Mechanics}, + VOLUME = {96}, + YEAR = {2016}, + NUMBER = {1}, + PAGES = {106--120}, + ISSN = {0044-2267}, + MRCLASS = {60H15 (35R60 65M70 65M75)}, + MRNUMBER = {3454317}, +MRREVIEWER = {Peter E. Kloeden}, + DOI = {10.1002/zamm.201400080}, + URL = {https://doi.org/10.1002/zamm.201400080}, +} + +@article {KamraniHosseini2012, + AUTHOR = {Kamrani, Minoo and Hosseini, S. Mohammad}, + TITLE = {Spectral collocation method for stochastic {B}urgers equation + driven by additive noise}, + JOURNAL = {Math. Comput. Simulation}, + FJOURNAL = {Mathematics and Computers in Simulation}, + VOLUME = {82}, + YEAR = {2012}, + NUMBER = {9}, + PAGES = {1630--1644}, + ISSN = {0378-4754}, + MRCLASS = {65M70 (60H15 65M75)}, + MRNUMBER = {2927720}, + DOI = {10.1016/j.matcom.2012.03.007}, + URL = {https://doi.org/10.1016/j.matcom.2012.03.007}, +} + + + +@book {ZhangKarniadakis2017, + AUTHOR = {Zhang, Zhongqiang and Karniadakis, George Em}, + TITLE = {Numerical methods for stochastic partial differential + equations with white noise}, + SERIES = {Applied Mathematical Sciences}, + VOLUME = {196}, + PUBLISHER = {Springer, Cham}, + YEAR = {2017}, + PAGES = {xv+394}, + ISBN = {978-3-319-57510-0; 978-3-319-57511-7}, + MRCLASS = {60-02 (60H10 60H15 60H35 65-02 65M75 65N75)}, + MRNUMBER = {3700670}, +MRREVIEWER = {Jos\'{e} Eduardo Souza de Cursi}, + DOI = {10.1007/978-3-319-57511-7}, + URL = {https://doi.org/10.1007/978-3-319-57511-7}, +} + + +@article{CampbellLord2018, + title={{Adaptive time-stepping for Stochastic Partial Differential Equations with non-Lipschitz drift}}, + author={Campbell, Stuart and Lord, Gabriel}, + journal={arXiv:1812.09036}, + year={2018}, + pages={25 pages} +} + + + +@article{ChenGanWang2018arxiv, + title={{Mean-square approximations of L{\'e}vy noise driven SDEs with super-linearly growing diffusion and jump coefficients}}, + author= {Chen, Ziheng and Gan, Siqing and Wang, Xiaojie}, + journal={arXiv:1812.03069}, + year={2018}, + pages={19 pages} +} + +@article {GHB2017, + AUTHOR = {Ghayebi, Bakhtiyar and Hosseini, S. Mohammad and Bl\"omker, + Dirk}, + TITLE = {Numerical solution of the {B}urgers equation with {N}eumann + boundary noise}, + JOURNAL = {J. Comput. Appl. Math.}, + FJOURNAL = {Journal of Computational and Applied Mathematics}, + VOLUME = {311}, + YEAR = {2017}, + PAGES = {148--164}, + ISSN = {0377-0427}, + MRCLASS = {65M60 (60H10 60H15 60H35 65M75)}, + MRNUMBER = {3552692}, + DOI = {10.1016/j.cam.2016.07.005}, + URL = {http://dx.doi.org/10.1016/j.cam.2016.07.005}, +} + + + +@article{BeckerEtAl2018, + title={Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations}, + author={Becker, Sebastian and Gess, Benjamin and Jentzen, Arnulf and Kloeden, Peter E}, + journal={arXiv:1811.01725}, + year={2018}, + pages={20 pages} +} + + +@article {Breckner2000, + AUTHOR = {Breckner, Hannelore}, + TITLE = {Galerkin approximation and the strong solution of the + {N}avier-{S}tokes equation}, + JOURNAL = {J. Appl. Math. Stochastic Anal.}, + FJOURNAL = {Journal of Applied Mathematics and Stochastic Analysis}, + VOLUME = {13}, + YEAR = {2000}, + NUMBER = {3}, + PAGES = {239--259}, + ISSN = {1048-9533}, + MRCLASS = {60H15}, + MRNUMBER = {1782683}, +MRREVIEWER = {Constantin Tudor}, + DOI = {10.1155/S1048953300000228}, + URL = {https://doi.org/10.1155/S1048953300000228}, +} + + +@article{Mazzonetto2018, + title={Strong convergence for explicit space-time discrete numerical approximation for 2{D} stochastic {N}avier-{S}tokes equations}, + author={Mazzonetto, Sara}, + journal={arXiv:1809.01937}, + year={2018}, + pages={35 pages} +} + + + +@article{BlomkerKamrani2017, + author = {Bl\"omker, Dirk and Kamrani, Minoo}, + title = {{Numerically Computable A Posteriori-Bounds for stochastic Allen-Cahn equation}}, +journal = {arXiv:1702.01347}, + year = {2017}, + pages = {27 pages} +} + + +@article{BrehierGoudenege2018, + author = {Br{\'e}hier, Charles-Edouard and Gouden{\'e}ge, Ludovic}, + title = {Analysis of some splitting schemes for the stochastic {A}llen-{C}ahn equation}, +journal = {arXiv:1801.06455}, + year = {2018}, + pages = {23 pages} +} + +@article{BrehierCuiHong2018, + author = {Br{\'e}hier, Charles-Edouard and Cui, Jianbo and Hong, Jialin}, + title = {Strong convergence rates of semidiscrete splitting approximations for the stochastic {A}llen-{C}ahn equation}, +journal = {arXiv:1802.06372}, + year = {2018}, + pages = {33 pages} +} + +@article{LiLuo2003, +title = "Numerical solution of elliptic partial differential equation using radial basis function neural networks", +journal = "Neural Networks", +volume = "16", +number = "5", +pages = "729 - 734", +year = "2003", +issn = "0893-6080", +doi = "https://doi.org/10.1016/S0893-6080(03)00083-2", +url = "http://www.sciencedirect.com/science/article/pii/S0893608003000832", +author = "Li Jianyu and Luo Siwei and Qi Yingjian and Huang Yaping", +keywords = "Radial basis function neural networks, Partial differential equation, Multi-quadrics" +} + + +@article{BeccariHutzenthaler2019, + author = {Matteo Beccari and Martin Hutzenthaler and Arnulf Jentzen and Ryan Kurniawan and Felix Lindner and Diyora Salimova}, + title = {Strong and weak divergence of exponential and linear-implicit {E}uler approximations for stochastic partial differential equations with superlinearly growing nonlinearities}, + journal = {arXiv:1903.06066}, + year = {2019}, + pages = {65 pages} +} + + + + +@article{GrohsHornungJentzen2019, + author = {Philipp Grohs and Fabian Hornung and Arnulf Jentzen and Philipp Zimmermann}, + title = {Space-time error estimates for deep neural network approximations for differential equations}, + journal = {arXiv:1908.03833}, + year = {2019}, + pages = {86 pages} +} + + + +@article{ElbraechterSchwab2018, + author = {Elbr{\"a}chter, Dennis and Grohs, Philipp and Jentzen, Arnulf and Schwab, Christoph}, + title = {{DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing}}, + journal = {arXiv:1809.07669}, + year = {2018}, + pages = {50 pages} +} + + + + + +@article {EWang2018, + AUTHOR = {E, Weinan and Wang, Qingcan}, + TITLE = {Exponential convergence of the deep neural network + approximation for analytic functions}, + JOURNAL = {Sci. China Math.}, + FJOURNAL = {Science China. Mathematics}, + VOLUME = {61}, + YEAR = {2018}, + NUMBER = {10}, + PAGES = {1733--1740}, + ISSN = {1674-7283}, + MRCLASS = {65D15}, + MRNUMBER = {3856963}, + DOI = {10.1007/s11425-018-9387-x}, + URL = {https://doi.org/10.1007/s11425-018-9387-x}, +} + + + + + + +@article {MontanelliDu2017, + AUTHOR = {Montanelli, Hadrien and Du, Qiang}, + TITLE = {New error bounds for deep {R}e{LU} networks using sparse + grids}, + JOURNAL = {SIAM J. Math. Data Sci.}, + FJOURNAL = {SIAM Journal on Mathematics of Data Science}, + VOLUME = {1}, + YEAR = {2019}, + NUMBER = {1}, + PAGES = {78--92}, + ISSN = {2577-0187}, + MRCLASS = {41A63 (68T05 82C32)}, + MRNUMBER = {3949701}, + DOI = {10.1137/18M1189336}, + URL = {https://doi.org/10.1137/18M1189336}, +} + + + +@INPROCEEDINGS{Taigman2014, +author={Y. Taigman and M. Yang and M. Ranzato and L. Wolf}, +booktitle={IEEE Conference on Computer Vision and Pattern Recognition}, +title={DeepFace: Closing the Gap to Human-Level Performance in Face Verification}, +year={2014}, +volume={}, +number={}, +pages={1701-1708}, +keywords={face recognition;image representation;neural nets;DeepFace;human-level performance;face verification;face recognition;alignment step;representation step;3D face modeling;piecewise affine transformation;face representation;deep neural network;labeled faces in the wild;LFW dataset;Face;Three-dimensional displays;Training;Face recognition;Agriculture;Solid modeling;Shape}, +doi={10.1109/CVPR.2014.220}, +ISSN={1063-6919},} + +@inproceedings{HuConvolutional2014, + author = {Hu, Baotian and Lu, Zhengdong and Li, Hang and Chen, Qingcai}, + title = {Convolutional Neural Network Architectures for Matching Natural Language Sentences}, + booktitle = {Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2}, + year = {2014}, + pages = {2042--2050}, +} + + + + +@INPROCEEDINGS{Kalchbrenner14aconvolutional, + author = {Nal Kalchbrenner and Edward Grefenstette and Phil Blunsom}, + title = {A convolutional neural network for modelling sentences}, + booktitle = {Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics}, + year = {2014}, +pages={655--665} +} + + +@InProceedings{wangfacerecognition2015, +author="Wang, Weihong +and Yang, Jie +and Xiao, Jianwei +and Li, Sheng +and Zhou, Dixin", +title="Face Recognition Based on Deep Learning", +booktitle="Human Centered Computing", +year="2015", +pages="812--820", +abstract="As one of the non-contact biometrics, face representation had been widely used in many circumstances. However conventional methods could no longer satisfy the demand at present, due to its low recognition accuracy and restrictions of many occasions. In this paper, we presented the deep learning method to achieve facial landmark detection and unrestricted face recognition. To solve the face landmark detection problem, this paper proposed a layer-by-layer training method of a deep convolutional neural network to help the convolutional neural network to converge and proposed a sample transformation method to avoid over-fitting. This method had reached an accuracy of 91{\%} on ORL face database. To solve the face recognition problem, this paper proposed a SIAMESE convolutional neural network which was trained on different parts and scales of a face and concatenated the face representation. The face recognition algorithm had reached an accuracy of 91{\%} on ORL and 81{\%} on LFW face database.", +isbn="978-3-319-15554-8" +} + + + + + +@article {SchwabZech2018, + AUTHOR = {Schwab, Christoph and Zech, Jakob}, + TITLE = {Deep learning in high dimension: neural network expression + rates for generalized polynomial chaos expansions in {UQ}}, + JOURNAL = {Anal. Appl. (Singap.)}, + FJOURNAL = {Analysis and Applications}, + VOLUME = {17}, + YEAR = {2019}, + NUMBER = {1}, + PAGES = {19--55}, + ISSN = {0219-5305}, + MRCLASS = {68Q32 (35R60 41A25 41A46)}, + MRNUMBER = {3894732}, + DOI = {10.1142/S0219530518500203}, + URL = {https://doi.org/10.1142/S0219530518500203}, +} + + + + +@book {DaPratoZabczyk92, + AUTHOR = {Da Prato, Giuseppe and Zabczyk, Jerzy}, + TITLE = {Stochastic equations in infinite dimensions}, + SERIES = {Encyclopedia of Mathematics and its Applications}, + VOLUME = {44}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {1992}, + PAGES = {xviii+454}, + ISBN = {0-521-38529-6}, + MRCLASS = {60H15 (34F05 34Gxx 35R15 35R60 46G12 47N20)}, + MRNUMBER = {1207136}, + MRREVIEWER = {P. 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Comput. Harmon. Anal.}, + FJOURNAL = {Applied and Computational Harmonic Analysis. Time-Frequency + and Time-Scale Analysis, Wavelets, Numerical Algorithms, and + Applications}, + VOLUME = {44}, + YEAR = {2018}, + NUMBER = {3}, + PAGES = {537--557}, + ISSN = {1063-5203}, + MRCLASS = {94C15 (94A20)}, + MRNUMBER = {3768851}, +MRREVIEWER = {Michael M. Dediu}, + DOI = {10.1016/j.acha.2016.04.003}, + URL = {https://doi.org/10.1016/j.acha.2016.04.003}, +} + + +@article{Cybenko1989, + Abstract = {In this paper we demonstrate that finite linear combinations of compositions of a fixed, univariate function and a set of affine functionals can uniformly approximate any continuous function ofn real variables with support in the unit hypercube; only mild conditions are imposed on the univariate function. Our results settle an open question about representability in the class of single hidden layer neural networks. 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Amer. Math. Soc., arXiv:1809.02362}, + Year = {2018}, + Pages = {124 pages} +} + + +@article {DarbonOsher2016, + AUTHOR = {Darbon, J\'{e}r\^{o}me and Osher, Stanley}, + TITLE = {Algorithms for overcoming the curse of dimensionality for + certain {H}amilton-{J}acobi equations arising in control + theory and elsewhere}, + JOURNAL = {Res. Math. Sci.}, + FJOURNAL = {Research in the Mathematical Sciences}, + VOLUME = {3}, + YEAR = {2016}, + PAGES = {Paper No. 19, 26}, + ISSN = {2522-0144}, + MRCLASS = {65M06 (35F21 35F25 49L20 65K10)}, + MRNUMBER = {3543239}, +MRREVIEWER = {Tore Fl\aa tten}, + DOI = {10.1186/s40687-016-0068-7}, + URL = {https://doi.org/10.1186/s40687-016-0068-7}, +} + +@article {chang2016branching, + AUTHOR = {Chang, Dejian and Liu, Huili and Xiong, Jie}, + TITLE = {A branching particle system approximation for a class of + {FBSDE}s}, + JOURNAL = {Probab. Uncertain. Quant. Risk}, + FJOURNAL = {Probability, Uncertainty and Quantitative Risk}, + VOLUME = {1}, + YEAR = {2016}, + PAGES = {Paper No. 9, 34}, + MRCLASS = {60H35 (60H15 60J80)}, + MRNUMBER = {3583180}, +MRREVIEWER = {Jos\'{e} Villa-Morales}, + DOI = {10.1186/s41546-016-0007-y}, + URL = {https://doi.org/10.1186/s41546-016-0007-y}, +} + +@article {le2017particle, + AUTHOR = {Le Cavil, Anthony and Oudjane, Nadia and Russo, Francesco}, + TITLE = {Particle system algorithm and chaos propagation related to + non-conservative {M}c{K}ean type stochastic differential + equations}, + JOURNAL = {Stoch. Partial Differ. Equ. Anal. Comput.}, + FJOURNAL = {Stochastic Partial Differential Equations. Analysis and + Computations}, + VOLUME = {5}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {1--37}, + ISSN = {2194-0401}, + MRCLASS = {65C30 (35K59 58J35 60H10 60H30 60J60 65C35 68U20)}, + MRNUMBER = {3607725}, +MRREVIEWER = {Gabriela Mircea}, + DOI = {10.1007/s40072-016-0079-9}, + URL = {https://doi.org/10.1007/s40072-016-0079-9}, +} + +@article {le2018monte, + AUTHOR = {Le Cavil, Anthony and Oudjane, Nadia and Russo, Francesco}, + TITLE = {Monte-{C}arlo algorithms for a forward {F}eynman--{K}ac-type + representation for semilinear nonconservative partial + differential equations}, + JOURNAL = {Monte Carlo Methods Appl.}, + FJOURNAL = {Monte Carlo Methods and Applications}, + VOLUME = {24}, + YEAR = {2018}, + NUMBER = {1}, + PAGES = {55--70}, + ISSN = {0929-9629}, + MRCLASS = {60H10 (35K55 60H30 60J60 65C05 65C35 68U20)}, + MRNUMBER = {3770031}, + DOI = {10.1515/mcma-2018-0005}, + URL = {https://doi.org/10.1515/mcma-2018-0005}, +} + +@article {LeCavilOudjaneRusso2019, + AUTHOR = {Le Cavil, Anthony and Oudjane, Nadia and Russo, Francesco}, + TITLE = {Forward {F}eynman-{K}ac type representation for semilinear + non-conservative partial differential equations}, + JOURNAL = {Stochastics}, + FJOURNAL = {Stochastics. An International Journal of Probability and + Stochastic Processes}, + VOLUME = {91}, + YEAR = {2019}, + NUMBER = {8}, + PAGES = {1206--1248}, + ISSN = {1744-2508}, + MRCLASS = {60H10 (35K55 60H30 60J60 65C05 65C35)}, + MRNUMBER = {4023515}, + DOI = {10.1080/17442508.2019.1594809}, + URL = {https://doi.org/10.1080/17442508.2019.1594809}, +} + +@article {BenderDenk2007, + AUTHOR = {Bender, Christian and Denk, Robert}, + TITLE = {A forward scheme for backward {SDE}s}, + JOURNAL = {Stochastic Process. Appl.}, + FJOURNAL = {Stochastic Processes and their Applications}, + VOLUME = {117}, + YEAR = {2007}, + NUMBER = {12}, + PAGES = {1793--1812}, + ISSN = {0304-4149}, + MRCLASS = {65C30 (60H10 60H35 91A28)}, + MRNUMBER = {2437729}, +MRREVIEWER = {Edward J. Allen}, + DOI = {10.1016/j.spa.2007.03.005}, + URL = {https://doi.org/10.1016/j.spa.2007.03.005}, +} + +@article {bender2008time, + AUTHOR = {Bender, Christian and Zhang, Jianfeng}, + TITLE = {Time discretization and {M}arkovian iteration for coupled + {FBSDE}s}, + JOURNAL = {Ann. Appl. Probab.}, + FJOURNAL = {The Annals of Applied Probability}, + VOLUME = {18}, + YEAR = {2008}, + NUMBER = {1}, + PAGES = {143--177}, + ISSN = {1050-5164}, + MRCLASS = {60H10 (60H35 65C30)}, + MRNUMBER = {2380895}, +MRREVIEWER = {Rainer Buckdahn}, + DOI = {10.1214/07-AAP448}, + URL = {https://doi.org/10.1214/07-AAP448}, +} + +@article {GobetLabart2010, + AUTHOR = {Gobet, Emmanuel and Labart, C\'{e}line}, + TITLE = {Solving {BSDE} with adaptive control variate}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {48}, + YEAR = {2010}, + NUMBER = {1}, + PAGES = {257--277}, + ISSN = {0036-1429}, + MRCLASS = {65C30 (60H10 60H35)}, + MRNUMBER = {2608369}, +MRREVIEWER = {Peter E. Kloeden}, + DOI = {10.1137/090755060}, + URL = {https://doi.org/10.1137/090755060}, +} + +@article {labart2013parallel, + AUTHOR = {Labart, C\'{e}line and Lelong, J\'{e}r\^{o}me}, + TITLE = {A parallel algorithm for solving {BSDE}s}, + JOURNAL = {Monte Carlo Methods Appl.}, + FJOURNAL = {Monte Carlo Methods and Applications}, + VOLUME = {19}, + YEAR = {2013}, + NUMBER = {1}, + PAGES = {11--39}, + ISSN = {0929-9629}, + MRCLASS = {65C30 (60H10 60H35 65Y05 91G20 91G60)}, + MRNUMBER = {3039401}, +MRREVIEWER = {Edward J. Allen}, + DOI = {10.1515/mcma-2013-0001}, + URL = {https://doi.org/10.1515/mcma-2013-0001}, +} + + + +@article {henry2019branching, + AUTHOR = {Henry-Labord\`ere, Pierre and Oudjane, Nadia and Tan, Xiaolu and + Touzi, Nizar and Warin, Xavier}, + TITLE = {Branching diffusion representation of semilinear {PDE}s and + {M}onte {C}arlo approximation}, + JOURNAL = {Ann. Inst. Henri Poincar\'{e} Probab. Stat.}, + FJOURNAL = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et + Statistiques}, + VOLUME = {55}, + YEAR = {2019}, + NUMBER = {1}, + PAGES = {184--210}, + ISSN = {0246-0203}, + MRCLASS = {60J85 (35K58 35R25 60H07 65C05)}, + MRNUMBER = {3901645}, + DOI = {10.1214/17-aihp880}, + URL = {https://doi.org/10.1214/17-aihp880}, +} + +@article {warin2018nesting, + AUTHOR = {Warin, Xavier}, + TITLE = {Nesting {M}onte {C}arlo for high-dimensional non-linear + {PDE}s}, + JOURNAL = {Monte Carlo Methods Appl.}, + FJOURNAL = {Monte Carlo Methods and Applications}, + VOLUME = {24}, + YEAR = {2018}, + NUMBER = {4}, + PAGES = {225--247}, + ISSN = {0929-9629}, + MRCLASS = {65C05 (35G20 49L25)}, + MRNUMBER = {3891748}, +MRREVIEWER = {Charles-Edouard Br\'{e}hier}, + DOI = {10.1515/mcma-2018-2020}, + URL = {https://doi.org/10.1515/mcma-2018-2020}, +} + +@article{warin2018monte, + title={Monte {C}arlo for high-dimensional degenerated semi linear and full non linear {PDE}s}, + author={Warin, Xavier}, + journal={arXiv:1805.05078}, + year={2018}, + pages={23 pages}, +} + +@article {skorokhod1964branching, + AUTHOR = {Skorohod, A. V.}, + TITLE = {Branching diffusion processes}, + JOURNAL = {Teor. Verojatnost. i Primenen.}, + FJOURNAL = {Akademija Nauk SSSR. Teorija Verojatnoste\u{\i} i ee Primenenija}, + VOLUME = {9}, + YEAR = {1964}, + PAGES = {492--497}, + ISSN = {0040-361x}, + MRCLASS = {60.67}, + MRNUMBER = {0168030}, +MRREVIEWER = {K. Dietz}, +} + +@article {watanabe1965branching, + AUTHOR = {Watanabe, Shinzo}, + TITLE = {On the branching process for {B}rownian particles with an + absorbing boundary}, + JOURNAL = {J. Math. Kyoto Univ.}, + FJOURNAL = {Journal of Mathematics of Kyoto University}, + VOLUME = {4}, + YEAR = {1965}, + PAGES = {385--398}, + ISSN = {0023-608X}, + MRCLASS = {60.67 (60.62)}, + MRNUMBER = {178505}, +MRREVIEWER = {P. E. Ney}, + DOI = {10.1215/kjm/1250524667}, + URL = {https://doi.org/10.1215/kjm/1250524667}, +} + +@Article{HutzenthalerJentzen2014, + Title = {{On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients}}, + Author = {{Hutzenthaler}, M. and {Jentzen}, A.}, + Journal = {arXiv:1401.0295}, + Year = {2014}, + Pages = {41 pages}, + Note = {To appear in \textit{The Annals of Probability}}, +} + + + + + +@book {MR3729416, + AUTHOR = {Einsiedler, Manfred and Ward, Thomas}, + TITLE = {Functional analysis, spectral theory, and applications}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {276}, + PUBLISHER = {Springer, Cham}, + YEAR = {2017}, + PAGES = {xiv+614}, + ISBN = {978-3-319-58539-0; 978-3-319-58540-6}, + MRCLASS = {46-01 (35J25 35P10 35P20 37A99 47-01 47A60)}, + MRNUMBER = {3729416}, +} + +@Book{h81, + Title = {{Geometric Theory of Semilinear Parabolic Equations}}, + Author = {Daniel Henry}, + Publisher = {Springer-Verlag}, + Year = {1981}, + Note = {348 pages}, + Address = {Berlin}, + Series = {Lecture Notes in Mathematics}, + Volume = {840}, + + ISBN = {3-540-10557-3}, + Mrclass = {35K55 (34G20 58D25)}, + Mrnumber = {MR610244 (83j:35084)}, + Mrreviewer = {J. A. Goldstein}, + Pages = {iv+348} +} + + +@Book{PrevotRoeckner2007, + AUTHOR = {Claudia Pr{\'e}v{\^o}t and Michael R{\"o}ckner}, + TITLE = "A Concise Course on Stochastic Partial Differential Equations", + SERIES = "Lecture Notes in Mathematics", + VOLUME = "1905", + PUBLISHER = "Springer", + ADDRESS = "Berlin", + YEAR = "2007", + PAGES = "vi+144", + ISBN = "978-3-540-70780-6; 3-540-70780-8", + MRCLASS = "60H15 (60-01)", + MRNUMBER = "MR2329435 (2009a:60069)", + MRREVIEWER = "Sandra Cerrai" +} + +@Book{Kuehn2004, + Title = {Stochastische Analysis mit Finanzmathematik}, + Author = {Christoph K{\"u}hn}, + Year = {2004}, + + Journal = {Lecture notes}, + Pages = {86}, + Url = {http://www.math.uni-frankfurt.de/~ismi/kuehn/stoch_ana.pdf} +} + +@Book{Kallenberg, + Title = {Foundations of modern probability}, + Author = {Kallenberg, Olav}, + Publisher = {Springer-Verlag}, + Year = {2002}, + + Address = {New York}, + Edition = {Second}, + Series = {Probability and its Applications (New York)}, + + ISBN = {0-387-95313-2}, + Mrclass = {60-01}, + Mrnumber = {1876169 (2002m:60002)}, + Mrreviewer = {Klaus D. Schmidt}, + Pages = {xx+638} +} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%From viscosity solutions + +@Article{CrandallLions1983, + Title = {Viscosity solutions of {H}amilton-{J}acobi equations}, + Author = {Crandall, Michael G. and Lions, Pierre-Louis}, + Journal = {Trans. Amer. Math. Soc.}, + Year = {1983}, + Number = {1}, + Pages = {1--42}, + Volume = {277}, + + Coden = {TAMTAM}, + Doi = {10.2307/1999343}, + Fjournal = {Transactions of the American Mathematical Society}, + ISSN = {0002-9947}, + Mrclass = {35F20}, + Mrnumber = {690039 (85g:35029)}, + Mrreviewer = {Moshe Marcus}, + Url = {http://dx.doi.org/10.2307/1999343} +} + + + + +@book {MR2604525, + AUTHOR = {M\"orters, Peter and Peres, Yuval}, + TITLE = {Brownian motion}, + SERIES = {Cambridge Series in Statistical and Probabilistic Mathematics}, + VOLUME = {30}, + NOTE = {With an appendix by Oded Schramm and Wendelin Werner}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2010}, + PAGES = {xii+403}, + ISBN = {978-0-521-76018-8}, + MRCLASS = {60J65 (28A78 60H05 60J45 60J55 60J67)}, + MRNUMBER = {2604525}, +MRREVIEWER = {Ren\'e L. Schilling}, + DOI = {10.1017/CBO9780511750489}, + URL = {https://doi.org/10.1017/CBO9780511750489}, +} + + +@Article{Evans1978, + Title = {A convergence theorem for solutions of nonlinear second-order elliptic equations}, + Author = {Evans, Lawrence C.}, + Journal = {Indiana Univ. Math. J.}, + Year = {1978}, + Number = {5}, + Pages = {875--887}, + Volume = {27}, + + Coden = {IUMJAB}, + Doi = {10.1512/iumj.1978.27.27059}, + Fjournal = {Indiana University Mathematics Journal}, + ISSN = {0022-2518}, + Mrclass = {35J60}, + Mrnumber = {503721 (80e:35023)}, + Mrreviewer = {P. C. Fife}, + Url = {http://dx.doi.org/10.1512/iumj.1978.27.27059} +} + +@Article{Evans1980, + Title = {On solving certain nonlinear partial differential equations by accretive operator methods}, + Author = {Evans, Lawrence C.}, + Journal = {Israel J. Math.}, + Year = {1980}, + Number = {3-4}, + Pages = {225--247}, + Volume = {36}, + + Coden = {ISJMAP}, + Doi = {10.1007/BF02762047}, + Fjournal = {Israel Journal of Mathematics}, + ISSN = {0021-2172}, + Mrclass = {35G20 (47H06 47H15)}, + Mrnumber = {597451 (82b:35032)}, + Mrreviewer = {J. Kolom{\'y}}, + Url = {http://dx.doi.org/10.1007/BF02762047} +} + +@Article{CrandallLions1981, + Title = {Condition d'unicit\'e pour les solutions g\'en\'eralis\'ees des \'equations de {H}amilton-{J}acobi du premier ordre}, + Author = {Crandall, Michael G. and Lions, Pierre-Louis}, + Journal = {C. R. Acad. Sci. Paris S\'er. I Math.}, + Year = {1981}, + Number = {3}, + Pages = {183--186}, + Volume = {292}, + + Coden = {CRSMDY}, + Fjournal = {Comptes Rendus des S\'eances de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique}, + ISSN = {0151-0509}, + Mrclass = {49C10}, + Mrnumber = {610314 (82c:49020)} +} + +@Article{CrandallIshiiLions1992, + Title = {User's guide to viscosity solutions of second order partial differential equations}, + Author = {Crandall, Michael G. and Ishii, Hitoshi and Lions, Pierre-Louis}, + Journal = {Bull. Amer. Math. Soc. (N.S.)}, + Year = {1992}, + Number = {1}, + Pages = {1--67}, + Volume = {27}, + + Coden = {BAMOAD}, + Doi = {10.1090/S0273-0979-1992-00266-5}, + Fjournal = {American Mathematical Society. Bulletin. New Series}, + ISSN = {0273-0979}, + Mrclass = {35J60 (35B05 35D05 35G20)}, + Mrnumber = {1118699 (92j:35050)}, + Mrreviewer = {P. Szeptycki}, + Url = {http://dx.doi.org/10.1090/S0273-0979-1992-00266-5} +} + +@Book{Evans2010, + Title = {Partial differential equations}, + Author = {Evans, Lawrence C.}, + Publisher = {American Mathematical Society}, + Year = {2010}, + + Address = {Providence, RI}, + Edition = {Second}, + Series = {Graduate Studies in Mathematics}, + Volume = {19}, + + ISBN = {978-0-8218-4974-3}, + Mrclass = {35-01}, + Mrnumber = {2597943 (2011c:35002)}, + Mrreviewer = {Diego M. Maldonado}, + Pages = {xxii+749} +} + +@article {MR3766391, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Wang, Xiaojie}, + TITLE = {Exponential integrability properties of numerical + approximation processes for nonlinear stochastic differential + equations}, + JOURNAL = {Math. Comp.}, + FJOURNAL = {Mathematics of Computation}, + VOLUME = {87}, + YEAR = {2018}, + NUMBER = {311}, + PAGES = {1353--1413}, + ISSN = {0025-5718}, + MRCLASS = {65C30 (60H35)}, + MRNUMBER = {3766391}, + DOI = {10.1090/mcom/3146}, + URL = {https://doi.org/10.1090/mcom/3146}, +} + +@Article{Peng2010, + Title = {Nonlinear Expectations and Stochastic Calculus under Uncertainty}, + Author = {Shige Peng}, + Journal = {arXiv:1002.4546v1}, + Year = {2010}, + Pages = {149 pages} +} + + + +@article{HairerHutzenthalerJentzenAOP, +author = {Hairer, Martin and Hutzenthaler, Martin and Jentzen, Arnulf}, +doi = {10.1214/13-AOP838}, +fjournal = {The Annals of Probability}, +journal = {Ann. Probab.}, +month = {03}, +number = {2}, +pages = {468--527}, +publisher = {The Institute of Mathematical Statistics}, +title = {Loss of regularity for Kolmogorov equations}, +url = {https://doi.org/10.1214/13-AOP838}, +volume = {43}, +year = {2015} +} + + + + + + + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Deep learning +@book{GoodfellowBengioCourville16, + title={Deep learning}, + author={Goodfellow, Ian and Bengio, Yoshua and Courville, Aaron}, + volume={1}, + year={2016}, + publisher={MIT press Cambridge} +} + +@article{LeCunBengioHinton15, + title={Deep learning}, + author={LeCun, Yann and Bengio, Yoshua and Hinton, Geoffrey}, + journal={nature}, + volume={521}, + number={7553}, + pages={436}, + year={2015}, + publisher={Nature Publishing Group} +} + +@incollection{LeCunBottouOrrMueller98, + title={Efficient backprop}, + author={LeCun, Yann and Bottou, L{\'e}on and Orr, Genevieve B and M{\"u}ller, Klaus-Robert}, + booktitle={Neural networks: Tricks of the trade}, + pages={9--50}, + year={1998}, + publisher={Springer} +} + + + +% Other articles +@article{DereichGronbach17, + Title = {General multilevel adaptations for stochastic approximation algorithms}, + Author = {Steffen Dereich and Thomas Mueller-Gronbach}, + Journal = {arXiv:1506.05482}, + Year = {2017}, + Pages = {33 pages} +} + +@article{KingmaBa14, + author = {Diederik P. Kingma and + Jimmy Ba}, + title = {Adam: {A} Method for Stochastic Optimization}, + journal = {arXiv:1412.6980}, + year = {2014}, + pages = {15 pages}, + url = {http://arxiv.org/abs/1412.6980}, + archivePrefix = {arXiv}, + eprint = {1412.6980}, + timestamp = {Wed, 07 Jun 2017 14:40:52 +0200}, + biburl = {http://dblp.org/rec/bib/journals/corr/KingmaB14}, + bibsource = {dblp computer science bibliography, http://dblp.org} +} + +@article{JentzenWurstemberger2018, + author = {Jentzen, Arnulf and von Wurstemberger, Philippe}, + title = {Lower error bounds for the stochastic gradient descent optimization algorithm: Sharp convergence rates for slowly and fast decaying learning rates}, + journal = {arXiv:1803.08600}, + year = {2018}, + pages = {42 pages}, +} + + +@article{Ruder16, + author = {Sebastian Ruder}, + title = {An overview of gradient descent optimization algorithms}, + journal = {arXiv:1609.04747}, + year = {2016}, + pages = {12 pages}, + timestamp = {Wed, 07 Jun 2017 14:40:06 +0200}, + biburl = {http://dblp.uni-trier.de/rec/bib/journals/corr/Ruder16}, + bibsource = {dblp computer science bibliography, http://dblp.org} +} + +@article{Zeiler12, + author = {Matthew D. Zeiler}, + title = {ADADELTA: An Adaptive Learning Rate Method}, + journal = {arXiv:1212.5701}, + year = {2012}, + pages = {6 pages}, + url = {http://arxiv.org/abs/1212.5701}, + archivePrefix = {arXiv}, + eprint = {1212.5701}, + timestamp = {Wed, 07 Jun 2017 14:43:02 +0200}, + biburl = {http://dblp.org/rec/bib/journals/corr/abs-1212-5701}, + bibsource = {dblp computer science bibliography, http://dblp.org} +} + +@article{DuchiHazanSinger11, + author = {Duchi, John and Hazan, Elad and Singer, Yoram}, + title = {Adaptive Subgradient Methods for Online Learning and Stochastic Optimization}, + journal = {J. Mach. Learn. 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Math. Stat.}, + FJOURNAL = {Communications in Mathematics and Statistics}, + VOLUME = {5}, + YEAR = {2017}, + NUMBER = {4}, + PAGES = {349--380}, + ISSN = {2194-6701}, + MRCLASS = {65M75 (60H35 65C30)}, + MRNUMBER = {3736669}, +MRREVIEWER = {Samuel Bernard}, + DOI = {10.1007/s40304-017-0117-6}, + URL = {https://doi.org/10.1007/s40304-017-0117-6}, +} + +@article{Silvester2000, + title={Determinants of block matrices}, + author={Silvester, John R}, + journal={The Mathematical Gazette}, + volume={84}, + number={501}, + pages={460--467}, + year={2000}, + publisher={JSTOR} +} + +@inproceedings{Tropp01, + title={An Elementary Proof of the Spectral Radius Formula for Matrices}, + author={Joel A. Tropp}, + year ={2001}, + note = {\url{http://users.cms.caltech.edu/~jtropp/notes/Tro01-Spectral-Radius.pdf}. [Accessed 16-February-2018]}, +} + +@article{EinsiedlerWard15, + title={Functional Analysis, Spectral Theory, and Applications}, + author={Einsiedler, Manfred and Ward, Thomas}, + journal={draft version}, + year={2015}, + publisher={Springer} +} + +@article{Mypaper, + author = {Jentzen, Arnulf and Kuckuck, Benno and Neufeld, Ariel and von Wurstemberger,Philippe}, + title = {Strong error analysis for stochastic gradient descent optimization algorithms}, + journal = {arXiv:1801.09324}, + year = {2018}, + pages = {75 pages}, +} + + + + +%Averaging +@techreport{Ruppert88, + title={Efficient estimations from a slowly convergent Robbins-Monro process}, + author={Ruppert, David}, + year={1988}, + institution={Cornell University Operations Research and Industrial Engineering} +} + +@article{PolyakJuditsky92, + title={Acceleration of stochastic approximation by averaging}, + author={Polyak, Boris T and Juditsky, Anatoli B}, + journal={SIAM Journal on Control and Optimization}, + volume={30}, + number={4}, + pages={838--855}, + year={1992}, + publisher={SIAM} +} + +@article{Polyak90, + title={New stochastic approximation type procedures}, + author={Polyak, Boris T}, + journal={Automat. i Telemekh}, + volume={7}, + number={98-107}, + pages={2}, + year={1990} +} + + + + + +%websites +@misc{TensorflowMomentum, + title={Tensorflow: Momentum Optimizer}, + howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/train/MomentumOptimizer}", + note = "[Accessed 6-December-2017]" +} + +@misc{TensorflowRMSprop, + title={Tensorflow: RMSProp}, + howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/train/RMSPropOptimizer}", + note = "[Accessed 6-December-2017]" +} + + +@misc{TensorflowNN, + title={Tensorflow: Neural Networks}, + howpublished = "\url{https://www.tensorflow.org/api_guides/python/nn}", + note = "[Accessed 06-April-2018]" +} + +@misc{TensorflowRelu, + title={Tensorflow: Relu}, + howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/nn/relu}", + note = "[Accessed 06-April-2018]" +} + + +@misc{TensorflowSigmoid, + title={Tensorflow: Sigmoid}, + howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/sigmoid}", + note = "[Accessed 06-April-2018]" +} + + + + +@misc{TensorflowSoftplus, + title={Tensorflow: Softplus}, + howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/nn/softplus}", + note = "[Accessed 06-April-2018]" +} + + +@misc{TensorflowSoftmax, + title={Tensorflow: Softmax}, + howpublished = "\url{https://www.tensorflow.org/api_docs/python/tf/nn/softmax}", + note = "[Accessed 06-April-2018]" +} + + + +@misc{CaffeOptimizer, + title={Caffe: Solver}, + howpublished = "\url{http://caffe.berkeleyvision.org/tutorial/solver.html}", + note = "[Accessed 6-December-2017]" +} + +@misc{LasagneOptimizer, + title={Lasagne: Updates}, + howpublished = "\url{http://lasagne.readthedocs.io/en/latest/modules/updates.html#lasagne.updates.rmsprop}", + note = "[Accessed 6-December-2017]" +} + + +@misc{NgCoursera, + title={COURSERA: Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization}, + author={Ng, Andrew}, + howpublished = "\url{https://www.coursera.org/learn/deep-neural-network}", + note = "[Accessed 6-December-2017]" +} + +@misc{HintonCoursera, + title={COURSERA: Neural Networks for Machine learning}, + author={Hinton, Geoffrey}, + howpublished = "\url{https://www.coursera.org/learn/neural-networks}", + note = "[Accessed 6-December-2017]" +} + +@misc{WikipediaGronwall, + title={Gronwall's inequality - Wikipedia}, + howpublished = "\url{https://en.wikipedia.org/wiki/Gronwall%27s_inequality}", + note = "[Accessed 10-January-2018]" +} + +@misc{WikipediaSpectralRadius, + title={Spectral Radius - Wikipedia}, + howpublished = "\url{https://en.wikipedia.org/wiki/Spectral_radius#Gelfand's_formula}", + note = "[Accessed 16-February-2018]" +} + + +@misc{WikipediaAdditionstheorem, + title={List of trigonometric identities - Wikipedia}, + howpublished = "\url{https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities}", + note = "[Accessed 06-March-2018]" +} + +@misc{WikipediaLBlog, + title={Lower Bound of Natural Logarithm - Pr$\infty$fWiki}, + howpublished = "\url{https://proofwiki.org/wiki/Lower_Bound_of_Natural_Logarithm}", + note = "[Accessed 08-March-2018]" +} + +@misc{WikipediaRelu, + title={Rectifier (neural networks) - Wikipedia}, + howpublished = "\url{https://en.wikipedia.org/wiki/Rectifier_(neural_networks)}", + note = "[Accessed 06-April-2018]" +} + +@misc{WikipediaSoftmax, + title={Softmax function - Wikipedia}, + howpublished = "\url{https://en.wikipedia.org/wiki/Softmax_function}", + note = "[Accessed 06-April-2018]" +} + +@misc{WikipediaSigmoid, + title={Sigmoid function - Wikipedia}, + howpublished = "\url{https://en.wikipedia.org/wiki/Sigmoid_function}", + note = "[Accessed 06-April-2018]" +} + + + + + + + + + + + + + + +%Reference books + +@book{Coleman12, + title={Calculus on normed vector spaces}, + author={Coleman, Rodney}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +@book{Petersen12, + title={Linear Algebra}, + author={Petersen, P.}, + isbn={9781461436119}, + lccn={2012938237}, + series={Undergraduate Texts in Mathematics}, + url={https://books.google.ch/books?id=3klgLwEACAAJ}, + year={2012}, + publisher={Springer New York} +} + + + + + + + + + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 +@article{Dockhorn2019, + title={A Discussion on Solving Partial Differential Equations using Neural Networks}, + author={Tim Dockhorn}, + journal={arXiv:1904.07200}, + year={2019}, + PAGES = {9 pages}, + +} + + +@article {YangZhang2017, + AUTHOR = {Yang, Li and Zhang, Yanzhi}, + TITLE = {Convergence of the spectral {G}alerkin method for the + stochastic reaction-diffusion-advection equation}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {446}, + YEAR = {2017}, + NUMBER = {2}, + PAGES = {1230--1254}, + ISSN = {0022-247X}, + MRCLASS = {65M60 (60H35 65M12 65M75)}, + MRNUMBER = {3563032}, +MRREVIEWER = {Josep Vives}, + DOI = {10.1016/j.jmaa.2016.09.028}, + URL = {http://dx.doi.org/10.1016/j.jmaa.2016.09.028}, +} + + +@article {KamraniBloemker2017, + AUTHOR = {Kamrani, Minoo and Bl\"omker, Dirk}, + TITLE = {Pathwise convergence of a numerical method for stochastic + partial differential equations with correlated noise and local + {L}ipschitz condition}, + JOURNAL = {J. Comput. Appl. Math.}, + FJOURNAL = {Journal of Computational and Applied Mathematics}, + VOLUME = {323}, + YEAR = {2017}, + PAGES = {123--135}, + ISSN = {0377-0427}, + MRCLASS = {60H35 (60H10 60H15 65M12 65M60 65M75)}, + MRNUMBER = {3649320}, + DOI = {10.1016/j.cam.2017.04.012}, + URL = {http://dx.doi.org/10.1016/j.cam.2017.04.012}, +} + +@article {Kamrani2015, + AUTHOR = {Kamrani, Minoo}, + TITLE = {Numerical solution of stochastic fractional differential + equations}, + JOURNAL = {Numer. Algorithms}, + FJOURNAL = {Numerical Algorithms}, + VOLUME = {68}, + YEAR = {2015}, + NUMBER = {1}, + PAGES = {81--93}, + ISSN = {1017-1398}, + MRCLASS = {65C30 (26A33 60H15)}, + MRNUMBER = {3296700}, +MRREVIEWER = {Gabriela Mircea}, + DOI = {10.1007/s11075-014-9839-7}, + URL = {http://dx.doi.org/10.1007/s11075-014-9839-7}, +} + +@article {HairerMatetski2016, + AUTHOR = {Hairer, M. and Matetski, K.}, + TITLE = {Optimal rate of convergence for stochastic {B}urgers-type + equations}, + JOURNAL = {Stoch. Partial Differ. Equ. Anal. Comput.}, + FJOURNAL = {Stochastic Partial Differential Equations. Analysis and + Computations}, + VOLUME = {4}, + YEAR = {2016}, + NUMBER = {2}, + PAGES = {402--437}, + ISSN = {2194-0401}, + MRCLASS = {35R60 (35Q53 60H15 60H99)}, + MRNUMBER = {3498987}, + DOI = {10.1007/s40072-015-0067-5}, + URL = {http://dx.doi.org/10.1007/s40072-015-0067-5}, +} + +@article {FurihataKovacsLarssonLindgren2018, + AUTHOR = {Furihata, Daisuke and Kov\'acs, Mih\'aly and Larsson, Stig and + Lindgren, Fredrik}, + TITLE = {Strong {C}onvergence of a {F}ully {D}iscrete {F}inite + {E}lement {A}pproximation of the {S}tochastic + {C}ahn--{H}illiard {E}quation}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {56}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {708--731}, + ISSN = {0036-1429}, + MRCLASS = {60H15 (60H35 65M75)}, + MRNUMBER = {3780120}, + DOI = {10.1137/17M1121627}, + URL = {https://doi.org/10.1137/17M1121627}, +} + + +@article {CuiHongLiuZhou2019, + AUTHOR = {Cui, Jianbo and Hong, Jialin and Liu, Zhihui and Zhou, Weien}, + TITLE = {Strong convergence rate of splitting schemes for stochastic + nonlinear {S}chr\"{o}dinger equations}, + JOURNAL = {J. Differential Equations}, + FJOURNAL = {Journal of Differential Equations}, + VOLUME = {266}, + YEAR = {2019}, + NUMBER = {9}, + PAGES = {5625--5663}, + ISSN = {0022-0396}, + MRCLASS = {60H35 (35Q55 35R60 60G05 60H15)}, + MRNUMBER = {3912762}, + DOI = {10.1016/j.jde.2018.10.034}, + URL = {https://doi.org/10.1016/j.jde.2018.10.034}, +} + + +@article {CuiHongLiu2017, + AUTHOR = {Cui, Jianbo and Hong, Jialin and Liu, Zhihui}, + TITLE = {Strong convergence rate of finite difference approximations + for stochastic cubic {S}chr\"odinger equations}, + JOURNAL = {J. Differential Equations}, + FJOURNAL = {Journal of Differential Equations}, + VOLUME = {263}, + YEAR = {2017}, + NUMBER = {7}, + PAGES = {3687--3713}, + ISSN = {0022-0396}, + MRCLASS = {60H35 (35Q55 35R60 60H15 65M06 65M12)}, + MRNUMBER = {3670034}, + DOI = {10.1016/j.jde.2017.05.002}, + URL = {http://dx.doi.org/10.1016/j.jde.2017.05.002}, +} + + +@article{AnderssonJentzen2017, + title={Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values}, + author={Adam Andersson and Arnulf Jentzen and Ryan Kurniawan}, + journal={arXiv:1512.06899}, + year={2015}, + PAGES = {35 pages}, +note = {Revision requested from \emph{J.\ Math.\ Anal.\ Appl.}} + +} + + +@article {AnderssonKruse2017, + AUTHOR = {Andersson, Adam and Kruse, Raphael}, + TITLE = {Mean-square convergence of the {BDF}2-{M}aruyama and backward + {E}uler schemes for {SDE} satisfying a global monotonicity + condition}, + JOURNAL = {BIT}, + FJOURNAL = {BIT. Numerical Mathematics}, + VOLUME = {57}, + YEAR = {2017}, + NUMBER = {1}, + PAGES = {21--53}, + ISSN = {0006-3835}, + MRCLASS = {65C30 (60H35)}, + MRNUMBER = {3608309}, +MRREVIEWER = {Carlos M. Mora Gonz\'{a}lez}, + DOI = {10.1007/s10543-016-0624-y}, + URL = {https://doi.org/10.1007/s10543-016-0624-y}, +} + + +@article {HefterJentzen2019, + AUTHOR = {Hefter, Mario and Jentzen, Arnulf}, + TITLE = {On arbitrarily slow convergence rates for strong numerical + approximations of {C}ox-{I}ngersoll-{R}oss processes and + squared {B}essel processes}, + JOURNAL = {Finance Stoch.}, + FJOURNAL = {Finance and Stochastics}, + VOLUME = {23}, + YEAR = {2019}, + NUMBER = {1}, + PAGES = {139--172}, + ISSN = {0949-2984}, + MRCLASS = {60H10 (65C30 91G60)}, + MRNUMBER = {3898403}, + DOI = {10.1007/s00780-018-0375-5}, + URL = {https://doi.org/10.1007/s00780-018-0375-5}, +} + +@book {Novak2010, + AUTHOR = {Novak, Erich and Wo\'{z}niakowski, Henryk}, + TITLE = {Tractability of multivariate problems. {V}olume {II}: + {S}tandard information for functionals}, + SERIES = {EMS Tracts in Mathematics}, + VOLUME = {12}, + PUBLISHER = {European Mathematical Society (EMS), Z\"{u}rich}, + YEAR = {2010}, + PAGES = {xviii+657}, + ISBN = {978-3-03719-084-5}, + MRCLASS = {46N40 (41A55 41A63 47A50 47N40 65Y20 68Q25)}, + MRNUMBER = {2676032}, +MRREVIEWER = {Arthur G. Werschulz}, + DOI = {10.4171/084}, + URL = {https://doi.org/10.4171/084}, +} + + +@book {Bellman1957, + AUTHOR = {Bellman, Richard}, + TITLE = {Dynamic programming}, + PUBLISHER = {Princeton University Press, Princeton, N. J.}, + YEAR = {1957}, + PAGES = {xxv+342}, + MRCLASS = {90.0X}, + MRNUMBER = {0090477}, +MRREVIEWER = {J. Kiefer}, +} + +@article{HutzenthalerKruse2017, + title={Multilevel {P}icard Approximations of High-Dimensional Semilinear Parabolic Differential Equations with Gradient-Dependent Nonlinearities}, + author={Hutzenthaler, Martin and Kruse, Thomas}, + journal={SIAM J.\ Numer.\ Anal.}, + volume={58}, + number={2}, + pages={929--961}, + year={2020}, + publisher={SIAM} +} + + + +@article{HutzenthalerJentzenKruse2018, +author = {Hutzenthaler, Martin and Jentzen, Arnulf and Kruse, Thomas and Nguyen, Tuan Anh and von Wurstemberger, Philippe }, +title = {Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations}, +fjournal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, +journal = {Proc. Roy. Soc. London A}, +volume = {476}, +number = {2244}, +pages = {20190630}, +year = {2020}, +doi = {10.1098/rspa.2019.0630}, +URL = {https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2019.0630}, +eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0630} +, + abstract = { For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy. } +} + +@article {EHutzenthaler2019, + AUTHOR = {E, Weinan and Hutzenthaler, Martin and Jentzen, Arnulf and + Kruse, Thomas}, + TITLE = {On multilevel {P}icard numerical approximations for + high-dimensional nonlinear parabolic partial differential + equations and high-dimensional nonlinear backward stochastic + differential equations}, + JOURNAL = {J. Sci. Comput.}, + FJOURNAL = {Journal of Scientific Computing}, + VOLUME = {79}, + YEAR = {2019}, + NUMBER = {3}, + PAGES = {1534--1571}, + ISSN = {0885-7474}, + MRCLASS = {65C05 (65C30 65M99 91G60)}, + MRNUMBER = {3946468}, + DOI = {10.1007/s10915-018-00903-0}, + URL = {https://doi.org/10.1007/s10915-018-00903-0}, +} + + +@article{EHutzenthaler2016, + title={{Multilevel Picard iterations for solving smooth semilinear parabolic heat equations}}, + author={Weinan E and Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse}, + journal={arXiv:1607.03295}, + note={\emph{Accepted in Partial Differ. Equ. Appl.}}, + year={2016}, + pages={19 pages}, +} + + + +@article {Kruse2014, + AUTHOR = {Kruse, Raphael}, + TITLE = {Optimal error estimates of {G}alerkin finite element methods + for stochastic partial differential equations with + multiplicative noise}, + JOURNAL = {IMA J. Numer. Anal.}, + FJOURNAL = {IMA Journal of Numerical Analysis}, + VOLUME = {34}, + YEAR = {2014}, + NUMBER = {1}, + PAGES = {217--251}, + ISSN = {0272-4979}, + MRCLASS = {65M60 (35R60 60H15 65M75)}, + MRNUMBER = {3168284}, +MRREVIEWER = {Mohammad Asadzadeh}, + DOI = {10.1093/imanum/drs055}, + URL = {https://doi.org/10.1093/imanum/drs055}, +} + +@article {Deya2012, + AUTHOR = {Deya, Aur\'{e}lien}, + TITLE = {Numerical schemes for rough parabolic equations}, + JOURNAL = {Appl. Math. Optim.}, + FJOURNAL = {Applied Mathematics and Optimization}, + VOLUME = {65}, + YEAR = {2012}, + NUMBER = {2}, + PAGES = {253--292}, + ISSN = {0095-4616}, + MRCLASS = {65M75 (35R60 60G17 60G22 60H15)}, + MRNUMBER = {2891223}, + DOI = {10.1007/s00245-011-9157-6}, + URL = {https://doi.org/10.1007/s00245-011-9157-6}, +} + + +@article {WiegmannBube2000, + AUTHOR = {Wiegmann, Andreas and Bube, Kenneth P.}, + TITLE = {The explicit-jump immersed interface method: finite difference + methods for {PDE}s with piecewise smooth solutions}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {37}, + YEAR = {2000}, + NUMBER = {3}, + PAGES = {827--862}, + ISSN = {0036-1429}, + MRCLASS = {65N06 (31A10 31C20)}, + MRNUMBER = {1740384}, +MRREVIEWER = {Zheng Hui Xie}, + DOI = {10.1137/S0036142997328664}, + URL = {https://doi.org/10.1137/S0036142997328664}, +} + +@article {Furihata2001, + AUTHOR = {Furihata, Daisuke}, + TITLE = {A stable and conservative finite difference scheme for the + {C}ahn-{H}illiard equation}, + JOURNAL = {Numer. Math.}, + FJOURNAL = {Numerische Mathematik}, + VOLUME = {87}, + YEAR = {2001}, + NUMBER = {4}, + PAGES = {675--699}, + ISSN = {0029-599X}, + MRCLASS = {65M06 (35K55 65M12 82C26)}, + MRNUMBER = {1815731}, +MRREVIEWER = {Bruno Scheurer}, + DOI = {10.1007/PL00005429}, + URL = {https://doi.org/10.1007/PL00005429}, +} + +@article {YusteAcedo2005, + AUTHOR = {Yuste, S. B. and Acedo, L.}, + TITLE = {An explicit finite difference method and a new von + {N}eumann-type stability analysis for fractional diffusion + equations}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {42}, + YEAR = {2005}, + NUMBER = {5}, + PAGES = {1862--1874}, + ISSN = {0036-1429}, + MRCLASS = {45K05 (65M06)}, + MRNUMBER = {2139227}, +MRREVIEWER = {\v{Z}ivorad Tomovski}, + DOI = {10.1137/030602666}, + URL = {https://doi.org/10.1137/030602666}, +} + +@article {Meerschaert2006, + AUTHOR = {Meerschaert, Mark M. and Tadjeran, Charles}, + TITLE = {Finite difference approximations for two-sided + space-fractional partial differential equations}, + JOURNAL = {Appl. Numer. Math.}, + FJOURNAL = {Applied Numerical Mathematics. An IMACS Journal}, + VOLUME = {56}, + YEAR = {2006}, + NUMBER = {1}, + PAGES = {80--90}, + ISSN = {0168-9274}, + MRCLASS = {65M06}, + MRNUMBER = {2186432}, +MRREVIEWER = {Matthew Edward Hubbard}, + DOI = {10.1016/j.apnum.2005.02.008}, + URL = {https://doi.org/10.1016/j.apnum.2005.02.008}, +} + +@article {Babuska1978, + AUTHOR = {Babu\v{s}ka, I. and Rheinboldt, W. C.}, + TITLE = {Error estimates for adaptive finite element computations}, + JOURNAL = {SIAM J. Numer. Anal.}, + FJOURNAL = {SIAM Journal on Numerical Analysis}, + VOLUME = {15}, + YEAR = {1978}, + NUMBER = {4}, + PAGES = {736--754}, + ISSN = {0036-1429}, + MRCLASS = {65J05 (65N30)}, + MRNUMBER = {0483395}, +MRREVIEWER = {William W. Hager}, + DOI = {10.1137/0715049}, + URL = {https://doi.org/10.1137/0715049}, +} + + +@book {Braess2007, + AUTHOR = {Braess, Dietrich}, + TITLE = {Finite elements}, + EDITION = {Third}, + NOTE = {Theory, fast solvers, and applications in elasticity theory, + Translated from the German by Larry L. Schumaker}, + PUBLISHER = {Cambridge University Press, Cambridge}, + YEAR = {2007}, + PAGES = {xviii+365}, + ISBN = {978-0-521-70518-9; 0-521-70518-5}, + MRCLASS = {65N30 (65-02 74S05)}, + MRNUMBER = {2322235}, + DOI = {10.1017/CBO9780511618635}, + URL = {https://doi.org/10.1017/CBO9780511618635}, +} + + +@book {BensoussanLions1982, + AUTHOR = {Bensoussan, Alain and Lions, Jacques-Louis}, + TITLE = {Applications of variational inequalities in stochastic + control}, + SERIES = {Studies in Mathematics and its Applications}, + VOLUME = {12}, + PUBLISHER = {North-Holland Publishing Co., Amsterdam-New York}, + YEAR = {1982}, + PAGES = {xi+564}, + ISBN = {0-444-86358-3}, + MRCLASS = {49A29 (49-02)}, + MRNUMBER = {653144}, +} + + +@inproceedings{uchiyama1993solving, + title={Solving inverse problems in nonlinear {PDE}s by recurrent neural networks}, + author={Uchiyama, Tadasu and Sonehara, Noboru}, + booktitle={IEEE International Conference on Neural Networks}, + pages={99--102}, + year={1993}, + organization={IEEE} +} + + +@article {MeadeFernandez1994, + AUTHOR = {Meade, Jr., A. J. and Fern\'{a}ndez, A. A.}, + TITLE = {The numerical solution of linear ordinary differential + equations by feedforward neural networks}, + JOURNAL = {Math. Comput. Modelling}, + FJOURNAL = {Mathematical and Computer Modelling}, + VOLUME = {19}, + YEAR = {1994}, + NUMBER = {12}, + PAGES = {1--25}, + ISSN = {0895-7177}, + MRCLASS = {65L05}, + MRNUMBER = {1284175}, + DOI = {10.1016/0895-7177(94)90095-7}, + URL = {https://doi.org/10.1016/0895-7177(94)90095-7}, +} + + +@article{Lagaris1998ArtificialNN, + title={Artificial neural networks for solving ordinary and partial differential equations}, + author={Isaac E. Lagaris and Aristidis Likas and Dimitrios I. Fotiadis}, + journal={IEEE transactions on neural networks}, + year={1998}, + volume={9 (5)}, + pages={ + 987--1000 + } +} + +@article{GrohsJentzenSalimova2019, + title={{Deep neural network approximations for Monte Carlo algorithms}}, + author={Grohs, Philipp and Jentzen, Arnulf and Salimova, Diyora}, + journal={arXiv:1908.10828}, + year={2019}, + PAGES = {45 pages}, +} + +@article {JentzenSalimovaWeltiBurgers, + AUTHOR = {Jentzen, Arnulf and Salimova, Diyora and Welti, Timo}, + TITLE = {Strong convergence for explicit space-time discrete numerical + approximation methods for stochastic {B}urgers equations}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {469}, + YEAR = {2019}, + NUMBER = {2}, + PAGES = {661--704}, + ISSN = {0022-247X}, + MRCLASS = {60H35 (35Q53 35R60 65M75)}, + MRNUMBER = {3860443}, +MRREVIEWER = {Romeo I. Negrea}, + DOI = {10.1016/j.jmaa.2018.09.032}, + URL = {https://doi.org/10.1016/j.jmaa.2018.09.032}, +} + +@article{JentzenSalimovaWeltiBurgersArxiv, + title={Strong convergence for explicit space-time discrete numerical + approximation methods for stochastic {B}urgers equations}, + author={Jentzen, Arnulf and Salimova, Diyora and Welti, Timo}, + journal={arXiv:1710.07123}, + year={2017}, + PAGES = {60 pages}, +} + + +@article {HutzenthalerJentzenSalimova2018, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Salimova, Diyora}, + TITLE = {Strong convergence of full-discrete nonlinearity-truncated + accelerated exponential {E}uler-type approximations for + stochastic {K}uramoto-{S}ivashinsky equations}, + JOURNAL = {Commun. Math. Sci.}, + FJOURNAL = {Communications in Mathematical Sciences}, + VOLUME = {16}, + YEAR = {2018}, + NUMBER = {6}, + PAGES = {1489--1529}, + ISSN = {1539-6746}, + MRCLASS = {60H35 (35Q53 35R60)}, + MRNUMBER = {3919602}, + DOI = {10.4310/CMS.2018.v16.n6.a2}, + URL = {https://doi.org/10.4310/CMS.2018.v16.n6.a2}, +} + + +@incollection {GjerdeHolden1995, + AUTHOR = {Gjerde, Jon and Holden, Helge and \O ksendal, Bernt and Ub\o e, + Jan and Zhang, Tu Sheng}, + TITLE = {An equation modelling transport of a substance in a stochastic + medium}, + BOOKTITLE = {Seminar on {S}tochastic {A}nalysis, {R}andom {F}ields and + {A}pplications ({A}scona, 1993)}, + SERIES = {Progr. Probab.}, + VOLUME = {36}, + PAGES = {123--134}, + PUBLISHER = {Birkh\"{a}user, Basel}, + YEAR = {1995}, + MRCLASS = {60H15 (35K10 35R60 76R50)}, + MRNUMBER = {1360271}, +MRREVIEWER = {Krystyna Twardowska}, +} + + +@article {Shandarin1989, + AUTHOR = {Shandarin, S. F. and Zeldovich, Ya. B.}, + TITLE = {The large-scale structure of the universe: turbulence, + intermittency, structures in a self-gravitating medium}, + JOURNAL = {Rev. Modern Phys.}, + FJOURNAL = {Reviews of Modern Physics}, + VOLUME = {61}, + YEAR = {1989}, + NUMBER = {2}, + PAGES = {185--220}, + ISSN = {0034-6861}, + MRCLASS = {85A05 (58C27 58F40)}, + MRNUMBER = {989562}, + DOI = {10.1103/RevModPhys.61.185}, + URL = {https://doi.org/10.1103/RevModPhys.61.185}, +} + + + +@incollection {LeGland1992, + AUTHOR = {Le Gland, Fran\c{c}ois}, + TITLE = {Splitting-up approximation for {SPDE}s and {SDE}s with + application to nonlinear filtering}, + BOOKTITLE = {Stochastic partial differential equations and their + applications ({C}harlotte, {NC}, 1991)}, + SERIES = {Lect. Notes Control Inf. Sci.}, + VOLUME = {176}, + PAGES = {177--187}, + PUBLISHER = {Springer, Berlin}, + YEAR = {1992}, + MRCLASS = {60H15 (60G35 93E11)}, + MRNUMBER = {1176783}, + DOI = {10.1007/BFb0007332}, + URL = {https://doi.org/10.1007/BFb0007332}, +} + + +@book {Wu1996, + AUTHOR = {Wu, Jianhong}, + TITLE = {Theory and applications of partial functional-differential + equations}, + SERIES = {Applied Mathematical Sciences}, + VOLUME = {119}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1996}, + PAGES = {x+429}, + ISBN = {0-387-94771-X}, + MRCLASS = {35R10 (35Bxx 35K99 35Q80)}, + MRNUMBER = {1415838}, +MRREVIEWER = {Yang Kuang}, + DOI = {10.1007/978-1-4612-4050-1}, + URL = {https://doi.org/10.1007/978-1-4612-4050-1}, +} + +@article {BusenbergMahaffy1985, + AUTHOR = {Busenberg, Stavros and Mahaffy, Joseph}, + TITLE = {Interaction of spatial diffusion and delays in models of + genetic control by repression}, + JOURNAL = {J. Math. Biol.}, + FJOURNAL = {Journal of Mathematical Biology}, + VOLUME = {22}, + YEAR = {1985}, + NUMBER = {3}, + PAGES = {313--333}, + ISSN = {0303-6812}, + MRCLASS = {92A09}, + MRNUMBER = {813402}, + DOI = {10.1007/BF00276489}, + URL = {https://doi.org/10.1007/BF00276489}, +} + +@Inbook{Levin1978, +author="Levin, Simon A.", +title="Pattern Formation in Ecological Communities", +bookTitle="Spatial Pattern in Plankton Communities", +year="1978", +publisher="Springer US", +address="Boston, MA", +pages="433--465", +abstract="The primary concern of this volume is the study of spatial pattern in plankton communities, but the basic issues are fundamental ones in all ecosystems (Levin, 1976b). The plankton has emerged as perhaps the prototypical system for the discussion of spatial pattern principally because of the ubiquitous nature of patchiness, independent of the scale of observation (Platt and Denman, 1975). Population densities are sufficiently high that pattern is easily detected, and planktonic systems have been intensively studied from a variety of aspects.", +isbn="978-1-4899-2195-6", +doi="10.1007/978-1-4899-2195-6_16", +url="https://doi.org/10.1007/978-1-4899-2195-6_16" +} + + +@article {MourratWeber2017, + AUTHOR = {Mourrat, Jean-Christophe and Weber, Hendrik}, + TITLE = {Convergence of the two-dimensional dynamic {I}sing-{K}ac model + to {$\Phi^4_2$}}, + JOURNAL = {Comm. Pure Appl. Math.}, + FJOURNAL = {Communications on Pure and Applied Mathematics}, + VOLUME = {70}, + YEAR = {2017}, + NUMBER = {4}, + PAGES = {717--812}, + ISSN = {0010-3640}, + MRCLASS = {82C20 (35R60 60H15 60K35 82C22)}, + MRNUMBER = {3628883}, +MRREVIEWER = {Arvind Ayyer}, + DOI = {10.1002/cpa.21655}, + URL = {https://doi.org/10.1002/cpa.21655}, +} + +@article {Kushner1964, + AUTHOR = {Kushner, Harold J.}, + TITLE = {On the differential equations satisfied by conditional + probablitity densities of {M}arkov processes, with + applications}, + JOURNAL = {J. Soc. Indust. Appl. Math. Ser. A Control}, + FJOURNAL = {Journal of the Society for Industrial and Applied Mathematics. + Series A. Control}, + VOLUME = {2}, + YEAR = {1964}, + PAGES = {106--119}, + MRCLASS = {60.75}, + MRNUMBER = {0180407}, +MRREVIEWER = {A. V. Balakrishnan}, +} + + +@article {Zakai1969, + AUTHOR = {Zakai, Moshe}, + TITLE = {On the optimal filtering of diffusion processes}, + JOURNAL = {Z. Wahrscheinlichkeitstheorie und Verw. Gebiete}, + VOLUME = {11}, + YEAR = {1969}, + PAGES = {230--243}, + MRCLASS = {93.60 (60.00)}, + MRNUMBER = {0242552}, +MRREVIEWER = {R. S. Bucy}, + DOI = {10.1007/BF00536382}, + URL = {https://doi.org/10.1007/BF00536382}, +} + +@book {LeimkuhlerMatthews2015, + AUTHOR = {Leimkuhler, Ben and Matthews, Charles}, + TITLE = {Molecular dynamics}, + SERIES = {Interdisciplinary Applied Mathematics}, + VOLUME = {39}, + NOTE = {With deterministic and stochastic numerical methods}, + PUBLISHER = {Springer, Cham}, + YEAR = {2015}, + PAGES = {xxii+443}, + ISBN = {978-3-319-16374-1; 978-3-319-16375-8}, + MRCLASS = {82Bxx (35Q82 65-02 70-02 82Cxx)}, + MRNUMBER = {3362507}, +} + +@article {HarmsStefanovits2018, + AUTHOR = {Harms, Philipp and Stefanovits, David and Teichmann, Josef and + W\"{u}thrich, Mario V.}, + TITLE = {Consistent recalibration of yield curve models}, + JOURNAL = {Math. Finance}, + FJOURNAL = {Mathematical Finance. An International Journal of Mathematics, + Statistics and Financial Economics}, + VOLUME = {28}, + YEAR = {2018}, + NUMBER = {3}, + PAGES = {757--799}, + ISSN = {0960-1627}, + MRCLASS = {91G30}, + MRNUMBER = {3818710}, + DOI = {10.1111/mafi.12159}, + URL = {https://doi.org/10.1111/mafi.12159}, +} + + +@article {FilipovicTappe2010, + AUTHOR = {Filipovi\'{c}, Damir and Tappe, Stefan and Teichmann, Josef}, + TITLE = {Term structure models driven by {W}iener processes and + {P}oisson measures: existence and positivity}, + JOURNAL = {SIAM J. Financial Math.}, + FJOURNAL = {SIAM Journal on Financial Mathematics}, + VOLUME = {1}, + YEAR = {2010}, + NUMBER = {1}, + PAGES = {523--554}, + ISSN = {1945-497X}, + MRCLASS = {91G30 (60H10 60H15 60H30)}, + MRNUMBER = {2669403}, +MRREVIEWER = {Guohe Deng}, + DOI = {10.1137/090758593}, + URL = {https://doi.org/10.1137/090758593}, +} + +@article{ReisingerZhang2019, + title={Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems}, + author={Christoph Reisinger and Yufei Zhang}, + journal={arXiv:1903.06652}, + year={2019}, + PAGES = {34 pages}, +} + +@article{KutyniokPeterseb2019, + title={A Theoretical Analysis of Deep Neural Networks and Parametric {PDE}s}, + author={Gitta Kutyniok and Philipp Petersen and Mones Raslan and Reinhold Schneider}, + journal={arXiv:1904.00377}, + year={2019}, + PAGES = {43 pages}, +} + +@article{BernerGrohsJentzen2018, + title={{Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations}}, + author={Julius Berner and Philipp Grohs and Arnulf Jentzen}, + journal={Minor revision requested from SIAM J. Math. Data Sci., arXiv:1809.03062}, + year={2018}, + PAGES = {35 pages}, +} + +@article{Raissi2018DeepHP, + title={Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations}, + author={Maziar Raissi}, + journal={J. Mach. Learn. Res.}, + year={2018}, + volume={19}, + pages={25:1-25:24} +} + +@article{PhamWarin2019, + title={{Neural networks-based backward scheme for fully nonlinear PDEs}}, + author={Huy{\^e}n Pham and Xavier Warin}, + journal={arXiv:1908.00412}, + year={2019}, + PAGES = {15 pages}, +} + + +@inproceedings{Magill2018NeuralNT, + title={Neural Networks Trained to Solve Differential Equations Learn General Representations}, + author={Martin Magill and Faisal Qureshi and Hendrick W. de Haan}, + booktitle={Advances in Neural Information Processing Systems}, + year={2018}, +pages = {4071--4081}, +} + +@article{LyeMishraRay2019, + title={Deep learning observables in computational fluid dynamics}, + author={Lye, Kjetil O. and Siddhartha Mishra and Deep Ray}, + journal={arXiv:1903.03040}, + year={2019}, + PAGES = {57 pages}, +} + + + + + +@InProceedings{LongLuMaDong2018, + title = {{PDE-Net: Learning PDEs from Data}}, + author = {Long, Zichao and Lu, Yiping and Ma, Xianzhong and Dong, Bin}, + booktitle = {Proceedings of the 35th International Conference on Machine Learning}, + pages = {3208--3216}, + year = {2018}, + pdf = {http://proceedings.mlr.press/v80/long18a/long18a.pdf}, + url = {http://proceedings.mlr.press/v80/long18a.html}, + abstract = {Partial differential equations (PDEs) play a prominent role in many disciplines of science and engineering. PDEs are commonly derived based on empirical observations. However, with the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of physical laws. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. Comparing with existing approaches, our approach has the most flexibility by learning both differential operators and the nonlinear response function of the underlying PDE model. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.} +} + + +@article{JacquierOumgari2019, + title={{Deep PPDEs for rough local stochastic volatility}}, + author={Antoine Jacquier and Mugad Oumgari}, + journal={arXiv:1906.02551}, + year={2019}, + PAGES = {21 pages}, +} + +@article{HurePhamWarin2019, + title={{Some machine learning schemes for high-dimensional nonlinear PDEs}}, + author={C{\^o}me Hur{\'e} and Huy{\^e}n Pham and Xavier Warin}, + journal={arXiv:1902.01599}, + year={2019}, + PAGES = {33 pages}, +} + + +@article{HanLong2018, + title={{Convergence of the Deep BSDE Method for Coupled FBSDEs}}, + author={Jiequn Han and Jihao Long}, + journal={arXiv:1811.01165}, + year={2018}, + PAGES = {26 pages}, +} + + +@article{GoudenegeMolent2019, + title={{Machine Learning for Pricing American Options in High Dimension}}, + author={Gouden{\`e}ge, Ludovic and Molent, Andrea and Zanette, Antonino}, + journal={arXiv:1903.11275}, + year={2019}, + PAGES = {11 pages}, +} + + + +@Article{FujiiTakahashi2019, +author="Fujii, Masaaki +and Takahashi, Akihiko +and Takahashi, Masayuki", +title="Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional {BSDE}s", +journal="Asia-Pacific Financial Markets", +year="2019", +month="Mar", +day="18", +abstract="We demonstrate that the use of asymptotic expansion as prior knowledge in the ``deep BSDE solver'', which is a deep learning method for high dimensional BSDEs proposed by Weinan et al. (Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, 2017b. arXiv:1706.04702), drastically reduces the loss function and accelerates the speed of convergence. We illustrate the technique and its implications by using Bergman's model with different lending and borrowing rates as a typical model for FVA as well as a class of solvable BSDEs with quadratic growth drivers. We also present an extension of the deep BSDE solver for reflected BSDEs representing American option prices.", +issn="1573-6946", +doi="10.1007/s10690-019-09271-7", +url="https://doi.org/10.1007/s10690-019-09271-7" +} + +@article{Farahmand2017DeepRL, + title={Deep reinforcement learning for partial differential equation control}, + author={Farahmand, Amir-massoud and Saleh Nabi and Daniel Nikovski}, + journal={2017 American Control Conference (ACC)}, + year={2017}, + pages={3120-3127} +} + +@article {ChanMikaelWarin2019, + AUTHOR = {Chan-Wai-Nam, Quentin and Mikael, Joseph and Warin, Xavier}, + TITLE = {Machine learning for semi linear {PDE}s}, + JOURNAL = {J. Sci. Comput.}, + FJOURNAL = {Journal of Scientific Computing}, + VOLUME = {79}, + YEAR = {2019}, + NUMBER = {3}, + PAGES = {1667--1712}, + ISSN = {0885-7474}, + MRCLASS = {68T05 (65C05 65M99)}, + MRNUMBER = {3946472}, + DOI = {10.1007/s10915-019-00908-3}, + URL = {https://doi.org/10.1007/s10915-019-00908-3}, +} + +@article{Berg2018AUD, + title={A unified deep artificial neural network approach to partial differential equations in complex geometries}, + author={Jens Berg and Kaj Nystr{\"o}m}, + journal={Neurocomputing}, + year={2018}, + volume={317}, + pages={28-41} +} + + +@article{BeckerCheriditoJentzen2019, + title={Solving high-dimensional optimal stopping problems using deep learning}, + author={Becker, Sebastian and Cheridito, Patrick and Jentzen, Arnulf and Welti, Timo}, + journal={Minor revision requested from European J. Appl. Math., arXiv:1908.01602}, + year={2019}, + PAGES = {42 pages}, +} + + + +@article{BeckBeckerCheridito2019, + title={{Deep splitting method for parabolic PDEs}}, + author={Beck, Christian and Becker, Sebastian and Cheridito, Patrick and Jentzen, Arnulf and Neufeld, Ariel}, + journal={Revision requested from SIAM J. Sci. Comput., arXiv:1907.03452}, + year={2019}, + PAGES = {40 pages}, +} + + + + +@article {Sirignano2018, + AUTHOR = {Sirignano, Justin and Spiliopoulos, Konstantinos}, + TITLE = {D{GM: A} deep learning algorithm for solving partial + differential equations}, + JOURNAL = {J. Comput. Phys.}, + FJOURNAL = {Journal of Computational Physics}, + VOLUME = {375}, + YEAR = {2018}, + PAGES = {1339--1364}, + ISSN = {0021-9991}, + MRCLASS = {65M99 (68T05)}, + MRNUMBER = {3874585}, + DOI = {10.1016/j.jcp.2018.08.029}, + URL = {https://doi.org/10.1016/j.jcp.2018.08.029}, +} + + + @article {sirignano2017dgm, + AUTHOR = {Sirignano, Justin and Spiliopoulos, Konstantinos}, + TITLE = {{D{GM}: A deep learning algorithm for solving partial + differential equations}}, + JOURNAL = {J. Comput. Phys.}, + FJOURNAL = {Journal of Computational Physics}, + VOLUME = {375}, + YEAR = {2018}, + PAGES = {1339--1364}, + ISSN = {0021-9991}, + MRCLASS = {65M99 (68T05)}, + MRNUMBER = {3874585}, + DOI = {10.1016/j.jcp.2018.08.029}, + URL = {https://doi.org/10.1016/j.jcp.2018.08.029}, + } + + + +@article{mishra2018machine, + title={A machine learning framework for data driven acceleration of computations of differential equations}, + author={Mishra, Siddhartha}, + journal={arXiv:1807.09519}, + year={2018}, + PAGES = {23 pages}, +} + + + +@article{khoo2017solving, + title={Solving parametric {PDE} problems with artificial neural networks}, + author={Khoo, Yuehaw and Lu, Jianfeng and Ying, Lexing}, + journal={arXiv:1707.03351}, + year={2017}, + PAGES = {17 pages}, +} + + + +@article {KhooLuYing2019, + AUTHOR = {Khoo, Yuehaw and Lu, Jianfeng and Ying, Lexing}, + TITLE = {Solving for high-dimensional committor functions using + artificial neural networks}, + JOURNAL = {Res. Math. Sci.}, + FJOURNAL = {Research in the Mathematical Sciences}, + VOLUME = {6}, + YEAR = {2019}, + NUMBER = {1}, + PAGES = {Paper No. 1, 13}, + ISSN = {2522-0144}, + MRCLASS = {60H10 (35Q84 35R60 65C30)}, + MRNUMBER = {3887217}, + DOI = {10.1007/s40687-018-0160-2}, + URL = {https://doi.org/10.1007/s40687-018-0160-2}, +} + + + +@article{henry2017deep, + title={Deep {Primal}-{Dual} {Algorithm} for {BSDEs}: {Applications} of {Machine} {Learning} to {CVA} and {IM}}, + author={Henry-Labord\`ere, Pierre}, + year={2017}, +pages= {(November 15, 2017), 16 pages}, +note= {Available at SSRN: \url{https://ssrn.com/abstract=3071506}}, +} + +@article {MR3881695, + AUTHOR = {Raissi, M. and Perdikaris, P. and Karniadakis, G. E.}, + TITLE = {{Physics-informed neural networks: A deep learning framework + for solving forward and inverse problems involving nonlinear + partial differential equations}}, + JOURNAL = {J. Comput. Phys.}, + FJOURNAL = {Journal of Computational Physics}, + VOLUME = {378}, + YEAR = {2019}, + PAGES = {686--707}, + ISSN = {0021-9991}, + MRCLASS = {65M70 (68T05)}, + MRNUMBER = {3881695}, + DOI = {10.1016/j.jcp.2018.10.045}, + URL = {https://doi.org/10.1016/j.jcp.2018.10.045}, +} + +@article {Han2018PNAS, + AUTHOR = {Han, Jiequn and Jentzen, Arnulf and E, Weinan}, + TITLE = {Solving high-dimensional partial differential equations using + deep learning}, + JOURNAL = {Proc. Natl. Acad. Sci. USA}, + FJOURNAL = {Proceedings of the National Academy of Sciences of the United + States of America}, + VOLUME = {115}, + YEAR = {2018}, + NUMBER = {34}, + PAGES = {8505--8510}, + ISSN = {0027-8424}, + MRCLASS = {35K55 (68T05 91A26 92C20)}, + MRNUMBER = {3847747}, + DOI = {10.1073/pnas.1718942115}, + URL = {https://doi.org/10.1073/pnas.1718942115}, +} + +@article{fujii2017asymptotic, + title={Asymptotic {Expansion} as {Prior} {Knowledge} in {Deep} {Learning} {Method} for high dimensional {BSDEs}}, + author={Fujii, Masaaki and Takahashi, Akihiko and Takahashi, Masayuki}, + journal={arXiv:1710.07030}, + year={2017}, + PAGES = {16 pages}, +} + + + + +@article {weinan2018deep, + AUTHOR = {E, Weinan and Yu, Bing}, + TITLE = {The deep {R}itz method: {A} deep learning-based numerical + algorithm for solving variational problems}, + JOURNAL = {Commun. Math. 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Stat.}, + FJOURNAL = {Communications in Mathematics and Statistics}, + VOLUME = {5}, + YEAR = {2017}, + NUMBER = {4}, + PAGES = {349--380}, + ISSN = {2194-6701}, + MRCLASS = {65M75 (60H35 65C30)}, + MRNUMBER = {3736669}, + MRREVIEWER = {Samuel Bernard}, + DOI = {10.1007/s40304-017-0117-6}, + URL = {https://doi.org/10.1007/s40304-017-0117-6}, +} + +@article{BeckerCheridito2019, + author = {Sebastian Becker and Patrick Cheridito and Arnulf Jentzen}, + title = {Deep Optimal Stopping}, + fjournal = {Journal of Machine Learning Research}, + journal={J. Mach. Learn. Res.}, + year = {2019}, + volume = {20}, + number = {74}, + pages = {1--25}, + url = {http://jmlr.org/papers/v20/18-232.html} +} + + + + + +@article{Kolmogorov, + AUTHOR = {Beck, Christian and Becker, Sebastian and Grohs, Philipp and + Jaafari, Nor and Jentzen, Arnulf}, + TITLE = {Solving the {K}olmogorov {PDE} by means of deep learning}, + JOURNAL = {J. Sci. Comput.}, + FJOURNAL = {Journal of Scientific Computing}, + VOLUME = {88}, + YEAR = {2021}, + NUMBER = {3}, + PAGES = {Paper No. 73, 28}, + ISSN = {0885-7474}, + MRCLASS = {65C30 (35R60 60H10 62M45 65M75)}, + MRNUMBER = {4293960}, + DOI = {10.1007/s10915-021-01590-0}, + URL = {https://doi.org/10.1007/s10915-021-01590-0}, +} + +@inproceedings{Zhai2016, + author = {Zhai, Shuangfei and Chang, Keng-hao and Zhang, Ruofei and Zhang, Zhongfei Mark}, + title = {DeepIntent: Learning Attentions for Online Advertising with Recurrent Neural Networks}, + booktitle = {Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining}, + year = {2016}, + isbn = {978-1-4503-4232-2}, + pages = {1295--1304}, + url = {http://doi.acm.org/10.1145/2939672.2939759}, + doi = {10.1145/2939672.2939759}, + keywords = {BM25, RNN, attention mechanism, deep learning, online advertising, query rewriting, sponsored search}, +} + + +@inproceedings{Wang2017Ad, + author = {Wang, Ruoxi and Fu, Bin and Fu, Gang and Wang, Mingliang}, + title = {Deep \& Cross Network for Ad Click Predictions}, + booktitle = {Proceedings of the ADKDD'17}, + year = {2017}, + keywords = {CTR Prediction, Deep Learning, Feature Crossing, Neural Networks}, +} + +@INPROCEEDINGS{Roy2018, +author={A. 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Beling}, +booktitle={2018 Systems and Information Engineering Design Symposium (SIEDS)}, +title={Deep learning detecting fraud in credit card transactions}, +year={2018}, +volume={}, +number={}, +pages={129-134}, +keywords={cloud computing;credit transactions;data mining;fraud;learning (artificial intelligence);neural nets;regression analysis;security of data;North American financial institutions;fraudulent activity;credit card fraud detection;real-time transaction details;prevailing fraud detection methods;common fraud detection problems;digital payment systems;deep learning;artificial neural network;credit card transactions;distributed cloud computing environment;Machine learning;Credit cards;Topology;Artificial neural networks;Computer architecture;Network topology;Hidden Markov models;Deep Learning;Fraud Detection;Neural Networks;Sensitivity Analysis},} + + +@article{CHOUIEKH2018, +title = "ConvNets for Fraud Detection analysis", +journal = "Procedia Computer Science", +volume = "127", +pages = "133--138", +year = "2018", +author = "Alae Chouiekh and EL Hassane Ibn EL Haj", +keywords = "Fraud, Machine learning, Deep learning, CNN" +} + +@inproceedings{wu2016stimulated, +author={Chunyang Wu and Penny Karanasou and Mark J.F. Gales and Khe Chai Sim}, +title={Stimulated Deep Neural Network for Speech Recognition}, +year=2016, +booktitle={Interspeech 2016}, +doi={10.21437/Interspeech.2016-580}, +url={http://dx.doi.org/10.21437/Interspeech.2016-580}, +pages={400--404} +} + + + + + +@article{JentzenSalimovaWelti2018, + title={A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of {K}olmogorov partial differential equations with constant diffusion and nonlinear drift coefficients}, + author={Jentzen, Arnulf and Salimova, Diyora and Welti, Timo}, + journal={arXiv:1809.07321}, + year={2018}, + pages={48 pages} +} + + + +@Article{jp15, + Title = {{Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities}}, + Author = {{Jentzen}, A. and {Pu{\v s}nik}, P.}, + Journal={arXiv:1504.03523}, + Year = {2015}, + Pages = {38 pages}, + Note = {To appear in \emph{IMA J.\ Num.\ Anal.}}, +} + + + +@article{beckergess2017, + title={Strong convergence rates for explicit space-time discrete numerical approximations of stochastic {A}llen-{C}ahn equations}, + author={Becker, Sebastian and Gess, Benjamin and Jentzen, Arnulf and Kloeden, Peter E}, + journal={arXiv:1711.02423}, + year={2017} +} + + + + +@article {Yaroslavtseva2017, + AUTHOR = {Yaroslavtseva, Larisa}, + TITLE = {On non-polynomial lower error bounds for adaptive strong + approximation of {SDE}s}, + JOURNAL = {J. Complexity}, + FJOURNAL = {Journal of Complexity}, + VOLUME = {42}, + YEAR = {2017}, + PAGES = {1--18}, + ISSN = {0885-064X}, + MRCLASS = {65C30 (65Y20)}, + MRNUMBER = {3683255}, +MRREVIEWER = {Judith M. Ford}, + DOI = {10.1016/j.jco.2017.04.002}, + URL = {https://doi.org/10.1016/j.jco.2017.04.002}, +} + +@article {GerencserJentzen2017, + AUTHOR = {Gerencs\'{e}r, M\'{a}t\'{e} and Jentzen, Arnulf and Salimova, Diyora}, + TITLE = {On stochastic differential equations with arbitrarily slow + convergence rates for strong approximation in two space + dimensions}, + JOURNAL = {Proc. A.}, + FJOURNAL = {Proceedings A}, + VOLUME = {473}, + YEAR = {2017}, + NUMBER = {2207}, + PAGES = {20170104, 16}, + ISSN = {1364-5021}, + MRCLASS = {60H10 (34F05)}, + MRNUMBER = {3741822}, +MRREVIEWER = {Daniel C. Biles}, + DOI = {10.1098/rspa.2017.0104}, + URL = {https://doi.org/10.1098/rspa.2017.0104}, +} + +@article {QiWang2017, + AUTHOR = {Qi, Ruisheng and Wang, Xiaojie}, + TITLE = {An accelerated exponential time integrator for semi-linear + stochastic strongly damped wave equation with additive noise}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {447}, + YEAR = {2017}, + NUMBER = {2}, + PAGES = {988--1008}, + ISSN = {0022-247X}, + MRCLASS = {35R60 (35A35 35L35 35L76)}, + MRNUMBER = {3573128}, + DOI = {10.1016/j.jmaa.2016.09.052}, + URL = {https://doi.org/10.1016/j.jmaa.2016.09.052}, +} + +@article {MullerGronbachYaroslavtseva2018, + AUTHOR = {M\"{u}ller-Gronbach, Thomas and Yaroslavtseva, Larisa}, + TITLE = {A note on strong approximation of {SDE}s with smooth + coefficients that have at most linearly growing derivatives}, + JOURNAL = {J. Math. Anal. Appl.}, + FJOURNAL = {Journal of Mathematical Analysis and Applications}, + VOLUME = {467}, + YEAR = {2018}, + NUMBER = {2}, + PAGES = {1013--1031}, + ISSN = {0022-247X}, + MRCLASS = {60H10 (35R60 60H35)}, + MRNUMBER = {3842417}, + DOI = {10.1016/j.jmaa.2018.07.041}, + URL = {https://doi.org/10.1016/j.jmaa.2018.07.041}, +} + +@Article{DaPratoJentzenRoeckner_Mild, + Title = {A mild {I}t\^{o} formula for {SPDE}s}, + Author = {Giuseppe {Da Prato} and Arnulf Jentzen and Michael R{\"o}ckner}, + Journal = {arXiv:1009.3526}, + Year = {2010}, + Pages = {39 pages}, + Note = {To appear in \textit{Trans.~Amer.~Math.~Soc.}} +} + +@book{Hytonen2016analysis, + AUTHOR={Hyt{\"o}nen, Tuomas and van Neerven, Jan and Veraar, Mark and Weis, Lutz}, + TITLE={Analysis in Banach Spaces: Volume I: Martingales and Littlewood-Paley Theory}, + PUBLISHER={Springer}, + YEAR={2016} +} + + +@Article{GerencserJentzenSalimova2017, + Title = {On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions}, + Author = {{Gerencs\'er}, M. and {Jentzen}, A. and {Salimova}, D. }, + Journal = {arXiv:1702.03229}, + Year = {2017}, + Pages = {25 pages}, +} + +@Article{MullerGronbachYaroslavtseva2017, +author = {Thomas M{\"u}ller-Gronbach and Larisa Yaroslavtseva}, +title = {On sub-polynomial lower error bounds for quadrature of {SDE}s with bounded smooth coefficients}, +journal = {Stochastic Analysis and Applications}, +volume = {0}, +number = {0}, +pages = {1-29}, +year = {2017}, +doi = {10.1080/07362994.2016.1263157} +} + + + + + + + + +@article{HutzenthalerJentzenKruseNguyen2020, + title={{Multilevel Picard approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities}}, + author={Martin Hutzenthaler and Arnulf Jentzen and Thomas Kruse and Tuan Anh Nguyen}, + journal={arXiv:2009.02484}, + year={2020}, + pages={37 pages}, +} + + + + +@article{GrohsHerrmann2020, + title={Deep neural network approximation for high-dimensional elliptic {PDE}s with boundary conditions}, + author={Philipp Grohs and Lukas Herrmann}, + year={2020}, + journal={arXiv:2007.05384}, + archivePrefix={arXiv}, + primaryClass={math.NA}, + pages = {22 pages} +} + + +@article{HornungJentzenSalimova2020, + title={Space-time deep neural network approximations for high-dimensional partial differential equations}, + author={Fabian Hornung and Arnulf Jentzen and Diyora Salimova}, + year={2020}, + journal={arXiv:2006.02199}, + archivePrefix={arXiv}, + pages={52 pages}, + primaryClass={math.PR} +} + +@Article{Yaroslavtseva2016, + Title = {On non-polynomial lower error bounds for adaptive strong approximation of {SDE}s}, + Author = {{Yaroslavtseva}, L.}, + Journal = {arXiv:1609.08073}, + Year = {2016}, + Pages = {19 pages}, +} + +@Article{MullerGronbachYaroslavtseva2016, + Title = {On hard quadrature problems for marginal distributions of {SDE}s with bounded smooth coefficients}, + Author = {{M{\"u}ller-Gronbach}, T. and {Yaroslavtseva}, L.}, + Journal = {arXiv:1603.08686}, + Year = {2016}, + Pages = {38 pages}, +} + +@Article{Muller2017, + Title = {On sub-polynomial error bounds for quadrature of {SDE}s with bounded smooth coefficients}, + Author = {M{\"u}ller-Gronbach, Thomas and Yaroslavtseva, Larisa}, + Journal = {Stoch. 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We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong non-linearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.}, + doi = {10.1093/rfs/9.2.385}, + eprint = {http://rfs.oxfordjournals.org/content/9/2/385.full.pdf+html}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://rfs.oxfordjournals.org/content/9/2/385.abstract}, +} + +@Article{Akamatsu1988, + author = {Akamatsu, Toyohiro}, + journal = {Osaka J. 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Summer School held in Cetraro, August 29--September 3, 2005, Edited by Giuseppe Da Prato and Michael R{\"o}ckner}, + series = {Lecture Notes in Mathematics}, + volume = {1942}, + doi = {10.1007/978-3-540-78493-7}, + groups = {sdiyora:6}, + mrclass = {76M35 (60-06 60H15 76-06 76D05)}, + mrnumber = {2459087 (2009g:76120)}, + owner = {sdiyora}, + pages = {viii+166}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-3-540-78493-7}, +} + +@Article{Albeverio1991, + author = {Sergio Albeverio and Zbignew Haba and Francesco Russo}, + journal = {Probab. Theory Related Fields}, + title = {A two-space dimensional semilinear heat equation perturbed by (Gaussian) white noise}, + year = {1991}, + number = {3}, + pages = {347--386}, + volume = {89}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Albeverio1991a, + author = {Sergio Albeverio and Michael R{\"o}ckner}, + journal = {Probab. 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Y. and Danchin, R.}, + publisher = {Springer}, + title = {Fourier analysis and nonlinear partial differential equations}, + year = {2011}, + volume = {343}, + file = {Snapshot:/Users/Hao/Library/Application Support/Zotero/Profiles/qg7ecoj6.default/zotero/storage/54M2I3UQ/books.html:text/html}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://books.google.com/books?hl=en\&lr=\&id=CcTnaveQkn0C\&oi=fnd\&pg=PR7\&dq=H.+BAHOURI,+J.+CHEMIN,+and+R.+DANCHIN.+Fourier+analysis+and+nonlinear+partial+differential+equations\&ots=FL9lit60VE\&sig=lelyO-4nVVAjcJxDACaOI6zrxZ8}, + urldate = {2013-01-21}, +} + +@Article{Bakshi1997, + author = {Bakshi, Gurdip and Cao, Charles and Chen, Zhiwu}, + journal = {The Journal of Finance}, + title = {Empirical Performance of Alternative Option Pricing Models}, + year = {1997}, + number = {5}, + pages = {pp. 2003-2049}, + volume = {52}, + copyright = {Copyright © 1997 American Finance Association}, + groups = {sdiyora:6}, + jstor_formatteddate = {Dec., 1997}, + owner = {sdiyora}, + publisher = {Wiley for the American Finance Association}, + timestamp = {2016.02.02}, + url = {http://www.jstor.org/stable/2329472}, +} + +@Article{Baldeaux2014, + author = {Baldeaux, Jan and Badran, Alexander}, + journal = {Appl. 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Optim.}, + title = {Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear {S}chr{\"o}dinger equation}, + year = {2006}, + issn = {0095-4616}, + number = {3}, + pages = {369--399}, + volume = {54}, + coden = {AMOMBN}, + fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H35 (35R60 60H15 65C30)}, + mrnumber = {MR2268663 (2008g:60208)}, + mrreviewer = {Carlos M. Mora Gonz{\'a}lez}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{dd06, + Title = {Weak and strong order of convergence of a semidiscrete scheme + for the stochastic nonlinear {S}chr\"odinger equation}, + Author = {de Bouard, Anne and Debussche, Arnaud}, + Journal = {Appl. Math. Optim.}, + Year = {2006}, + Number = {3}, + Pages = {369--399}, + Volume = {54}, + + Coden = {AMOMBN}, + Fjournal = {Applied Mathematics and Optimization. An International Journal + with Applications to Stochastics}, + ISSN = {0095-4616}, + Mrclass = {60H35 (35R60 60H15 65C30)}, + Mrnumber = {MR2268663 (2008g:60208)}, + Mrreviewer = {Carlos M. Mora Gonz{\'a}lez} +} + +@Article{Bouard2004, + author = {A. de Bouard and A. Debussche}, + journal = {Numer. Math.}, + title = {A semi-discrete scheme for the stochastic nonlinear {S}chr{\"o}dinger equation}, + year = {2004}, + issn = {0029-599X}, + number = {4}, + pages = {733--770}, + volume = {96}, + coden = {NUMMA7}, + fjournal = {Numerische Mathematik}, + groups = {sdiyora:6}, + mrclass = {60H15 (35Q55 35R60 65M12)}, + mrnumber = {MR2036364 (2004m:60134)}, + mrreviewer = {Krystyna Twardowska}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{dd04, + Title = {A semi-discrete scheme for the stochastic nonlinear + {S}chr\"odinger equation}, + Author = {de Bouard, A. and Debussche, A.}, + Journal = {Numer. Math.}, + Year = {2004}, + Number = {4}, + Pages = {733--770}, + Volume = {96}, + + Coden = {NUMMA7}, + Fjournal = {Numerische Mathematik}, + ISSN = {0029-599X}, + Mrclass = {60H15 (35Q55 35R60 65M12)}, + Mrnumber = {MR2036364 (2004m:60134)}, + Mrreviewer = {Krystyna Twardowska} +} + +@InCollection{Bouard2001, + author = {Anne de Bouard and Arnaud Debussche and Laurent {Di Menza}}, + booktitle = {Journ{\'e}es ``{\'E}quations aux {D}{\'e}riv{\'e}es {P}artielles'' ({P}lestin-les-{G}r{\`e}ves, 2001)}, + publisher = {Univ. Nantes}, + title = {Theoretical and numerical aspects of stochastic nonlinear {S}chr{\"o}dinger equations}, + year = {2001}, + address = {Nantes}, + pages = {Exp. No. III, 13}, + groups = {sdiyora:6}, + mrclass = {60H15 (35B40 35R60 65C30)}, + mrnumber = {MR1843404 (2002i:60117)}, + mrreviewer = {Hongjun Gao}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Bouard2001a, + author = {A. de Bouard and A. Debussche and L. {Di Menza}}, + journal = {Monte Carlo Methods Appl.}, + title = {Theoretical and numerical aspects of stochastic nonlinear {S}chr{\"o}dinger equations}, + year = {2001}, + issn = {0929-9629}, + note = {Monte Carlo and probabilistic methods for partial differential equations (Monte Carlo, 2000)}, + number = {1-2}, + pages = {55--63}, + volume = {7}, + fjournal = {Monte Carlo Methods and Applications}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H35)}, + mrnumber = {MR1828196 (2002b:65014)}, + mrreviewer = {Madalina Deaconu}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{ddd01, + Title = {Theoretical and numerical aspects of stochastic nonlinear + {S}chr\"odinger equations}, + Author = {de Bouard, Anne and Debussche, Arnaud and Di Menza, Laurent}, + Booktitle = {Journ\'ees ``\'{E}quations aux {D}\'eriv\'ees {P}artielles'' + ({P}lestin-les-{G}r\`eves, 2001)}, + Publisher = {Univ. Nantes}, + Year = {2001}, + + Address = {Nantes}, + Pages = {Exp. No. III, 13}, + + Mrclass = {60H15 (35B40 35R60 65C30)}, + Mrnumber = {MR1843404 (2002i:60117)}, + Mrreviewer = {Hongjun Gao} +} + +@Article{ddd01b, + Title = {Theoretical and numerical aspects of stochastic nonlinear + {S}chr\"odinger equations}, + Author = {de Bouard, A. and Debussche, A. and Di Menza, L.}, + Journal = {Monte Carlo Methods Appl.}, + Year = {2001}, + Note = {Monte Carlo and probabilistic methods for partial differential + equations (Monte Carlo, 2000)}, + Number = {1-2}, + Pages = {55--63}, + Volume = {7}, + + Fjournal = {Monte Carlo Methods and Applications}, + ISSN = {0929-9629}, + Mrclass = {65C30 (60H35)}, + Mrnumber = {MR1828196 (2002b:65014)}, + Mrreviewer = {Madalina Deaconu} +} + +@Article{Bouchard2004, + author = {Bouchard, Bruno and Touzi, Nizar}, + journal = {Stochastic Processes and their applications}, + title = {Discrete-time approximation and {Monte-Carlo} simulation of backward stochastic differential equations}, + year = {2004}, + number = {2}, + pages = {175--206}, + volume = {111}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Elsevier}, + timestamp = {2016.02.02}, +} + +@Article{Bouchaud1999, + author = {Jean-Philippe Bouchaud and Rama Cont and Nicole El-Karoui and Marc Potters and Nicolas Sagna and Centre D'etudes De Saclay}, + journal = {Applied Mathematical Finance}, + title = {Phenomenology Of The Interest Rate Curve}, + year = {1999}, + pages = {209--231}, + volume = {6}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Bouchaud1998, + author = {Jean-Philippe Bouchaud and Rama Cont and Nicole El-Karoui and Marc Potters and Nicolas Sagna and Centre D'etudes De Saclay}, + journal = {RISK}, + title = {Strings attached}, + year = {1998}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Boufoussi2005, + author = {Brahim Boufoussi and Ciprian A. Tudor}, + journal = {Rev. Roumaine Math. Pures Appl.}, + title = {Kramers-{S}moluchowski approximation for stochastic evolution equations with {FBM}}, + year = {2005}, + issn = {0035-3965}, + number = {2}, + pages = {125--136}, + volume = {50}, + fjournal = {Revue Roumaine de Math{\'e}matiques Pures et Appliqu{\'e}es. Romanian Journal of Pure and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H20 (60H10)}, + mrnumber = {MR2156107 (2006m:60086)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{bt05, + Title = {Kramers-{S}moluchowski approximation for stochastic evolution + equations with {FBM}}, + Author = {Boufoussi, Brahim and Tudor, Ciprian A.}, + Journal = {Rev. Roumaine Math. Pures Appl.}, + Year = {2005}, + Number = {2}, + Pages = {125--136}, + Volume = {50}, + + Fjournal = {Revue Roumaine de Math\'ematiques Pures et Appliqu\'ees. + Romanian Journal of Pure and Applied Mathematics}, + ISSN = {0035-3965}, + Mrclass = {60H20 (60H10)}, + Mrnumber = {MR2156107 (2006m:60086)} +} + +@Article{Bou-Rabee2013, + author = {Nawaf Bou-Rabee and Martin Hairer}, + journal = {IMA J. Numer. Anal.}, + title = {Non-asymptotic mixing of the {MALA} algorithm}, + year = {2013}, + pages = {80--110}, + fjournal = {IMA Journal of Numerical Analysis}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Bou-Rabee2010, + author = {Nawaf Bou-Rabee and Eric Vanden-Eijnden}, + journal = {Comm. Pure Appl. Math.}, + title = {Pathwise accuracy and ergodicity of metropolized integrators for {SDE}s}, + year = {2010}, + issn = {0010-3640}, + number = {5}, + pages = {655--696}, + volume = {63}, + coden = {CPAMA}, + fjournal = {Communications on Pure and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H10 (60H35)}, + mrnumber = {2583309}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Brehier2014, + author = {Charles-Edouard Br\'{e}hier}, + journal = {Potential Analysis}, + title = {Approximation of the Invariant Measure with an {E}uler Scheme for Stochastic {PDE}s Driven by Space-Time White Noise}, + year = {2014}, + issn = {0926-2601}, + number = {1}, + pages = {1-40}, + volume = {40}, + doi = {10.1007/s11118-013-9338-9}, + groups = {sdiyora:6}, + keywords = {Stochastic partial differential equations; Invariant measures and Ergodicity; Weak approximation; Euler scheme; 35A40; 37L40; 60H15; 60H35}, + language = {English}, + owner = {sdiyora}, + publisher = {Springer Netherlands}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s11118-013-9338-9}, +} + +@Article{Brehier2012, + author = {Charles-Edouard Br\'{e}hier}, + journal = {arXiv:1202.2707v1}, + title = {Approximation of the invariant measure with an {E}uler scheme for Stochastic {PDE}'s driven by {S}pace-{T}ime {W}hite Noise}, + year = {2012}, + pages = {31 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Brehier2013, + author = {Charles-Edouard Br\'{e}hier and Marie Kopec}, + journal = {arXiv:1311.7030}, + title = {Approximation of the invariant law of {SPDE}s: error analysis using a Poisson equation for a full-discretization scheme}, + year = {2013}, + pages = {50 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{b97, + Title = {Approximation of the solution of a stochastic evolution + equation}, + Author = {Breckner, Hannelore}, + Journal = {Studia Univ. 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Finance}, + title = {Tracking errors from discrete hedging in exponential {L}\'evy models}, + year = {2011}, + issn = {0219-0249}, + number = {6}, + pages = {803--837}, + volume = {14}, + doi = {10.1142/S0219024911006760}, + fjournal = {International Journal of Theoretical and Applied Finance}, + groups = {sdiyora:6}, + mrclass = {91G10 (60J75 91G20)}, + mrnumber = {2845218}, + mrreviewer = {Jean-Luc Prigent}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1142/S0219024911006760}, +} + +@Article{Bruti-Liberati2007, + author = {Bruti-Liberati, Nicola and Platen, Eckhard}, + journal = {J. Comput. Appl. 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Math.}, + volume = {245}, + doi = {10.1201/9781420028720.ch5}, + groups = {sdiyora:6}, + mrclass = {60H15 (35Q53 35R60 37H10 37L55 76D05 76M35)}, + mrnumber = {2227218 (2007k:60178)}, + mrreviewer = {Marek Capi{\'n}ski}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1201/9781420028720.ch5}, +} + +@Article{Brzezniak1997, + author = {Brze{\'z}niak, Zdzis{\l}aw}, + journal = {Stochastics Stochastics Rep.}, + title = {On stochastic convolution in {B}anach spaces and applications}, + year = {1997}, + issn = {1045-1129}, + number = {3-4}, + pages = {245--295}, + volume = {61}, + coden = {STOCBS}, + fjournal = {Stochastics and Stochastics Reports}, + groups = {sdiyora:6}, + mrclass = {35R60 (47D07 60H15)}, + mrnumber = {1488138}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Brzezniak1992, + author = {Brze{\'z}niak, Z. and Capi{\'n}ski, M. and Flandoli, F.}, + journal = {Stochastic Anal. 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Appl.}, + title = {Almost sure approximation of {W}ong-{Z}akai type for stochastic partial differential equations}, + year = {1995}, + issn = {0304-4149}, + number = {2}, + pages = {329--358}, + volume = {55}, + coden = {STOPB7}, + fjournal = {Stochastic Processes and their Applications}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R60)}, + mrnumber = {MR1313027 (96b:60156)}, + mrreviewer = {Krystyna Twardowska}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Brzezniak2014, + author = {Brze{\'z}niak, Zdzis{\l}aw and Liu, Wei and Zhu, Jiahui}, + journal = {Nonlinear Anal. Real World Appl.}, + title = {Strong solutions for {SPDE} with locally monotone coefficients driven by {L}\'evy noise}, + year = {2014}, + issn = {1468-1218}, + pages = {283--310}, + volume = {17}, + doi = {10.1016/j.nonrwa.2013.12.005}, + fjournal = {Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal}, + groups = {sdiyora:6}, + mrclass = {35R60 (35D35 60H15)}, + mrnumber = {3158475}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.nonrwa.2013.12.005}, +} + +@Article{Brzezniak2003, + author = {Zdzis{\l}aw Brze{\'z}niak and Jan van Neerven}, + journal = {J. Math. Kyoto Univ.}, + title = {Space-time regularity for linear stochastic evolution equations driven by spatially homogeneous noise}, + year = {2003}, + issn = {0023-608X}, + number = {2}, + pages = {261--303}, + volume = {43}, + coden = {JMKYAZ}, + fjournal = {Journal of Mathematics of Kyoto University}, + groups = {sdiyora:6}, + mrclass = {60H15 (35B65 35R60 47D06 60G15 60H05)}, + mrnumber = {MR2051026 (2005c:60077)}, + mrreviewer = {Isamu D{\^o}ku}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{bv03, + Title = {Space-time regularity for linear stochastic evolution + equations driven by spatially homogeneous noise}, + Author = {Brze{\'z}niak, Zdzis{\l}aw and van Neerven, Jan}, + Journal = {J. Math. Kyoto Univ.}, + Year = {2003}, + Number = {2}, + Pages = {261--303}, + Volume = {43}, + + Coden = {JMKYAZ}, + Fjournal = {Journal of Mathematics of Kyoto University}, + ISSN = {0023-608X}, + Mrclass = {60H15 (35B65 35R60 47D06 60G15 60H05)}, + Mrnumber = {MR2051026 (2005c:60077)}, + Mrreviewer = {Isamu D{\^o}ku} +} + +@Article{Brzezniak2008, + author = {Z. Brze{\'z}niak and J. M. A. M. van Neerven and M. C. Veraar and L. Weis}, + journal = {J. Differential Equations}, + title = {It{\^o}'s formula in {UMD} {B}anach spaces and regularity of solutions of the {Z}akai equation}, + year = {2008}, + issn = {0022-0396}, + number = {1}, + pages = {30--58}, + volume = {245}, + coden = {JDEQAK}, + doi = {10.1016/j.jde.2008.03.026}, + fjournal = {Journal of Differential Equations}, + groups = {sdiyora:6}, + mrclass = {60B11 (28C20 35R60 46N30)}, + mrnumber = {MR2422709 (2010c:60014)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.jde.2008.03.026}, +} + +@Article{Buckdahn2010, + author = {Rainer Buckdahn and Ingo Bulla and Jin Ma}, + journal = {arXiv:1008.3221v1}, + title = {Pathwise {T}aylor {E}xpansions for {I}t{\^o} {R}andom {F}ields}, + year = {2010}, + pages = {38 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{bm02, + Title = {Pathwise stochastic {T}aylor expansions and stochastic + viscosity solutions for fully nonlinear stochastic {PDE}s}, + Author = {Buckdahn, Rainer and Ma, Jin}, + Journal = {Ann. 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Probab.}, + title = {Pathwise stochastic {T}aylor expansions and stochastic viscosity solutions for fully nonlinear stochastic {PDE}s}, + year = {2002}, + issn = {0091-1798}, + number = {3}, + pages = {1131--1171}, + volume = {30}, + coden = {APBYAE}, + fjournal = {The Annals of Probability}, + groups = {sdiyora:6}, + mrclass = {60H07 (60H15 60H30)}, + mrnumber = {MR1920103 (2003e:60116)}, + mrreviewer = {Rimas Norvai\v{s}a}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Bungartz1999, + author = {Bungartz, Hans-Joachim and Griebel, Michael}, + journal = {Journal of Complexity}, + title = {A note on the complexity of solving {P}oisson's equation for spaces of bounded mixed derivatives}, + year = {1999}, + number = {2}, + pages = {167--199}, + volume = {15}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Elsevier}, + timestamp = {2016.02.02}, +} + +@Article{bb00, + Title = {Order conditions of stochastic {R}unge-{K}utta methods by + {$B$}-series}, + Author = {Burrage, K. and Burrage, P. M.}, + Journal = {SIAM J. Numer. Anal.}, + Year = {2000}, + Number = {5}, + Pages = {1626--1646 (electronic)}, + Volume = {38}, + + Fjournal = {SIAM Journal on Numerical Analysis}, + ISSN = {0036-1429}, + Mrclass = {65C30 (65L06)}, + Mrnumber = {MR1813248 (2001m:65016)}, + Mrreviewer = {Alexander Ostermann} +} + +@Article{Burrage1996, + author = {K. Burrage and P. M. Burrage}, + journal = {Appl. Numer. Math.}, + title = {High strong order explicit {R}unge-{K}utta methods for stochastic ordinary differential equations}, + year = {1996}, + issn = {0168-9274}, + note = {Special issue celebrating the centenary of Runge-Kutta methods}, + number = {1-3}, + pages = {81--101}, + volume = {22}, + coden = {ANMAEL}, + doi = {10.1016/S0168-9274(96)00027-X}, + fjournal = {Applied Numerical Mathematics. An IMACS Journal}, + groups = {sdiyora:6}, + mrclass = {65L06 (34A45 34F05 60H10)}, + mrnumber = {1424291 (97k:65166)}, + mrreviewer = {E. S. Lee}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/S0168-9274(96)00027-X}, +} + +@Article{Burrage2004, + author = {Burrage, K. and Burrage, P. M. and Tian, T.}, + journal = {Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, + title = {Numerical methods for strong solutions of stochastic differential equations: an overview}, + year = {2004}, + issn = {1364-5021}, + note = {Stochastic analysis with applications to mathematical finance}, + number = {2041}, + pages = {373--402}, + volume = {460}, + doi = {10.1098/rspa.2003.1247}, + fjournal = {Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H10)}, + mrnumber = {MR2052268 (2005c:65006)}, + mrreviewer = {Mikhail V. Tretyakov}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1098/rspa.2003.1247}, +} + +@Article{Burrage1994, + author = {K. Burrage and E. Platen}, + journal = {Ann. Numer. Math.}, + title = {Runge-{K}utta methods for stochastic differential equations}, + year = {1994}, + issn = {1021-2655}, + note = {Scientific computation and differential equations (Auckland, 1993)}, + number = {1-4}, + pages = {63--78}, + volume = {1}, + fjournal = {Annals of Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H10 (34F05 65L06 65U05)}, + mrnumber = {1340645 (96g:60075)}, + mrreviewer = {E. S. Lee}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cahn1959, + author = {John W. Cahn}, + journal = {J. Chem. Phys.}, + title = {Free energy of a nonuniform system. II. Thermodynamic basis}, + year = {1959}, + pages = {1121--1124}, + volume = {30}, + fjournal = {Journal of Chemical Physics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cahn1958, + author = {John W. Cahn and John E. Hilliard}, + journal = {J. Chem. Phys.}, + title = {Free energy of a nonuniform system. I. Interfacial free energy}, + year = {1958}, + number = {258}, + pages = {1121--1124}, + volume = {28}, + fjournal = {Journal of Chemical Physics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cambanis1996, + author = {Cambanis, Stamatis and Hu, Yaozhong}, + journal = {Stochastics Stochastics Rep.}, + title = {Exact convergence rate of the {E}uler-{M}aruyama scheme, with application to sampling design}, + year = {1996}, + issn = {1045-1129}, + number = {3-4}, + pages = {211--240}, + volume = {59}, + coden = {STOCBS}, + fjournal = {Stochastics and Stochastics Reports}, + groups = {sdiyora:6}, + mrclass = {60H10}, + mrnumber = {1427739 (97k:60159)}, + mrreviewer = {Eckhard Platen}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{ck06, + Title = {{T}he {P}athwise {N}umerical {A}pproximation of {S}tationary {S}olutions of {S}emilinear {S}tochastic {E}volution {E}quations}, + Author = {Caraballo, T. and Kloeden, P.E.}, + Journal = {Applied Mathematics and Optimization}, + Year = {2006}, + Number = {3}, + Pages = {401-415}, + Volume = {54}, + + Publisher = {Springer-Verlag} +} + +@Article{Caraballo2006, + author = {Caraballo, T. and Kloeden, P. E.}, + journal = {Appl. Math. Optim.}, + title = {The pathwise numerical approximation of stationary solutions of semilinear stochastic evolution equations}, + year = {2006}, + issn = {0095-4616}, + number = {3}, + pages = {401--415}, + volume = {54}, + coden = {AMOMBN}, + doi = {10.1007/s00245-006-0876-z}, + fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H35 (35R60 37H10 60H15 65C30)}, + mrnumber = {2268664 (2007m:60198)}, + mrreviewer = {Constantin Tudor}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s00245-006-0876-z}, +} + +@Article{Carelli2011, + author = {Carelli, Erich and Prohl, Andreas}, + journal = {Preprint}, + title = {Rates of convergence for discretizations of the stochastic incompressible {N}avier-{S}tokes equations}, + year = {2011}, + note = {available at http://na.uni-tuebingen.de/preprints.shtml}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Carelli2009, + author = {Carelli, Erich and Prohl, Andreas}, + journal = {J. Comput. Math.}, + title = {A note on pressure approximation of first and higher order projection schemes for the nonstationary incompressible {N}avier-{S}tokes equations}, + year = {2009}, + issn = {0254-9409}, + number = {2-3}, + pages = {338--347}, + volume = {27}, + coden = {JCMMEB}, + fjournal = {Journal of Computational Mathematics}, + groups = {sdiyora:6}, + mrclass = {65M60 (35Q30 76D05)}, + mrnumber = {2495064 (2010c:65174)}, + mrreviewer = {Jordi Blasco}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Carmona1994, + author = {Carmona, Ren{\'e} A. and Molchanov, S. A.}, + journal = {Mem. Amer. Math. Soc.}, + title = {Parabolic {A}nderson problem and intermittency}, + year = {1994}, + issn = {0065-9266}, + number = {518}, + pages = {viii+125}, + volume = {108}, + coden = {MAMCAU}, + doi = {10.1090/memo/0518}, + fjournal = {Memoirs of the American Mathematical Society}, + groups = {sdiyora:6}, + mrclass = {35K10 (35-02 35R60 60H25 82B44 82C44)}, + mrnumber = {1185878 (94h:35080)}, + mrreviewer = {Stanis{\l}aw W{\polhk{e}}drychowicz}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1090/memo/0518}, +} + +@Article{Carmona2009, + author = {Ren{\'e} Carmona and Lixin Wang}, + journal = {SIAM J. Appl. Math.}, + title = {Monte {C}arlo {M}alliavin computation of the sensitivities of solutions of {SPDE}s}, + year = {2009}, + issn = {0036-1399}, + number = {6}, + pages = {1682--1711}, + volume = {69}, + doi = {10.1137/050630519}, + fjournal = {SIAM Journal on Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H07 (35R60 60B11 60H15 65C05)}, + mrnumber = {2496713 (2010f:60165)}, + mrreviewer = {Roger Pettersson}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/050630519}, +} + +@Article{Cass2006, + author = {Thomas R. Cass and Peter K. Friz}, + journal = {arXiv:math/0604311}, + title = {The {B}ismut-{E}lworthy-{L}i formula for jump-diffusions and applications to {M}onte {C}arlo pricing in finance}, + year = {2006}, + pages = {1--30}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{ch98, + Title = {An introduction to semilinear evolution equations}, + Author = {Cazenave, Thierry and Haraux, Alain}, + Publisher = {The Clarendon Press Oxford University Press}, + Year = {1998}, + + Address = {New York}, + Note = {Translated from the 1990 French original by Yvan Martel and + revised by the authors}, + Series = {Oxford Lecture Series in Mathematics and its Applications}, + Volume = {13}, + + ISBN = {0-19-850277-X}, + Mrclass = {35-02 (34G20 35-01 47H20 47N20)}, + Mrnumber = {MR1691574 (2000e:35003)}, + Mrreviewer = {Sergiu Aizicovici}, + Pages = {xiv+186} +} + +@Article{Cerrai2003, + author = {Cerrai, Sandra}, + journal = {Probab. Theory Related Fields}, + title = {Stochastic reaction-diffusion systems with multiplicative noise and non-{L}ipschitz reaction term}, + year = {2003}, + issn = {0178-8051}, + number = {2}, + pages = {271--304}, + volume = {125}, + coden = {PTRFEU}, + doi = {10.1007/s00440-002-0230-6}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K57 35R60 47N30)}, + mrnumber = {1961346 (2004a:60117)}, + mrreviewer = {B. G. Pachpatte}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s00440-002-0230-6}, +} + +@Book{Cerrai2001, + Title = {Second order {PDE}'s in finite and infinite dimension}, + Author = {Cerrai, Sandra}, + Publisher = {Springer-Verlag, Berlin}, + Year = {2001}, + Note = {A probabilistic approach}, + Series = {Lecture Notes in Mathematics}, + Volume = {1762}, + + Doi = {10.1007/b80743}, + ISBN = {3-540-42136-X}, + Mrclass = {35R60 (35K57 47H20 47N20 60H10 60H15 60J35)}, + Mrnumber = {1840644 (2002j:35327)}, + Mrreviewer = {Krystyna Twardowska}, + Pages = {x+330}, + Url = {http://dx.doi.org/10.1007/b80743} +} + +@Article{Chancelier2002, + author = {Chancelier, Jean-Philippe and {\O}ksendal, Bernt and Sulem, Agn{\`e}s}, + journal = {Tr. Mat. Inst. Steklova}, + title = {Combined stochastic control and optimal stopping, and application to numerical approximation of combined stochastic and impulse control}, + year = {2002}, + number = {0}, + pages = {149--172}, + volume = {237}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Chassagneux2013, + author = {Chassagneux, JF and Crisan, D}, + journal = {Annals of Applied Probability}, + title = {Runge-Kutta schemes for BSDEs}, + year = {2013}, + groups = {sdiyora:6}, + owner = {martin}, + timestamp = {2014.07.25}, +} + +@Article{Chassagneux2014a, + author = {Chassagneux, Jean-Fran{\c{c}}ois and Crisan, Dan}, + journal = {Ann. Appl. Probab.}, + title = {Runge--{K}utta schemes for backward stochastic differential equations}, + year = {2014}, + issn = {1050-5164}, + number = {2}, + pages = {679--720}, + volume = {24}, + doi = {10.1214/13-AAP933}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + mrclass = {Preliminary Data}, + mrnumber = {3178495}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/13-AAP933}, +} + +@Article{Chassagneux2014, + author = {Jean-Francois Chassagneux and Antoine Jacquier and Ivo Mihaylov}, + journal = {arXiv:1405.3561}, + title = {An explicit {E}uler scheme with strong rate of convergence for non-{L}ipschitz {SDE}s}, + year = {2014}, + pages = {27 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Chen2015, + author = {{Chen}, Bohan and {Chong}, Carsten and {Kl{\"u}ppelberg}, Claudia}, + journal = {arXiv:1501.01645}, + title = {Simulation of stochastic {V}olterra equations driven by space-time {L}\'evy noise}, + year = {2015}, + pages = {1--20}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Chen2013, + author = {Chen, L. and Dalange, R.}, + journal = {arXiv:1307.0600}, + title = {Moments and growth indices for the nonlinear stochastic heat equation with rough initial value}, + year = {2013}, + pages = {45 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Chen2013a, + author = {Chen, L. and Dalange, R.}, + journal = {arXiv:1307.0600}, + title = {H\"{o}lder continuity for the nonlinear stochastic heat equation with rough initial conditions}, + year = {2013}, + pages = {45 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cheridito2007, + author = {Cheridito, Patrick and Soner, H Mete and Touzi, Nizar and Victoir, Nicolas}, + journal = {Communications on Pure and Applied Mathematics}, + title = {Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs}, + year = {2007}, + number = {7}, + pages = {1081--1110}, + volume = {60}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Wiley Online Library}, + timestamp = {2016.02.02}, +} + +@Article{Chorin2013, + author = {Chorin, Alexandre J. and Morzfeld, Matthias and Tu, Xuemin}, + journal = {Chin. Ann. Math. Ser. B}, + title = {Implicit sampling, with application to data assimilation}, + year = {2013}, + issn = {0252-9599}, + number = {1}, + pages = {89--98}, + volume = {34}, + doi = {10.1007/s11401-012-0757-5}, + fjournal = {Chinese Annals of Mathematics. Series B}, + groups = {sdiyora:6}, + mrclass = {62M20 (65C50)}, + mrnumber = {3011460}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s11401-012-0757-5}, +} + +@Book{Chow2007, + author = {Pao-Liu Chow}, + publisher = {Chapman \& Hall/CRC, Boca Raton, FL}, + title = {Stochastic partial differential equations}, + year = {2007}, + isbn = {978-1-58488-443-9; 1-58488-443-6}, + series = {Chapman \& Hall/CRC Applied Mathematics and Nonlinear Science Series}, + groups = {sdiyora:6}, + mrclass = {35R60 (60-02 60H15)}, + mrnumber = {2295103 (2008d:35243)}, + mrreviewer = {Nikita Y. Ratanov}, + owner = {sdiyora}, + pages = {x+281}, + timestamp = {2016.02.02}, +} + +@Article{Chow2009, + author = {P.-L Chow and A. Lang and J. Potthoff}, + journal = {Preprint, urn:nbn:de:bsz:180-madoc-23291}, + title = {Almost sure convergence of a semidiscrete {M}ilstein scheme for {SPDE}'s of {Z}akai type}, + year = {2009}, + fjournal = {Preprint}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Clark1980, + author = {Clark, J. M. C. and Cameron, R. J.}, + booktitle = {Stochastic differential systems ({P}roc. {IFIP}-{WG} 7/1 {W}orking {C}onf., {V}ilnius, 1978)}, + publisher = {Springer}, + title = {The maximum rate of convergence of discrete approximations for stochastic differential equations}, + year = {1980}, + address = {Berlin}, + pages = {162--171}, + series = {Lecture Notes in Control and Information Sci.}, + volume = {25}, + groups = {sdiyora:6}, + mrclass = {60H10}, + mrnumber = {609181 (82f:60133)}, + mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cliffe2000, + author = {Cliffe, K. A. and Graham, I. G. and Scheichl, R. and Stals, L.}, + journal = {J. Comput. Phys.}, + title = {Parallel computation of flow in heterogeneous media modelled by mixed finite elements}, + year = {2000}, + issn = {0021-9991}, + number = {2}, + pages = {258--282}, + volume = {164}, + coden = {JCTPAH}, + doi = {10.1006/jcph.2000.6593}, + fjournal = {Journal of Computational Physics}, + groups = {sdiyora:6}, + mrclass = {76M10}, + mrnumber = {1792513 (2001f:76059)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1006/jcph.2000.6593}, +} + +@Article{Cohen2011, + author = {David Cohen and Magdalena Sigg}, + journal = {Numer. Math.}, + title = {Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations}, + year = {2011}, + doi = {10.1007/s00211-011-0426-8}, + fjournal = {Numerische Mathematik}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.springerlink.com/content/ukt7211324175207/}, +} + +@Article{Cohen2007, + author = {Cohen, Serge and Rosi{\'n}ski, Jan}, + journal = {Bernoulli}, + title = {Gaussian approximation of multivariate {L}\'evy processes with applications to simulation of tempered stable processes}, + year = {2007}, + issn = {1350-7265}, + number = {1}, + pages = {195--210}, + volume = {13}, + doi = {10.3150/07-BEJ6011}, + fjournal = {Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability}, + groups = {sdiyora:6}, + mrclass = {60G51 (60G52)}, + mrnumber = {2307403 (2008a:60120)}, + mrreviewer = {Sonia Fourati}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.3150/07-BEJ6011}, +} + +@Book{c93, + Title = {Measure theory}, + Author = {Cohn, Donald L.}, + Publisher = {Birkh\"auser Boston Inc.}, + Year = {1993}, + + Address = {Boston, MA}, + Note = {Reprint of the 1980 original}, + + ISBN = {0-8176-3003-1}, + Mrclass = {28-01}, + Mrnumber = {MR1454121 (98b:28001)}, + Pages = {x+373} +} + +@Article{Cont2005, + author = {Cont, Rama}, + journal = {International Journal of Theoretical and Applied Finance (IJTAF)}, + title = {Modeling Term Structure Dynamics: An Infinite Dimensional Approach}, + year = {2005}, + number = {03}, + pages = {357--380}, + volume = {08}, + abstract = {Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures +many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.}, + groups = {sdiyora:6}, + keywords = {Term structure of interest rates; forward rates; multifactor models; Hilbert space; stochastic PDE; random field}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://EconPapers.repec.org/RePEc:wsi:ijtafx:v:08:y:2005:i:03:p:357-380}, +} + +@Book{Cont2004, + author = {Cont, Rama and Tankov, Peter}, + publisher = {Chapman \& Hall/CRC, Boca Raton, FL}, + title = {Financial modelling with jump processes}, + year = {2004}, + isbn = {1-5848-8413-4}, + note = {535 pages}, + series = {Chapman \& Hall/CRC Financial Mathematics Series}, + groups = {sdiyora:6}, + mrclass = {91-02 (60J75 91B28 91B84)}, + mrnumber = {2042661 (2004m:91004)}, + owner = {sdiyora}, + pages = {xvi+535}, + timestamp = {2016.02.02}, +} + +@Book{Conus2008a, + author = {Daniel Conus}, + publisher = {{\'E}cole {P}olytechnique {F}{\'e}d{\'e}derale {D}e {L}ausanne}, + title = {The Non-linear {S}tochastic {W}ave {E}quation in {H}igh {D}imensions: {E}xistence, {H}{\"o}lder-continuity and It{\^o}-Taylor {E}xpansion}, + year = {2008}, + note = {Dissertation}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Conus2008, + author = {Daniel Conus and Robert C.~Dalang}, + journal = {Electronic Journal of Probability}, + title = {The non-linear stochastic wave equation in high dimensions. Electronic Journal of Probability}, + year = {2008}, + pages = {629-670}, + volume = {13}, + fjournal = {Electronic Journal of Probability}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Conus2014, + author = {Daniel Conus and Arnulf Jentzen and Ryan Kurniawan}, + journal = {working paper}, + title = {{W}eak convergence rates of spectral {G}alerkin approximations for {SPDE}s with nonlinear diffusion coefficients}, + year = {2014}, + pages = {26 pages}, + groups = {sdiyora:6}, + owner = {scox}, + timestamp = {2014.07.25}, +} + +@Article{Conus2014a, + author = {Daniel Conus and Arnulf Jentzen and Ryan Kurniawan}, + journal = {working paper}, + title = {Weak convergence rates of spectral {G}alerkin approximations for stochastic evolution equations with nonlinear diffusion coefficients}, + year = {2014}, + pages = {XX pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Conus2014b, + author = {Daniel Conus and Arnulf Jentzen and Ryan Kurniawan}, + journal = {arXiv:1408.1108}, + title = {Weak convergence rates of spectral {G}alerkin approximations for {SPDE}s with nonlinear diffusion coefficients}, + year = {2014}, + pages = {1--29}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cook1970, + author = {H. Cook}, + journal = {Acta Metallurgica}, + title = {Brownian motion in spinodal decomposition}, + year = {1970}, + number = {3}, + pages = {297--306}, + volume = {18}, + fjournal = {Acta Metallurgica}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Cox1985, + author = {John C. Cox and Jr. Jonathan E. Ingersoll and Stephen A. Ross}, + journal = {Econometrica}, + title = {A theory of the term structure of interest rates}, + year = {1985}, + issn = {0012-9682}, + number = {2}, + pages = {385--407}, + volume = {53}, + coden = {ECMTA7}, + doi = {10.2307/1911242}, + fjournal = {Econometrica. Journal of the Econometric Society}, + groups = {sdiyora:6}, + mrclass = {90A14}, + mrnumber = {785475}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.2307/1911242}, +} + +@Unpublished{Cox2014, + author = {Sonja Cox and Martin Hutzenthaler and Arnulf Jentzen and Jan van Neerven}, + note = {Unpublished manuscript}, + title = {Convergence in the H\"{o}lder norm with applications to Monte Carlo methods in infinite dimensions}, + month = March, + year = {2014}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{cn13, + Title = {Pathwise {H}{\"o}lder convergence of the implicit-linear {E}uler scheme for semi-linear {SPDE}s with multiplicative noise}, + Author = {Cox, Sonja and Neerven, Jan}, + Journal = {Numerische Mathematik}, + Year = {2013}, + Pages = {1-87}, + + Doi = {10.1007/s00211-013-0538-4}, + ISSN = {0029-599X}, + Keywords = {Primary 65C30; Secondary 60H15; 60H35; 65J08}, + Language = {English}, + Publisher = {Springer-Verlag} +} + +@Article{Cox2013, + author = {Cox, Sonja and van Neerven, Jan}, + journal = {Numer. Math.}, + title = {Pathwise {H}\"older convergence of the implicit-linear {E}uler scheme for semi-linear {SPDE}s with multiplicative noise}, + year = {2013}, + issn = {0029-599X}, + number = {2}, + pages = {259--345}, + volume = {125}, + coden = {NUMMA7}, + doi = {10.1007/s00211-013-0538-4}, + fjournal = {Numerische Mathematik}, + groups = {sdiyora:6}, + mrclass = {Preliminary Data}, + mrnumber = {3101829}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s00211-013-0538-4}, +} + +@Article{cvn13, + Title = {Pathwise {H}\"older convergence of the implicit-linear {E}uler + scheme for semi-linear {SPDE}s with multiplicative noise}, + Author = {Cox, Sonja and van Neerven, Jan}, + Journal = {Numer. 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Anal.}, + title = {Convergence rates of the splitting scheme for parabolic linear stochastic {C}auchy problems}, + year = {2010}, + issn = {0036-1429}, + number = {2}, + pages = {428--451}, + volume = {48}, + doi = {10.1137/090761835}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {65C30 (35R20 35R60 47D06 60J35)}, + mrnumber = {2646103 (2011e:65012)}, + mrreviewer = {Madalina Deaconu}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/090761835}, +} + +@Article{cvn10, + Title = {Convergence rates of the splitting scheme for parabolic linear + stochastic {C}auchy problems}, + Author = {Cox, Sonja and van Neerven, Jan}, + Journal = {SIAM J. Numer. 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An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (35Q53 35R60 47H06 76M35)}, + mrnumber = {2276524 (2008a:60153)}, + mrreviewer = {Wilhelm Stannat}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s11118-006-9021-5}, +} + +@Article{DaPrato2003, + author = {Da Prato, G. and Debussche, A.}, + journal = {Ann. Probab.}, + title = {Strong solutions to the stochastic quantization equations}, + year = {2003}, + number = {4}, + pages = {1900-1916}, + volume = {31}, + fjournal = {The Annals of Probability}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/euclid.aop/1068646370}, + urldate = {2012-11-12}, +} + +@Article{DaPrato2003a, + author = {Da Prato, Giuseppe and Debussche, Arnaud}, + journal = {J. Math. Pures Appl. 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Theory, Methods \& Applications. An International Multidisciplinary Journal. Series A: Theory and Methods}, + groups = {sdiyora:6}, + mrclass = {35R60 (35K55 35Q99 60H15)}, + mrnumber = {1359472 (96k:35200)}, + mrreviewer = {Ya. {\=I}. B{\={\i}}lopol{\cprime}s{\cprime}ka}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/0362-546X(94)00277-O}, +} + +@Article{DaPrato1994, + author = {Giuseppe {Da Prato} and Arnaud Debussche and Roger Temam}, + journal = {NoDEA Nonlinear Differential Equations Appl.}, + title = {Stochastic {B}urgers' equation}, + year = {1994}, + issn = {1021-9722}, + number = {4}, + pages = {389--402}, + volume = {1}, + coden = {NDEAFL}, + doi = {10.1007/BF01194987}, + fjournal = {NoDEA. Nonlinear Differential Equations and Applications}, + groups = {sdiyora:6}, + mrclass = {35R60 (35Q53 60H15)}, + mrnumber = {MR1300149 (95i:35326)}, + mrreviewer = {Jan I. 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Anal.}, + Year = {2006}, + Number = {6}, + Pages = {2505--2538 (electronic)}, + Volume = {44}, + + Fjournal = {SIAM Journal on Numerical Analysis}, + ISSN = {0036-1429}, + Mrclass = {60H35 (60G35 60H07 60H15 65C30)}, + Mrnumber = {MR2272604 (2008c:60063)}, + Mrreviewer = {Hisao Watanabe} +} + +@Article{Gruene2001, + author = {L. Gr{\"u}ne and P. E. Kloeden}, + journal = {BIT}, + title = {Pathwise approximation of random ordinary differential equations}, + year = {2001}, + issn = {0006-3835}, + number = {4}, + pages = {711--721}, + volume = {41}, + doi = {10.1023/A:1021995918864}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {65L05 (65C30)}, + mrnumber = {1881213 (2002k:65099)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1023/A:1021995918864}, +} + +@Article{gnt05, + Title = {Ito's- and {T}anaka's-type formulae for the stochastic heat + equation: the linear case}, + Author = {Gradinaru, Mihai and Nourdin, Ivan and Tindel, Samy}, + Journal = {J. Funct. Anal.}, + Year = {2005}, + Number = {1}, + Pages = {114--143}, + Volume = {228}, + + Coden = {JFUAAW}, + Fjournal = {Journal of Functional Analysis}, + ISSN = {0022-1236}, + Mrclass = {60H15 (60H05 60H07)}, + Mrnumber = {MR2170986 (2006i:60085)}, + Mrreviewer = {David Nualart} +} + +@Article{Gradinaru2005, + author = {Mihai Gradinaru and Ivan Nourdin and Samy Tindel}, + journal = {J. Funct. Anal.}, + title = {Ito's- and {T}anaka's-type formulae for the stochastic heat equation: the linear case}, + year = {2005}, + issn = {0022-1236}, + number = {1}, + pages = {114--143}, + volume = {228}, + coden = {JFUAAW}, + fjournal = {Journal of Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (60H05 60H07)}, + mrnumber = {MR2170986 (2006i:60085)}, + mrreviewer = {David Nualart}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Graham2013, + author = {Graham, Carl and Talay, Denis}, + publisher = {Springer, Heidelberg}, + title = {Stochastic {S}imulation and {M}onte {C}arlo {M}ethods}, + year = {2013}, + isbn = {978-3-642-39362-4; 978-3-642-39363-1}, + note = {Mathematical foundations of stochastic simulation}, + series = {Stochastic Modelling and Applied Probability}, + volume = {68}, + doi = {10.1007/978-3-642-39363-1}, + groups = {sdiyora:6}, + mrclass = {60-08 (60Fxx 60G42 60H35 60Jxx 65C05)}, + mrnumber = {3097957}, + mrreviewer = {Nicolas Privault}, + owner = {sdiyora}, + pages = {xvi+260}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-3-642-39363-1}, +} + +@Article{Graham2011, + author = {Graham, I. 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Soc.}, + Year = {1996}, + Number = {1}, + Pages = {79--85}, + Volume = {54}, + + Coden = {ALNBAB}, + Fjournal = {Bulletin of the Australian Mathematical Society}, + ISSN = {0004-9727}, + Mrclass = {60H15 (65U05)}, + Mrnumber = {MR1402994 (97g:60080)}, + Mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n} +} + +@Article{Grecksch1996, + author = {W. Grecksch and P. E. Kloeden}, + journal = {Bull. Austral. Math. Soc.}, + title = {Time-discretised {G}alerkin approximations of parabolic stochastic {PDE}s}, + year = {1996}, + issn = {0004-9727}, + number = {1}, + pages = {79--85}, + volume = {54}, + coden = {ALNBAB}, + fjournal = {Bulletin of the Australian Mathematical Society}, + groups = {sdiyora:6}, + mrclass = {60H15 (65U05)}, + mrnumber = {MR1402994 (97g:60080)}, + mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Grecksch1996a, + author = {Wilfried Grecksch and Constantin Tudor}, + publisher = {Wiley}, + title = {Stochastic Evolution Equations: A Hilbert Space Approach}, + year = {1996}, + address = {New York}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {179}, + timestamp = {2016.02.02}, +} + +@Article{Griebel1999, + author = {Griebel, Michael and Oswald, Peter and Schiekofer, Thomas}, + journal = {Numerische Mathematik}, + title = {Sparse grids for boundary integral equations}, + year = {1999}, + number = {2}, + pages = {279--312}, + volume = {83}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Springer}, + timestamp = {2016.02.02}, +} + +@Article{Grisvard1967, + author = {Grisvard, P.}, + journal = {Arch. Rational Mech. Anal.}, + title = {Caract\'erisation de quelques espaces d'interpolation}, + year = {1967}, + issn = {0003-9527}, + pages = {40--63}, + volume = {25}, + fjournal = {Archive for Rational Mechanics and Analysis}, + groups = {sdiyora:6}, + mrclass = {46.38 (47.00)}, + mrnumber = {0213864 (35 \#4718)}, + mrreviewer = {C. Goulaouic}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Grorud1992, + author = {Grorud, A. and Pardoux, {\'E}.}, + journal = {Appl. Math. Optim.}, + title = {Int\'egrales hilbertiennes anticipantes par rapport \`a un processus de {W}iener cylindrique et calcul stochastique associ\'e}, + year = {1992}, + issn = {0095-4616}, + number = {1}, + pages = {31--49}, + volume = {25}, + coden = {AMOMBN}, + doi = {10.1007/BF01184155}, + fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H05 (60H07)}, + mrnumber = {1133251 (93b:60114)}, + mrreviewer = {David Nualart}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF01184155}, +} + +@Article{Guillin2012, + author = {Guillin, Arnaud and Wang, Feng-Yu}, + journal = {J. Differential Equations}, + title = {Degenerate {F}okker-{P}lanck equations: {B}ismut formula, gradient estimate and {H}arnack inequality}, + year = {2012}, + issn = {0022-0396}, + number = {1}, + pages = {20--40}, + volume = {253}, + coden = {JDEQAK}, + doi = {10.1016/j.jde.2012.03.014}, + fjournal = {Journal of Differential Equations}, + groups = {sdiyora:6}, + mrclass = {60J75 (35B45 35B65 35Q84 60J45)}, + mrnumber = {2917400}, + mrreviewer = {Ante Mimica}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.jde.2012.03.014}, +} + +@Article{Guo2015, + author = {Guo, Wenjie and Zhang, Jianfeng and Zhuo, Jia}, + journal = {Ann. Appl. Probab.}, + title = {A monotone scheme for high-dimensional fully nonlinear {PDEs}}, + year = {2015}, + month = {06}, + number = {3}, + pages = {1540--1580}, + volume = {25}, + doi = {10.1214/14-AAP1030}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {The Institute of Mathematical Statistics}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/14-AAP1030}, +} + +@Article{Guyon2011, + author = {Julien Guyon and Pierre Henry-Labord{\`e}re}, + journal = {Journal of Computational Finance}, + title = {The uncertain volatility model: a Monte Carlo approach}, + year = {2011}, + number = {3}, + pages = {385--402}, + volume = {14}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{g02, + Title = {Approximations of stochastic partial differential equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n}, + Booktitle = {Stochastic partial differential equations and applications + ({T}rento, 2002)}, + Publisher = {Dekker}, + Year = {2002}, + + Address = {New York}, + Pages = {287--307}, + Series = {Lecture Notes in Pure and Appl. Math.}, + Volume = {227}, + + Mrclass = {60H15}, + Mrnumber = {MR1919514 (2003b:60096)} +} + +@Article{g99, + Title = {Lattice approximations for stochastic quasi-linear parabolic + partial differential equations driven by space-time white + noise. {II}}, + Author = {Gy{\"o}ngy, Istv{\'a}n}, + Journal = {Potential Anal.}, + Year = {1999}, + Number = {1}, + Pages = {1--37}, + Volume = {11}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (35K99 35R60)}, + Mrnumber = {MR1699161 (2000g:60106)}, + Mrreviewer = {David Nualart} +} + +@Article{Gyoengy1999, + author = {Istv{\'a}n Gy{\"o}ngy}, + journal = {Potential Anal.}, + title = {Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. {II}}, + year = {1999}, + issn = {0926-2601}, + number = {1}, + pages = {1--37}, + volume = {11}, + coden = {POANE2}, + fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K99 35R60)}, + mrnumber = {MR1699161 (2000g:60106)}, + mrreviewer = {David Nualart}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{g98, + Title = {Lattice approximations for stochastic quasi-linear parabolic + partial differential equations driven by space-time white + noise. {I}}, + Author = {Gy{\"o}ngy, Istv{\'a}n}, + Journal = {Potential Anal.}, + Year = {1998}, + Number = {1}, + Pages = {1--25}, + Volume = {9}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (35K55 35R60)}, + Mrnumber = {MR1644183 (99j:60091)}, + Mrreviewer = {David Nualart} +} + +@article {Gyoengy1998, + AUTHOR = {Gy\"ongy, Istv\'an}, + TITLE = {A note on {E}uler's approximations}, + JOURNAL = {Potential Anal.}, + FJOURNAL = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + VOLUME = {8}, + YEAR = {1998}, + NUMBER = {3}, + PAGES = {205--216}, + ISSN = {0926-2601}, + MRCLASS = {60H10}, + MRNUMBER = {1625576}, +MRREVIEWER = {Shigetoku Kawabata}, + DOI = {10.1023/A:1008605221617}, + URL = {http://dx.doi.org/10.1023/A:1008605221617}, +} + +@Article{Gyongy1998, + Title = {A note on {E}uler's approximations}, + Author = {Gy{\"o}ngy, Istv{\'a}n}, + Journal = {Potential Anal.}, + Year = {1998}, + Number = {3}, + Pages = {205--216}, + Volume = {8}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H10}, + Mrnumber = {MR1625576 (99d:60060)}, + Mrreviewer = {Shigetoku Kawabata} +} + +@Article{gk06c, + Title = {Expansion of solutions of parameterized equations and + acceleration of numerical methods}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Krylov, Nicolai}, + Journal = {Illinois J. Math.}, + Year = {2006}, + Number = {1-4}, + Pages = {473--514 (electronic)}, + Volume = {50}, + + Coden = {IJMTAW}, + Fjournal = {Illinois Journal of Mathematics}, + ISSN = {0019-2082}, + Mrclass = {65B05 (65L06 65M99)}, + Mrnumber = {MR2247837 (2008c:65003)}, + Mrreviewer = {I. K. Marek} +} + +@Article{Gyoengy2006, + author = {Istv{\'a}n Gy{\"o}ngy and Nicolai Krylov}, + journal = {Illinois J. Math.}, + title = {Expansion of solutions of parameterized equations and acceleration of numerical methods}, + year = {2006}, + issn = {0019-2082}, + number = {1-4}, + pages = {473--514 (electronic)}, + volume = {50}, + coden = {IJMTAW}, + fjournal = {Illinois Journal of Mathematics}, + groups = {sdiyora:6}, + mrclass = {65B05 (65L06 65M99)}, + mrnumber = {MR2247837 (2008c:65003)}, + mrreviewer = {I. K. Marek}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/getRecord?id=euclid.ijm/1258059483}, +} + +@Article{gk05, + Title = {An accelerated splitting-up method for parabolic equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Krylov, Nicolai}, + Journal = {SIAM J. Math. Anal.}, + Year = {2005}, + Number = {4}, + Pages = {1070--1097 (electronic)}, + Volume = {37}, + + Doi = {10.1137/S0036141003437903}, + Fjournal = {SIAM Journal on Mathematical Analysis}, + ISSN = {0036-1410}, + Mrclass = {35K15 (35C20 35K65 65M06)}, + Mrnumber = {MR2192288 (2006i:35125)}, + Mrreviewer = {Espen Robstad Jakobsen} +} + +@Article{gk03a, + Title = {On the splitting-up method and stochastic partial differential + equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Krylov, Nicolai}, + Journal = {Ann. Probab.}, + Year = {2003}, + Number = {2}, + Pages = {564--591}, + Volume = {31}, + + Coden = {APBYAE}, + Fjournal = {The Annals of Probability}, + ISSN = {0091-1798}, + Mrclass = {60H15 (65C30 93E11)}, + Mrnumber = {MR1964941 (2004c:60182)}, + Mrreviewer = {Frederi G. Viens} +} + +@InCollection{gk03b, + Title = {On the rate of convergence of splitting-up approximations for + {SPDE}s}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Krylov, Nicolai}, + Booktitle = {Stochastic inequalities and applications}, + Publisher = {Birkh\"auser}, + Year = {2003}, + + Address = {Basel}, + Pages = {301--321}, + Series = {Progr. Probab.}, + Volume = {56}, + + Mrclass = {65C30 (60H15 60H35 93E11)}, + Mrnumber = {MR2073438 (2005f:65012)}, + Mrreviewer = {Vivek S. Borkar} +} + +@Article{Gyoengy1996, + author = {Gy{\"o}ngy, Istv{\'a}n and Krylov, Nicolai}, + journal = {Probab. Theory Related Fields}, + title = {Existence of strong solutions for {I}t\^o's stochastic equations via approximations}, + year = {1996}, + issn = {0178-8051}, + number = {2}, + pages = {143--158}, + volume = {105}, + coden = {PTRFEU}, + doi = {10.1007/BF01203833}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H10}, + mrnumber = {1392450 (97h:60058)}, + mrreviewer = {Toshio Yamada}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF01203833}, +} + +@Article{Gyoengy1981/82, + author = {I. Gy{\"o}ngy and N. V. Krylov}, + journal = {Stochastics}, + title = {On stochastics equations with respect to semimartingales. {II}. {I}t{\^o} formula in {B}anach spaces}, + year = {1981/82}, + issn = {0090-9491}, + number = {3-4}, + pages = {153--173}, + volume = {6}, + coden = {STOCB2}, + fjournal = {Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H10}, + mrnumber = {665398 (84m:60070a)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{gk06, + Title = {On acceleration of approximation methods}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Krylov, Nicolai V.}, + Booktitle = {Stochastic partial differential equations and + applications---{VII}}, + Publisher = {Chapman \& Hall/CRC, Boca Raton, FL}, + Year = {2006}, + Pages = {149--167}, + Series = {Lect. Notes Pure Appl. Math.}, + Volume = {245}, + + Mrclass = {65C30 (65J20)}, + Mrnumber = {MR2227227 (2007g:65020)}, + Mrreviewer = {Anatoly V. Zherelo} +} + +@InCollection{gk81, + Title = {It\^o formula in {B}anach spaces}, + Author = {Gy{\"o}ngy, I. and Krylov, N. V.}, + Booktitle = {Stochastic differential systems ({V}isegr\'ad, 1980)}, + Publisher = {Springer}, + Year = {1981}, + + Address = {Berlin}, + Pages = {69--73}, + Series = {Lecture Notes in Control and Information Sci.}, + Volume = {36}, + + Mrclass = {60H05 (60B11)}, + Mrnumber = {MR653648 (83i:60071)} +} + +@Article{gm06, + Title = {On numerical solution of stochastic partial differential + equations of elliptic type}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Mart{\'{\i}}nez, Teresa}, + Journal = {Stochastics}, + Year = {2006}, + Number = {4}, + Pages = {213--231}, + Volume = {78}, + + Doi = {10.1080/17442500600805047}, + Fjournal = {Stochastics. An International Journal of Probability and + Stochastic Processes}, + ISSN = {1744-2508}, + Mrclass = {60H35 (60H15)}, + Mrnumber = {MR2261426 (2007m:60200)}, + Mrreviewer = {Roger Pettersson} +} + +@Article{Gyoengy2006b, + author = {Istv{\'a}n Gy{\"o}ngy and Teresa Mart{\'i}nez}, + journal = {Stochastics}, + title = {On numerical solution of stochastic partial differential equations of elliptic type}, + year = {2006}, + issn = {1744-2508}, + number = {4}, + pages = {213--231}, + volume = {78}, + doi = {10.1080/17442500600805047}, + fjournal = {Stochastics. An International Journal of Probability and Stochastic Processes}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H15)}, + mrnumber = {MR2261426 (2007m:60200)}, + mrreviewer = {Roger Pettersson}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1080/17442500600805047}, +} + +@Article{gm09, + Title = {Rate of convergence of space time approximations for + stochastic evolution equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Millet, Annie}, + Journal = {Potential Anal.}, + Year = {2009}, + Number = {1}, + Pages = {29--64}, + Volume = {30}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (60F99 60H35 65C30)}, + Mrnumber = {MR2465711} +} + +@Article{Gyoengy2009, + author = {Istv{\'a}n Gy{\"o}ngy and Annie Millet}, + journal = {Potential Anal.}, + title = {Rate of convergence of space time approximations for stochastic evolution equations}, + year = {2009}, + issn = {0926-2601}, + number = {1}, + pages = {29--64}, + volume = {30}, + coden = {POANE2}, + fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (60F99 60H35 65C30)}, + mrnumber = {MR2465711}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{gm07, + Title = {Rate of convergence of implicit approximations for stochastic + evolution equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Millet, Annie}, + Booktitle = {Stochastic differential equations: theory and applications}, + Publisher = {World Sci. Publ., Hackensack, NJ}, + Year = {2007}, + Pages = {281--310}, + Series = {Interdiscip. Math. Sci.}, + Volume = {2}, + + Mrclass = {60H15 (60H35)}, + Mrnumber = {MR2393581 (2009d:60207)}, + Mrreviewer = {Carlos M. Mora Gonz{\'a}lez} +} + +@InCollection{Gyoengy2007, + author = {Istv{\'a}n Gy{\"o}ngy and Annie Millet}, + booktitle = {Stochastic differential equations: theory and applications}, + publisher = {World Sci. Publ., Hackensack, NJ}, + title = {Rate of convergence of implicit approximations for stochastic evolution equations}, + year = {2007}, + pages = {281--310}, + series = {Interdiscip. Math. Sci.}, + volume = {2}, + groups = {sdiyora:6}, + mrclass = {60H15 (60H35)}, + mrnumber = {MR2393581 (2009d:60207)}, + mrreviewer = {Carlos M. Mora Gonz{\'a}lez}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{gm05, + Title = {On discretization schemes for stochastic evolution equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Millet, Annie}, + Journal = {Potential Anal.}, + Year = {2005}, + Number = {2}, + Pages = {99--134}, + Volume = {23}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (65C30)}, + Mrnumber = {MR2139212 (2006a:60115)}, + Mrreviewer = {Carlos M. Mora Gonz{\'a}lez} +} + +@Article{gn97, + Title = {Implicit scheme for stochastic parabolic partial differential + equations driven by space-time white noise}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Nualart, David}, + Journal = {Potential Anal.}, + Year = {1997}, + Number = {4}, + Pages = {725--757}, + Volume = {7}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (35K60 35R60 65M06)}, + Mrnumber = {MR1480861 (98m:60097)}, + Mrreviewer = {Ralf Manthey} +} + +@Article{Gyoengy1997, + author = {Istv{\'a}n Gy{\"o}ngy and David Nualart}, + journal = {Potential Anal.}, + title = {Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise}, + year = {1997}, + issn = {0926-2601}, + number = {4}, + pages = {725--757}, + volume = {7}, + coden = {POANE2}, + fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K60 35R60 65M06)}, + mrnumber = {MR1480861 (98m:60097)}, + mrreviewer = {Ralf Manthey}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{gn95, + Title = {Implicit scheme for quasi-linear parabolic partial + differential equations perturbed by space-time white noise}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Nualart, David}, + Journal = {Stochastic Process. Appl.}, + Year = {1995}, + Number = {1}, + Pages = {57--72}, + Volume = {58}, + + Coden = {STOPB7}, + Fjournal = {Stochastic Processes and their Applications}, + ISSN = {0304-4149}, + Mrclass = {65M12 (35R60 60H15 65U05)}, + Mrnumber = {MR1341554 (96b:65091)} +} + +@Article{Gyoengy1995, + author = {Istv{\'a}n Gy{\"o}ngy and David Nualart}, + journal = {Stochastic Process. Appl.}, + title = {Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise}, + year = {1995}, + issn = {0304-4149}, + number = {1}, + pages = {57--72}, + volume = {58}, + coden = {STOPB7}, + fjournal = {Stochastic Processes and their Applications}, + groups = {sdiyora:6}, + mrclass = {65M12 (35R60 60H15 65U05)}, + mrnumber = {MR1341554 (96b:65091)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Gyoengy2011, + author = {Gy{\"o}ngy, Istv{\'a}n and R{\'a}sonyi, Mikl{\'o}s}, + journal = {Stochastic Process. Appl.}, + title = {A note on {E}uler approximations for {SDE}s with {H}\"older continuous diffusion coefficients}, + year = {2011}, + issn = {0304-4149}, + number = {10}, + pages = {2189--2200}, + volume = {121}, + coden = {STOPB7}, + doi = {10.1016/j.spa.2011.06.008}, + fjournal = {Stochastic Processes and their Applications}, + groups = {sdiyora:6}, + mrclass = {60H35 (60F15)}, + mrnumber = {2822773}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.spa.2011.06.008}, +} + +@Article{GoengySabanisS2014, + Title = {{Convergence of tamed {E}uler schemes for a class of stochastic evolution equations}}, + Author = {{Gy{\"o}ngy}, I. and {Sabanis}, S. and {{\v S}i{\v s}ka}, D. }, + Journal = {arXiv:1308.1796}, + Year = {2015}, + Pages = {17 pages} +} + +@Article{GoengySabanisS2015, + author="Gy{\"o}ngy, Istv{\'a}n + and Sabanis, Sotirios + and {\v{S}}i{\v{s}}ka, David", + title="Convergence of tamed {E}uler schemes for a class of stochastic evolution equations", + journal="Stoch. Partial Differ. Equ. Anal. Comput.", + year="2015", + pages="1--21", + abstract="We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.", + issn="2194-041X", + doi="10.1007/s40072-015-0057-7", + url="http://dx.doi.org/10.1007/s40072-015-0057-7" +} + + +@Article{gs06, + Title = {Rate of convergence of {W}ong-{Z}akai approximations for + stochastic partial differential equations}, + Author = {Gy{\"o}ngy, Istv{\'a}n and Shmatkov, Anton}, + Journal = {Appl. Math. Optim.}, + Year = {2006}, + Number = {3}, + Pages = {315--341}, + Volume = {54}, + + Coden = {AMOMBN}, + Fjournal = {Applied Mathematics and Optimization. An International Journal + with Applications to Stochastics}, + ISSN = {0095-4616}, + Mrclass = {60H35 (35R60 60H15)}, + Mrnumber = {MR2268661 (2008e:60212)}, + Mrreviewer = {Carlos M. Mora Gonz{\'a}lez} +} + +@Article{Gyoengy2006c, + author = {Istv{\'a}n Gy{\"o}ngy and Anton Shmatkov}, + journal = {Appl. Math. Optim.}, + title = {Rate of convergence of {W}ong-{Z}akai approximations for stochastic partial differential equations}, + year = {2006}, + issn = {0095-4616}, + number = {3}, + pages = {315--341}, + volume = {54}, + coden = {AMOMBN}, + fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H35 (35R60 60H15)}, + mrnumber = {MR2268661 (2008e:60212)}, + mrreviewer = {Carlos M. Mora Gonz{\'a}lez}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Hoermander1990, + author = {H{\"o}rmander, Lars}, + publisher = {Springer-Verlag}, + title = {The analysis of linear partial differential operators. {I}: Distribution theory and {F}ourier analysis}, + year = {1990}, + address = {Berlin}, + edition = {Second}, + isbn = {3-540-52343-X}, + series = {Springer Study Edition}, + groups = {sdiyora:6}, + mrclass = {35-02 (42B10 46Fxx)}, + mrnumber = {1065136 (91m:35001b)}, + owner = {sdiyora}, + pages = {xii+440}, + timestamp = {2016.02.02}, +} + +@Article{Hoermander1967, + author = {H{\"o}rmander, Lars}, + journal = {Acta Math.}, + title = {Hypoelliptic second order differential equations}, + year = {1967}, + issn = {0001-5962}, + pages = {147--171}, + volume = {119}, + fjournal = {Acta Mathematica}, + groups = {sdiyora:6}, + mrclass = {35.48 (47.00)}, + mrnumber = {0222474 (36 \$5526)}, + mrreviewer = {J. Smoller}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{hnw93, + Title = {Solving ordinary differential equations. {I}}, + Author = {Hairer, E. and N{\o}rsett, S. P. and Wanner, G.}, + Publisher = {Springer-Verlag}, + Year = {1993}, + + Address = {Berlin}, + Edition = {Second}, + Note = {Nonstiff problems}, + Series = {Springer Series in Computational Mathematics}, + Volume = {8}, + + ISBN = {3-540-56670-8}, + Mrclass = {65-02 (34A50 65L99)}, + Mrnumber = {MR1227985 (94c:65005)}, + Pages = {xvi+528} +} + +@Article{Hairer, + author = {Martin Hairer}, + journal = {To appear in Ann. of Math. (2)}, + title = {Solving the {KPZ} equation}, + pages = {103 pages}, + fjournal = {To appear in Annals of Mathematics. Second Series}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2013, + author = {Hairer, Martin}, + journal = {arXiv:1303.5113}, + title = {A theory of regularity structures}, + year = {2013}, + pages = {178 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2011, + author = {Hairer, Martin}, + journal = {Bull. Sci. Math.}, + title = {On {M}alliavin's proof of {H}\"ormander's theorem}, + year = {2011}, + issn = {0007-4497}, + number = {6-7}, + pages = {650--666}, + volume = {135}, + coden = {BSMQA9}, + doi = {10.1016/j.bulsci.2011.07.007}, + fjournal = {Bulletin des Sciences Math\'ematiques}, + groups = {sdiyora:6}, + mrclass = {60H07 (60H30 60J60)}, + mrnumber = {2838095}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.bulsci.2011.07.007}, +} + +@Article{Hairer2011a, + author = {Martin Hairer}, + journal = {arXiv:1109.6811}, + title = {Solving the {KPZ} equation}, + year = {2011}, + pages = {102 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2009, + author = {Martin Hairer}, + journal = {arXiv:0907.4178v1}, + title = {{A}n {I}ntroduction to {S}tochastic {PDE}s}, + year = {2009}, + pages = {78 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2008, + author = {Hairer, M.}, + journal = {Unpublished lecture notes}, + title = {Ergodic theory for Stochastic PDEs}, + year = {2008}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@article {Hairer2015, + AUTHOR = {Hairer, Martin and Hutzenthaler, Martin and Jentzen, Arnulf}, + TITLE = {Loss of regularity for {K}olmogorov equations}, + JOURNAL = {Ann. Probab.}, + FJOURNAL = {The Annals of Probability}, + VOLUME = {43}, + YEAR = {2015}, + NUMBER = {2}, + PAGES = {468--527}, + ISSN = {0091-1798}, + MRCLASS = {35K15 (35B65)}, + MRNUMBER = {3305998}, + DOI = {10.1214/13-AOP838}, + URL = {http://dx.doi.org/10.1214/13-AOP838}, +} + +@Article{Hairer2014, + author = {Martin Hairer and Martin Hutzenthaler and Arnulf Jentzen}, + journal = {To appear in Ann. Probab.}, + title = {Loss of regularity for {K}olmogorov equations}, + year = {2014}, + pages = {arXiv:1209.6035}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2010, + author = {Martin Hairer and Jan Maas}, + journal = {arXiv:1011.0966v1}, + title = {A spatial version of the {I}to-{S}tratonovich correction}, + year = {2010}, + pages = {33 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2011b, + author = {Hairer, Martin and Mattingly, Jonathan C.}, + journal = {Electron. J. Probab.}, + title = {A theory of hypoellipticity and unique ergodicity for semilinear stochastic {PDE}s}, + year = {2011}, + issn = {1083-6489}, + pages = {no. 23, 658--738}, + volume = {16}, + doi = {10.1214/EJP.v16-875}, + fjournal = {Electronic Journal of Probability}, + groups = {sdiyora:6}, + mrclass = {60H15 (35H10 35R60 60H07)}, + mrnumber = {2786645 (2012e:60170)}, + mrreviewer = {Feng-Yu Wang}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/EJP.v16-875}, +} + +@Article{Hairer2006, + author = {Hairer, Martin and Mattingly, Jonathan C.}, + journal = {Ann. of Math. (2)}, + title = {Ergodicity of the 2{D} {N}avier-{S}tokes equations with degenerate stochastic forcing}, + year = {2006}, + issn = {0003-486X}, + number = {3}, + pages = {993--1032}, + volume = {164}, + coden = {ANMAAH}, + doi = {10.4007/annals.2006.164.993}, + fjournal = {Annals of Mathematics. Second Series}, + groups = {sdiyora:6}, + mrclass = {37L55 (35Q30 35R60 60H15 76D05 76M35)}, + mrnumber = {2259251 (2008a:37095)}, + mrreviewer = {Hakima Bessaih}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.4007/annals.2006.164.993}, +} + +@Article{Hairer2012, + author = {Martin Hairer and Marc D. Ryser and Hendrik Weber}, + journal = {Electron. J. Probab.}, + title = {Triviality of the 2{D} stochastic {A}llen-{C}ahn equation}, + year = {2012}, + number = {39}, + pages = {1--14}, + volume = {17}, + fjournal = {Electronic Journal of Probability}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Hairer2009a, + author = {Martin Hairer and Andrew Stuart and Jochen Vo{\ss}}, + booktitle = {Trends in stochastic analysis}, + publisher = {Cambridge Univ. Press}, + title = {Sampling conditioned diffusions}, + year = {2009}, + address = {Cambridge}, + pages = {159--185}, + series = {London Math. Soc. Lecture Note Ser.}, + volume = {353}, + groups = {sdiyora:6}, + mrclass = {60-08 (60J22 60J60 62M20 65C30)}, + mrnumber = {2562154}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairera, + author = {M. Hairer and A. M. Stuart and J. Voss}, + journal = {To appear in Ann. Appl. Probab.}, + title = {Sampling Conditioned Hypoelliptic Diffusions}, + pages = {22 pages}, + fjournal = {To appear in The Annals of Applied Probability}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hairer2007, + author = {M. Hairer and A. M. Stuart and J. Voss}, + journal = {Ann. Appl. Probab.}, + title = {Analysis of {SPDE}s arising in path sampling. {II}. {T}he nonlinear case}, + year = {2007}, + issn = {1050-5164}, + number = {5-6}, + pages = {1657--1706}, + volume = {17}, + doi = {10.1214/07-AAP441}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + mrclass = {60H15 (60G17 60G35 65C30)}, + mrnumber = {2358638 (2008i:60105)}, + mrreviewer = {Jan I. Seidler}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/07-AAP441}, +} + +@Article{Hairer2005, + author = {M. Hairer and A. M. Stuart and J. Voss and P. Wiberg}, + journal = {Commun. Math. Sci.}, + title = {Analysis of {SPDE}s arising in path sampling. {I}. {T}he {G}aussian case}, + year = {2005}, + issn = {1539-6746}, + number = {4}, + pages = {587--603}, + volume = {3}, + fjournal = {Communications in Mathematical Sciences}, + groups = {sdiyora:6}, + mrclass = {60H15 (60G15 60G35 60H10)}, + mrnumber = {2188686 (2006i:60086)}, + mrreviewer = {Krystyna Twardowska}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/getRecord?id=euclid.cms/1144429334}, +} + +@Article{Hairer2010a, + author = {Martin Hairer and Jochen Voss}, + journal = {arXiv:1005.4438v1}, + title = {Approximations to the Stochastic {B}urgers Equation}, + year = {2010}, + pages = {19 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Haken1985, + author = {Haken, H. and Kelso, J. A. S. and Bunz, H.}, + journal = {Biol. Cybernet.}, + title = {A theoretical model of phase transitions in human hand movements}, + year = {1985}, + issn = {0340-1200}, + number = {5}, + pages = {347--356}, + volume = {51}, + coden = {BICYAF}, + doi = {10.1007/BF00336922}, + fjournal = {Biological Cybernetics}, + groups = {sdiyora:6}, + mrclass = {92A09 (82A25)}, + mrnumber = {778098 (86d:92007)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF00336922}, +} + +@Article{Halidias2015, + Title = {{Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations}}, + Author = {Nikolaos Halidias}, + Journal = {{Discrete Contin. Dyn. Syst. Ser. B}}, + Year = {2015}, + Number = {1}, + Pages = {153-160}, + Volume = {20}, + + Doi = {10.3934/dcdsb.2015.20.153}, + ISSN = {1531-3492}, + Url = {http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10508} +} + +@Article{Halidias2013, + Title = {A novel approach to construct numerical methods for stochastic differential equations}, + Author = {Halidias, Nikolaos}, + Journal = {Numer. Algorithms}, + Year = {2013}, + Number = {1}, + Pages = {79--87}, + Volume = {66}, + + Abstract = {In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.}, + Doi = {10.1007/s11075-013-9724-9}, + ISSN = {1572-9265}, + Url = {http://dx.doi.org/10.1007/s11075-013-9724-9} +} + +@Article{Halidias2012, + author = {Halidias, Nikolaos}, + journal = {Int. J. Comput. Math.}, + title = {Semi-discrete approximations for stochastic differential equations and applications}, + year = {2012}, + issn = {0020-7160}, + number = {6}, + pages = {780--794}, + volume = {89}, + doi = {10.1080/00207160.2012.658380}, + fjournal = {International Journal of Computer Mathematics}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H35)}, + mrnumber = {2903590}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1080/00207160.2012.658380}, +} + +@Article{Halidias2008, + author = {Halidias, Nikolaos and Kloeden, Peter E.}, + journal = {BIT}, + title = {A note on the {E}uler-{M}aruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient}, + year = {2008}, + issn = {0006-3835}, + number = {1}, + pages = {51--59}, + volume = {48}, + doi = {10.1007/s10543-008-0164-1}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H10 (60H20)}, + mrnumber = {2386114 (2009c:60150)}, + mrreviewer = {Renato G. C. Spigler}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s10543-008-0164-1}, +} + +@Article{HalidiasStamatiou2013, + Title = {{On the numerical solution of some nonlinear stochastic differential equations using the semi-discrete method}}, + Author = {{Halidias}, N. and {Stamatiou}, I.~S.}, + Journal = {arXiv:1309.3189}, + Year = {2014}, + Pages = {36 pages} +} + +@Book{h76, + Title = {Naive {M}engenlehre}, + Author = {Halmos, Paul R.}, + Publisher = {Vandenhoeck \& Ruprecht}, + Year = {1976}, + + Address = {G\"ottingen}, + Note = {Vierte Auflage, + Aus dem Englischen {\"u}bersetzt von Manfred Armbrust und + Fritz Ostermann, + Moderne Mathematik in elementarer Darstellung, No. 6}, + + Mrclass = {04-01}, + Mrnumber = {MR0427066 (55 \#102)}, + Pages = {132} +} + +@Article{hans02, + Title = {Strong stability and non-smooth data error estimates for + discretizations of linear parabolic problems}, + Author = {Hansbo, Anita}, + Journal = {BIT}, + Year = {2002}, + Number = {2}, + Pages = {351--379}, + Volume = {42}, + + Doi = {10.1023/A:1021903109720}, + Fjournal = {BIT. Numerical Mathematics}, + ISSN = {0006-3835}, + Mrclass = {65M12 (65M15)}, + Mrnumber = {1912592 (2003d:65085)} +} + +@Book{Hardy1988, + author = {Hardy, G. H. and Littlewood, J. E. and P{\'o}lya, G.}, + publisher = {Cambridge University Press, Cambridge}, + title = {Inequalities}, + year = {1988}, + isbn = {0-521-35880-9}, + note = {Reprint of the 1952 edition}, + series = {Cambridge Mathematical Library}, + groups = {sdiyora:6}, + mrclass = {26Dxx (01A75)}, + mrnumber = {944909 (89d:26016)}, + owner = {sdiyora}, + pages = {xii+324}, + timestamp = {2016.02.02}, +} + +@Article{h07a, + Title = {Finite element approximation of stochastic partial + differential equations driven by {P}oisson random measures of + jump type}, + Author = {Hausenblas, Erika}, + Journal = {SIAM J. Numer. Anal.}, + Year = {2007/08}, + Number = {1}, + Pages = {437--471}, + Volume = {46}, + + Fjournal = {SIAM Journal on Numerical Analysis}, + ISSN = {0036-1429}, + Mrclass = {60H35 (60H15 65C30)}, + Mrnumber = {MR2377271 (2009g:60094)}, + Mrreviewer = {Fran{\c{c}}ois Delarue} +} + +@Article{Hausenblas2007/08a, + author = {Erika Hausenblas}, + journal = {SIAM J. Numer. Anal.}, + title = {Finite element approximation of stochastic partial differential equations driven by {P}oisson random measures of jump type}, + year = {2007/08}, + issn = {0036-1429}, + number = {1}, + pages = {437--471}, + volume = {46}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H15 65C30)}, + mrnumber = {MR2377271 (2009g:60094)}, + mrreviewer = {Fran\c{c}ois Delarue}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hausenblas2010, + author = {Erika Hausenblas}, + journal = {J. Comput. Appl. Math.}, + title = {Weak approximation of the stochastic wave equation}, + year = {2010}, + number = {2}, + pages = {33--58}, + volume = {235}, + fjournal = {Journal of Computational and Applied Mathematics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{h07b, + Title = {Wong-{Z}akai type approximation of {SPDE}s of {L}\'evy noise}, + Author = {Hausenblas, Erika}, + Journal = {Acta Appl. Math.}, + Year = {2007}, + Number = {2}, + Pages = {99--134}, + Volume = {98}, + + Coden = {AAMADV}, + Fjournal = {Acta Applicandae Mathematicae. An International Survey Journal + on Applying Mathematics and Mathematical Applications}, + ISSN = {0167-8019}, + Mrclass = {60H15 (60B12 60G55)}, + Mrnumber = {MR2336465 (2008g:60192)}, + Mrreviewer = {Nicholas Frangos} +} + +@Article{h03a, + Title = {Approximation for semilinear stochastic evolution equations}, + Author = {Hausenblas, Erika}, + Journal = {Potential Anal.}, + Year = {2003}, + Number = {2}, + Pages = {141--186}, + Volume = {18}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (65C30)}, + Mrnumber = {MR1953619 (2003m:60167)} +} + +@InCollection{h03b, + Title = {Weak approximation for semilinear stochastic evolution + equations}, + Author = {Hausenblas, Erika}, + Booktitle = {Stochastic analysis and related topics {VIII}}, + Publisher = {Birkh\"auser}, + Year = {2003}, + + Address = {Basel}, + Pages = {111--128}, + Series = {Progr. Probab.}, + Volume = {53}, + + Mrclass = {60H15 (60H07 60H35 65C50)}, + Mrnumber = {MR2189620 (2006k:60114)}, + Mrreviewer = {Michal Vyoral} +} + +@Article{h02, + Title = {Numerical analysis of semilinear stochastic evolution + equations in {B}anach spaces}, + Author = {Hausenblas, Erika}, + Journal = {J. Comput. Appl. Math.}, + Year = {2002}, + Number = {2}, + Pages = {485--516}, + Volume = {147}, + + Coden = {JCAMDI}, + Fjournal = {Journal of Computational and Applied Mathematics}, + ISSN = {0377-0427}, + Mrclass = {35R60 (34G20 35A35 47D06 60H15 65C30)}, + Mrnumber = {MR1933610 (2003j:35338)}, + Mrreviewer = {Peter E. Kloeden} +} + +@Article{Hausenblas2002, + author = {Erika Hausenblas}, + journal = {J. Comput. Appl. Math.}, + title = {Numerical analysis of semilinear stochastic evolution equations in {B}anach spaces}, + year = {2002}, + issn = {0377-0427}, + number = {2}, + pages = {485--516}, + volume = {147}, + coden = {JCAMDI}, + fjournal = {Journal of Computational and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {35R60 (34G20 35A35 47D06 60H15 65C30)}, + mrnumber = {MR1933610 (2003j:35338)}, + mrreviewer = {Peter E. Kloeden}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hausenblas2006a, + author = {Erika Hausenblas and Iuliana Marchis}, + journal = {BIT}, + title = {A numerical approximation of parabolic stochastic partial differential equations driven by a {P}oisson random measure}, + year = {2006}, + issn = {0006-3835}, + number = {4}, + pages = {773--811}, + volume = {46}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H35 (35R60 60H15 65C30)}, + mrnumber = {MR2285208 (2007k:60218)}, + mrreviewer = {B{\'e}n{\'e}dicte Puig}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{hm06a, + Title = {A numerical approximation of parabolic stochastic partial + differential equations driven by a {P}oisson random measure}, + Author = {Hausenblas, Erika and Marchis, Iuliana}, + Journal = {BIT}, + Year = {2006}, + Number = {4}, + Pages = {773--811}, + Volume = {46}, + + Fjournal = {BIT. 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Papanicolaou} +} + +@Article{Huang1985, + author = {Hai Huang and Harold J. Kushner}, + journal = {Stochastics}, + title = {Weak convergence and approximations for partial differential equations with stochastic coefficients}, + year = {1985}, + issn = {0090-9491}, + number = {3}, + pages = {209--245}, + volume = {15}, + coden = {STOCB2}, + fjournal = {Stochastics}, + groups = {sdiyora:6}, + mrclass = {60J60 (60F05 60H15)}, + mrnumber = {MR869200 (89a:60183)}, + mrreviewer = {G. Papanicolaou}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hurd2008, + author = {Hurd, T. R. and Kuznetsov, A.}, + journal = {Markov Process. Related Fields}, + title = {Explicit formulas for {L}aplace transforms of stochastic integrals}, + year = {2008}, + issn = {1024-2953}, + number = {2}, + pages = {277--290}, + volume = {14}, + fjournal = {Markov Processes and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60E10 (33C55 60G09 60H10 60J60 91B28)}, + mrnumber = {2437532 (2009g:60023)}, + mrreviewer = {Christian Bender}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@article {Hutzenthaler2015, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf}, + TITLE = {Numerical approximations of stochastic differential equations + with non-globally {L}ipschitz continuous coefficients}, + JOURNAL = {Mem. Amer. Math. Soc.}, + FJOURNAL = {Memoirs of the American Mathematical Society}, + VOLUME = {236}, + YEAR = {2015}, + NUMBER = {1112}, + PAGES = {v+99}, + ISSN = {0065-9266}, + ISBN = {978-1-4704-0984-5}, + MRCLASS = {65C30 (60H35)}, + MRNUMBER = {3364862}, + DOI = {10.1090/memo/1112}, + URL = {http://dx.doi.org/10.1090/memo/1112}, +} + +@Article{Hutzenthaler2013, + author = {Martin Hutzenthaler and Arnulf Jentzen}, + journal = {Unpublished manusscript}, + title = {A draft for "Numerical approximations of stochastic partial differential equations with non-globally Lipschitz continuous coefficients"}, + year = {2013}, + pages = {2 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Hutzenthaler2012, + author = {Martin Hutzenthaler and Arnulf Jentzen}, + journal = {arXiv:1203.5809}, + title = {Numerical approximations of stochastic differential equations with non-globally {L}ipschitz continuous coefficients}, + year = {2012}, + pages = {59 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{hj11, + author = {Martin Hutzenthaler and Arnulf Jentzen}, + journal = {Found. Comput. Math.}, + title = {Convergence of the stochastic {E}uler scheme for locally {L}ipschitz coefficients}, + year = {2011}, + number = {6}, + pages = {657--706}, + volume = {11}, + fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{hj09a, + Title = {Non-globally {L}ipschitz {C}ounterexamples for the stochastic {E}uler scheme}, + Author = {Hutzenthaler, M. and Jentzen, A}, + Journal = {arXiv:0905.027}, + Year = {2009}, + Number = {2}, + Pages = {22} +} + +@Article{hj09b, + Title = {Convergence of the stochastic {E}uler scheme for locally {L}ipschitz coefficients}, + Author = {Hutzenthaler, Martin and Jentzen, Arnulf}, + Journal = {Submitted}, + Year = {2009}, + Pages = {57}, + + Fjournal = {Submitted} +} + +@article {HutzenthalerJentzenKloeden2013, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Kloeden, Peter + E.}, + TITLE = {Divergence of the multilevel {M}onte {C}arlo {E}uler method + for nonlinear stochastic differential equations}, + JOURNAL = {Ann. Appl. Probab.}, + FJOURNAL = {The Annals of Applied Probability}, + VOLUME = {23}, + YEAR = {2013}, + NUMBER = {5}, + PAGES = {1913--1966}, + ISSN = {1050-5164}, + MRCLASS = {60H35}, + MRNUMBER = {3134726}, +MRREVIEWER = {Renato G. C. Spigler}, + DOI = {10.1214/12-AAP890}, + URL = {http://dx.doi.org/10.1214/12-AAP890}, +} + +@article {HutzenthalerJentzenKloeden2012, + AUTHOR = {Hutzenthaler, Martin and Jentzen, Arnulf and Kloeden, Peter + E.}, + TITLE = {Strong convergence of an explicit numerical method for {SDE}s + with nonglobally {L}ipschitz continuous coefficients}, + JOURNAL = {Ann. Appl. 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Series A. + Mathematical, Physical and Engineering Sciences}, + VOLUME = {467}, + YEAR = {2011}, + NUMBER = {2130}, + PAGES = {1563--1576}, + ISSN = {1364-5021}, + MRCLASS = {65C30 (60H10 60H35)}, + MRNUMBER = {2795791}, +MRREVIEWER = {Henri Schurz}, + DOI = {10.1098/rspa.2010.0348}, + URL = {http://dx.doi.org/10.1098/rspa.2010.0348}, +} + +@Article{Hutzenthaler2011b, + author = {Martin Hutzenthaler and Arnulf Jentzen and Peter E. Kloeden}, + journal = {To appear in Ann. Appl. Probab. 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Chuo Univ.}, + title = {Remarks on existence of viscosity solutions of {H}amilton-{J}acobi equations}, + year = {1983}, + pages = {5--24}, + volume = {26}, + fjournal = {Chuo University. Faculty of Science and Engineering. Bulletin}, + groups = {sdiyora:6}, + mrclass = {35F20 (49C10)}, + mrnumber = {742691 (85k:35045)}, + mrreviewer = {E. N. Barron}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Ishii1990, + author = {Ishii, H. and Lions, P.-L.}, + journal = {J. 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Statist.}, + title = {The approximate {E}uler method for {L}\'evy driven stochastic differential equations}, + year = {2005}, + issn = {0246-0203}, + number = {3}, + pages = {523--558}, + volume = {41}, + coden = {AHPBAR}, + doi = {10.1016/j.anihpb.2004.01.007}, + fjournal = {Annales de l'Institut Henri Poincar\'e. Probabilit\'es et Statistiques}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H05 60H10)}, + mrnumber = {2139032 (2005m:60149)}, + mrreviewer = {Alexander M. Kolodi{\u\i}}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.anihpb.2004.01.007}, +} + +@Article{JanVanNeerven2010, + author = {Jan Van Neerven, Mark Veraar and Lutz Weis}, + journal = {arXiv:1004.1309v3}, + title = {Stochastic Maximal Lp-Regularity}, + year = {2010}, + pages = {21 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Unpublished{Jentzen2015SPDElecturenotes, + Title = {Stochastic {P}artial {D}ifferential {E}quations: +{A}nalysis and {N}umerical {A}pproximations}, + Author = {Arnulf Jentzen}, + Note = {Lecture notes, ETH Zurich, summer semester 2015, available online at \url{https://people.math.ethz.ch/~grsam/NASPDE_MATH_FS15/script/NASPDE_130.pdf}}, + + Month = {September}, + Year = {2015}, + + Owner = {scox}, + Timestamp = {2015.09.14}, + Url = {http://www.math.ethz.ch/education/bachelor/lectures/fs2015/math/numsol} +} + +@Unpublished{Jentzen2018DNNlecturenotes, + Title = {{A Mathematical Introduction to Machine +Learning Approximation Algorithms}}, + Author = {Arnulf Jentzen}, + Note = {Lecture notes, ETH Zurich, summer semester 2018}, + + Month = {December}, + Year = {2018}, +} + +@Unpublished{Jentzen2014SPDElecturenotes, + Title = {Numerical Analysis of Stochastic Partial Differential Equations}, + Author = {Arnulf Jentzen}, + Note = {Lecture notes, ETH Zurich, summer semester 2014, available online at http://www.math.ethz.ch/education/bachelor/lectures/fs2014/math/numsol}, + + Month = April, + Year = {2014}, + + Owner = {scox}, + Timestamp = {2014.07.25}, + Url = {http://www.math.ethz.ch/education/bachelor/lectures/fs2014/math/numsol} +} + +@Article{j08a, + Title = {Higher {O}rder {P}athwise {N}umerical {A}pproximations of {SPDE}s with {A}dditive {N}oise}, + Author = {Jentzen, A.}, + Journal = {SIAM Journal on Numerical Analysis}, + Year = {2011}, + Number = {2}, + Pages = {642-667}, + Volume = {49} +} + +@Article{Jentzen2011, + author = {Arnulf Jentzen}, + journal = {SIAM J. Numer. Anal.}, + title = {Higher order pathwise numerical approximations of {SPDE}s with additive noise}, + year = {2011}, + number = {2}, + pages = {642--667 (electronic)}, + volume = {49}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{j09a, + Title = {Taylor expansions of solutions of stochastic partial + differential equations}, + Author = {Jentzen, Arnulf}, + Journal = {Discrete Contin. Dyn. Syst. Ser. B}, + Year = {2010}, + Number = {2}, + Pages = {515--557}, + Volume = {14}, + + Doi = {10.3934/dcdsb.2010.14.515}, + Fjournal = {Discrete and Continuous Dynamical Systems. Series B. A Journal + Bridging Mathematics and Sciences}, + ISSN = {1531-3492}, + Mrclass = {60H15 (35R60)}, + Mrnumber = {2660871 (2011k:60216)}, + Mrreviewer = {Jinqiao Duan} +} + +@Article{j08b, + Title = {Pathwise numerical approximations of {SPDE}S with additive noise under non-global {L}ipschitz coefficients}, + Author = {Jentzen, Arnulf}, + Journal = {Potential Anal.}, + Year = {2009}, + Number = {4}, + Pages = {375--404}, + Volume = {31}, + + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H15 (65C30 35K90)} +} + +@Book{j09b, + Title = {Taylor Expansions + for Stochastic Partial + Differential Equations}, + Author = {Jentzen, Arnulf}, + Publisher = {Johann Wolfgang Goethe University}, + Year = {2009}, + + Address = {Frankfurt am Main, Germany}, + Note = {Dissertation} +} + +@Book{Jentzen2009a, + author = {Arnulf Jentzen}, + publisher = {Johann Wolfgang Goethe University}, + title = {Taylor Expansions for Stochastic Partial Differential Equations}, + year = {2009}, + address = {Frankfurt am Main, Germany}, + note = {Dissertation}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{j07, + Title = {Numerische Verfahren + hoher Ordnung f{\"u}r + zuf{\"a}llige Differentialgleichungen}, + Author = {Jentzen, Arnulf}, + Publisher = {Johann Wolfgang Goethe University}, + Year = {2007}, + + Address = {Frankfurt am Main, Germany}, + Note = {Diplomarbeit} +} + +@Book{Jentzen2007, + author = {Arnulf Jentzen}, + publisher = {Johann Wolfgang Goethe University}, + title = {Numerische Verfahren hoher Ordnung f{\"u}r zuf{\"a}llige Differentialgleichungen}, + year = {2007}, + address = {Frankfurt am Main, Germany}, + note = {Diplomarbeit}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Jentzen2012, + author = {Jentzen, Arnulf and Kloeden, Peter}, + publisher = {Society for Industrial and Applied Mathematics (SIAM)}, + title = {{T}aylor {A}pproximations for {S}tochastic {P}artial {D}ifferential {E}quations}, + year = {2012}, + address = {Philadelphia, PA}, + isbn = {978-1-611972-00-9}, + series = {CBMS-NSF Regional Conference Series in Applied Mathematics}, + volume = {83}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {xvi+220}, + timestamp = {2016.02.02}, +} + +@Article{jkw09, + Title = {{Efficient simulation of nonlinear parabolic SPDEs with additive + noise.}}, + Author = {Jentzen, Arnulf and Kloeden, Peter and Winkel, Georg}, + Journal = {Ann. Appl. Probab. }, + Year = {2011}, + Number = {3}, + Pages = {908-950}, + Volume = {21}, + + Language = {English} +} + +@Book{JentzenKloeden2011, + Title = {Taylor approximations for stochastic partial differential equations}, + Author = {Jentzen, Arnulf and Kloeden, Peter E.}, + Publisher = {Society for Industrial and Applied Mathematics (SIAM)}, + Year = {2011}, + + Address = {Philadelphia, PA}, + Series = {CBMS-NSF Regional Conference Series in Applied Mathematics}, + Volume = {83}, + + Doi = {10.1137/1.9781611972016}, + ISBN = {978-1-611972-00-9}, + Mrclass = {60H15 (60H35 65C30)}, + Mrnumber = {2856611}, + Mrreviewer = {Sergey V. Lototsky}, + Pages = {xiv+220}, + Url = {http://dx.doi.org/10.1137/1.9781611972016} +} + +@Article{jk09c, + Title = {Taylor expansions of solutions of stochastic partial differential equations with additive noise}, + Author = {Jentzen, Arnulf and Kloeden, Peter E.}, + Journal = {Ann. Probab.}, + Year = {2010}, + Number = {2}, + Pages = {532--569}, + Volume = {38}, + + Fjournal = {The Annals of Probability} +} + +@Article{Jentzen2009c, + author = {Arnulf Jentzen and Peter E. Kloeden}, + journal = {Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, + title = {Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise}, + year = {2009}, + issn = {1364-5021}, + number = {2102}, + pages = {649--667}, + volume = {465}, + fjournal = {Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H15 60H35)}, + mrnumber = {MR2471778}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{jk09a, + Title = {Pathwise {T}aylor schemes for random ordinary differential + equations}, + Author = {Jentzen, Arnulf and Kloeden, Peter E.}, + Journal = {BIT}, + Year = {2009}, + Number = {1}, + Pages = {113--140}, + Volume = {49}, + + Fjournal = {BIT. Numerical Mathematics}, + ISSN = {0006-3835}, + Mrclass = {65C30}, + Mrnumber = {MR2486131} +} + + + +@Article{jk09d, + Title = {The numerical approximation of stochastic partial differential equations}, + Author = {Jentzen, Arnulf and Kloeden, Peter E.}, + Journal = {Milan J. Math.}, + Year = {2009}, + Number = {1}, + Pages = {205--244}, + Volume = {77}, + + Fjournal = {Milan Journal of Mathematics}, + ISSN = {1424-9286} +} + +@Article{jk09e, + Title = {A unified existence and uniqueness theorem for stochastic evolution equations}, + Author = {Jentzen, Arnulf and Kloeden, Peter E.}, + Journal = {Bull. Austral. Math. Soc.}, + Year = {2009}, + Note = {To appear}, + Pages = {17 pages}, + + Fjournal = {Bulletin of the Australian Mathematical Society} +} + +@Article{Jentzen2007a, + author = {Arnulf Jentzen and Peter E. Kloeden}, + journal = {Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, + title = {Pathwise convergent higher order numerical schemes for random ordinary differential equations}, + year = {2007}, + issn = {1364-5021}, + number = {2087}, + pages = {2929--2944}, + volume = {463}, + fjournal = {Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences}, + groups = {sdiyora:6}, + mrclass = {65C30 (65L07)}, + mrnumber = {MR2360184 (2008m:65017)}, + mrreviewer = {Domingo Hern{\'a}ndez-Abreu}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Jentzen2009e, + author = {A. Jentzen and P. E. Kloeden and A. Neuenkirch}, + journal = {Numer. Math.}, + title = {Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global {L}ipschitz coefficients}, + year = {2009}, + issn = {0029-599X}, + number = {1}, + pages = {41--64}, + volume = {112}, + coden = {NUMMA7}, + fjournal = {Numerische Mathematik}, + groups = {sdiyora:6}, + mrclass = {60H35}, + mrnumber = {MR2481529}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Jentzen2009f, + author = {A. Jentzen and P. E. Kloeden and A. Neuenkirch}, + booktitle = {Foundations of computational mathematics, {H}ong {K}ong 2008}, + publisher = {Cambridge Univ. Press}, + title = {Pathwise convergence of numerical schemes for random and stochastic differential equations}, + year = {2009}, + address = {Cambridge}, + pages = {140--161}, + series = {London Math. Soc. Lecture Note Ser.}, + volume = {363}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H10 60H35)}, + mrnumber = {2562499}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{jkn09a, + Title = {Pathwise approximation of stochastic differential equations on + domains: higher order convergence rates without global + {L}ipschitz coefficients}, + Author = {Jentzen, A. and Kloeden, P. E. and Neuenkirch, A.}, + Journal = {Numer. Math.}, + Year = {2009}, + Number = {1}, + Pages = {41--64}, + Volume = {112}, + + Coden = {NUMMA7}, + Fjournal = {Numerische Mathematik}, + ISSN = {0029-599X}, + Mrclass = {60H35}, + Mrnumber = {MR2481529} +} + +@InCollection{jkn09b, + Title = {Pathwise convergence of numerical schemes for random and stochastic differential equations}, + Author = {Jentzen, A. and Kloeden, P. E. and Neuenkirch, A.}, + Publisher = {Cambridge University Press}, + Year = {2009}, + Pages = {140--161}, + + Editors = {Cucker, F. and Pinkus, A. and Todd, M.}, + Fjournal = {Foundations of Computational Mathematics: Hong Kong 2008}, + Journal = {Found. Comput. Math.} +} + +@Article{Jentzen2011b, + author = {Arnulf Jentzen and Peter E. Kloeden and Georg Winkel}, + journal = {Ann. Appl. Probab.}, + title = {Efficient simulation of nonlinear parabolic {SPDE}s with additive noise}, + year = {2011}, + number = {3}, + pages = {908--950}, + volume = {21}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{JentzenKurniawan2015, + Title = {{Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients}}, + Author = {{Jentzen}, A. and {Kurniawan}, R.}, + Journal = {arXiv:1501.03539}, + Year = {2015}, + Pages = {51 pages} +} + +@Article{jlsbs09, + Title = {An improved Maximum Allowable Transfer Interval for Networked Control Systems}, + Author = {Jentzen, A. and Leber, F. and Schneisgen, D. and Berger, A. and Siegmund, S}, + Journal = {IEEE Trans. Automat. Control}, + Year = {2009}, + Note = {in revision}, + Pages = {10}, + + Fjournal = {Institute of Electrical and Electronics Engineers. + Transactions on Automatic Control} +} + +@Article{Jentzen2015b, + author = {Arnulf Jentzen and Thomas M{\"u}ller-Gronbach and Larisa Yaroslavtseva}, + journal = {arXiv:1506.02828}, + title = {On stochastic differential equations with arbitrary slow convergence rates for strong approximation}, + year = {2015}, + pages = {1--26}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{jn09, + Title = {A random {E}uler scheme for {C}arath\'eodory differential + equations}, + Author = {Jentzen, A. and Neuenkirch, A.}, + Journal = {J. Comput. Appl. Math.}, + Year = {2009}, + Number = {1}, + Pages = {346--359}, + Volume = {224}, + + Coden = {JCAMDI}, + Fjournal = {Journal of Computational and Applied Mathematics}, + ISSN = {0377-0427}, + Mrclass = {65L05}, + Mrnumber = {MR2474237} +} + +@Article{jp2015, + Title = {{Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities}}, + Author = {{Jentzen}, A. and {Pu{\v s}nik}, P.}, + Journal = {arXiv:1504.03523}, + Year = {2015}, + Pages = {38 pages}, +Note = {To appear in \emph{IMA J.\ Num.\ Anal.}} +} + +@Article{Jentzen2015c, + author = {Jentzen, Arnulf and R{\"o}ckner, Michael}, + journal = {Found. Comput. Math.}, + title = {A {M}ilstein scheme for {SPDE}s}, + year = {2015}, + issn = {1615-3375}, + number = {2}, + pages = {313--362}, + volume = {15}, + doi = {10.1007/s10208-015-9247-y}, + fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H35)}, + mrnumber = {3320928}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s10208-015-9247-y}, +} + +@Article{jr10, + Title = {Regularity analysis of stochastic partial differential equations with nonlinear multiplicative trace class noise}, + Author = {Jentzen, Arnulf and R{\"o}ckner, Michael}, + Journal = {Journal of Differential Equations}, + Year = {2012}, + Number = {1}, + Pages = {114--136}, + Volume = {252}, + + Fjournal = {Journal of Differential Equations} +} + +@Article{jr10b, + Title = {A {M}ilstein scheme for {SPDEs}}, + Author = {Jentzen, Arnulf and R{\"o}ckner, Michael}, + Journal = {arXiv:1001.2751v4}, + Year = {2012}, + Pages = {37 pages} +} + +@InCollection{j91, + Title = {Lattice approximation of a nonlinear stochastic partial + differential equation with white noise}, + Author = {Jetschke, Gottfried}, + Booktitle = {Random partial differential equations ({O}berwolfach, 1989)}, + Publisher = {Birkh\"auser}, + Year = {1991}, + + Address = {Basel}, + Pages = {107--126}, + Series = {Internat. Ser. Numer. Math.}, + Volume = {102}, + + Mrclass = {60H15 (35R60)}, + Mrnumber = {MR1185742 (93h:60091)}, + Mrreviewer = {Wilfried Grecksch} +} + +@InCollection{Jetschke1991, + author = {Gottfried Jetschke}, + booktitle = {Random partial differential equations ({O}berwolfach, 1989)}, + publisher = {Birkh{\"a}user}, + title = {Lattice approximation of a nonlinear stochastic partial differential equation with white noise}, + year = {1991}, + address = {Basel}, + pages = {107--126}, + series = {Internat. Ser. Numer. Math.}, + volume = {102}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R60)}, + mrnumber = {MR1185742 (93h:60091)}, + mrreviewer = {Wilfried Grecksch}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Jona-Lasinio1985, + author = {Jona-Lasinio, G. and Mitter, P. K.}, + journal = {Communications in Mathematical Physics}, + title = {On the stochastic quantization of field theory}, + year = {1985}, + number = {3}, + pages = {409�436}, + volume = {101}, + file = {Snapshot:/Users/Hao/Library/Application Support/Zotero/Profiles/qg7ecoj6.default/zotero/storage/9M26E4ZT/10.html:text/html}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.springerlink.com/index/L1G0588734G0R075.pdf}, + urldate = {2013-01-21}, +} + +@Book{k04a, + Title = {Analysis. 1}, + Author = {K{\"o}nigsberger, Konrad}, + Publisher = {Springer-Verlag}, + Year = {2004}, + + Address = {Berlin}, + Edition = {Sixth}, + Series = {Springer-Lehrbuch. [Springer Textbook]}, + + ISBN = {3-540-40371-X}, + Mrclass = {26-01}, + Mrnumber = {MR2374633}, + Pages = {xiv+412} +} + +@Book{k04b, + Title = {Analysis. 2}, + Author = {K{\"o}nigsberger, Konrad}, + Publisher = {Springer-Verlag}, + Year = {2004}, + + Address = {Berlin}, + Edition = {Fifth}, + Series = {Springer-Lehrbuch. [Springer Textbook]}, + + ISBN = {3-540-43580-8}, + Mrclass = {00A05 (26-01)}, + Mrnumber = {MR1251736 (94i:00001)}, + Pages = {x+365} +} + +@Book{Kuehn2004, + author = {Christoph K{\"u}hn}, + title = {Stochastische Analysis mit Finanzmathematik}, + year = {2004}, + groups = {sdiyora:6}, + journal = {Lecture notes}, + owner = {sdiyora}, + pages = {86}, + timestamp = {2016.02.02}, + url = {http://www.math.uni-frankfurt.de/~ismi/kuehn/stoch_ana.pdf}, +} + +@Book{Kahaner1989, + author = {David Kahaner and Cleve B. 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Schmidt}, + owner = {sdiyora}, + pages = {xx+638}, + timestamp = {2016.02.02}, +} + +@Book{Kallianpur1995, + author = {Gopinath Kallianpur and Jie Xiong}, + publisher = {Inst of Mathematical Statistic}, + title = {Stochastic {D}ifferential {E}quations in {I}nfinite {D}imensional {S}paces}, + year = {1995}, + series = {Lecture Notes-Monographs}, + volume = {26}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {345}, + timestamp = {2016.02.02}, +} + +@Article{Kallianpur1994, + author = {Kallianpur, G. and Xiong, J.}, + journal = {Adv. in Appl. Probab.}, + title = {Stochastic models of environmental pollution}, + year = {1994}, + issn = {0001-8678}, + number = {2}, + pages = {377--403}, + volume = {26}, + coden = {AAPBBD}, + doi = {10.2307/1427442}, + fjournal = {Advances in Applied Probability}, + groups = {sdiyora:6}, + mrclass = {60G55 (60H15 90A30)}, + mrnumber = {1272718 (95g:60061)}, + mrreviewer = {S. 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Katsoulakis and Georgios T. Kossioris and Omar Lakkis}, + journal = {Interfaces Free Bound.}, + title = {Noise regularization and computations for the 1-dimensional stochastic {A}llen-{C}ahn problem}, + year = {2007}, + issn = {1463-9963}, + number = {1}, + pages = {1--30}, + volume = {9}, + fjournal = {Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation}, + groups = {sdiyora:6}, + mrclass = {35R60 (35B25 35K55 65C30)}, + mrnumber = {MR2317297 (2008d:35244)}, + mrreviewer = {Alp O. Eden}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{kkl07, + Title = {Noise regularization and computations for the 1-dimensional + stochastic {A}llen-{C}ahn problem}, + Author = {Katsoulakis, Markos A. and Kossioris, Georgios T. and Lakkis, + Omar}, + Journal = {Interfaces Free Bound.}, + Year = {2007}, + Number = {1}, + Pages = {1--30}, + Volume = {9}, + + Fjournal = {Interfaces and Free Boundaries. Mathematical Modelling, + Analysis and Computation}, + ISSN = {1463-9963}, + Mrclass = {35R60 (35B25 35K55 65C30)}, + Mrnumber = {MR2317297 (2008d:35244)}, + Mrreviewer = {Alp O. Eden} +} + +@Article{Kebaier2005, + author = {Ahmed Kebaier}, + journal = {Ann. Appl. Probab.}, + title = {Statistical {R}omberg extrapolation: a new variance reduction method and applications to option pricing}, + year = {2005}, + issn = {1050-5164}, + number = {4}, + pages = {2681--2705}, + volume = {15}, + doi = {10.1214/105051605000000511}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + mrclass = {60H30 (60F05 60H10 65C05 91B28)}, + mrnumber = {2187308 (2006j:60072)}, + mrreviewer = {Pedro Mar{\'i}n Rubio}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/105051605000000511}, +} + +@Article{Keller2013, + author = {Raphael Keller}, + journal = {Semester thesis, ETH Zurich}, + title = {Exact Computation Approach for Approximations and Regularity of Convolutions of Powers of {O}rnstein-{U}hlenbeck Processes}, + year = {2013}, + pages = {18 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Khasminskii2001, + author = {Khasminskii, R. Z. and Klebaner, F. C.}, + journal = {Ann. Appl. Probab.}, + title = {Long term behavior of solutions of the {L}otka-{V}olterra system under small random perturbations}, + year = {2001}, + issn = {1050-5164}, + number = {3}, + pages = {952--963}, + volume = {11}, + doi = {10.1214/aoap/1015345354}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + mrclass = {34F05 (34C60 34E10 60H10 92D25)}, + mrnumber = {MR1865029 (2002h:34106)}, + mrreviewer = {Jan M. Swart}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/aoap/1015345354}, +} + +@Article{Kimura1955, + author = {Kimura, M}, + journal = {Proc. Natl. Acad. Sci. 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E. Kloeden}, + journal = {Milan J. Math.}, + title = {The systematic derivation of higher order numerical schemes for stochastic differential equations}, + year = {2002}, + issn = {1424-9286}, + pages = {187--207}, + volume = {70}, + doi = {10.1007/s00032-002-0006-6}, + fjournal = {Milan Journal of Mathematics}, + groups = {sdiyora:6}, + mrclass = {65C30 (35R60 60H10)}, + mrnumber = {1936751 (2003k:65007)}, + mrreviewer = {Madalina Deaconu}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s00032-002-0006-6}, +} + +@Article{kj07, + Title = {Pathwise convergent higher order numerical schemes for random + ordinary differential equations}, + Author = {Kloeden, Peter E. and Jentzen, Arnulf}, + Journal = {Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.}, + Year = {2007}, + Number = {2087}, + Pages = {2929--2944}, + Volume = {463}, + + Fjournal = {Proceedings of The Royal Society of London. Series A. + Mathematical, Physical and Engineering Sciences}, + ISSN = {1364-5021}, + Mrclass = {65C30 (65L07)}, + Mrnumber = {MR2360184 (2008m:65017)}, + Mrreviewer = {Domingo Hern{\'a}ndez-Abreu} +} + +@Article{Kloeden2011, + author = {Kloeden, P. E. and Lord, G. J. and Neuenkirch, A. and Shardlow, T.}, + journal = {J. Comput. Appl. Math.}, + title = {The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds}, + year = {2011}, + issn = {0377-0427}, + number = {5}, + pages = {1245--1260}, + volume = {235}, + coden = {JCAMDI}, + doi = {10.1016/j.cam.2010.08.011}, + fjournal = {Journal of Computational and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H15)}, + mrnumber = {2728063}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.cam.2010.08.011}, +} + +@Article{klns09, + Title = {The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds}, + Author = {Kloeden, P. E. and Lord, G. J. and Neuenkirch, A. and Shardlow, T.}, + Year = {2009}, + Note = {Submitted (2009)} +} + +@Article{Kloeden2013, + author = {Kloeden, P. E. and Neuenkirch, A.}, + journal = {Recent Developments in Computational Finance, ed. Thomas Gerstner and Peter Kloeden, World Scientific}, + title = {Convergence of numerical methods for stochastic differential equations in mathematical finance}, + year = {2013}, + pages = {49--80}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kloeden2012, + author = {Kloeden, P. E. and Neuenkirch, A.}, + journal = {preprint}, + title = {Convergence of numerical methods for stochastic differential equations in mathematical finance}, + year = {2012}, + pages = {1--28}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kloeden2007, + author = {Peter E. Kloeden and Andreas Neuenkirch}, + journal = {LMS J. Comput. Math.}, + title = {The pathwise convergence of approximation schemes for stochastic differential equations}, + year = {2007}, + issn = {1461-1570}, + pages = {235--253}, + volume = {10}, + fjournal = {LMS Journal of Computation and Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H10 65C30)}, + mrnumber = {2320830 (2008e:60213)}, + mrreviewer = {Xue Rong Mao}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{kp92, + Title = {Numerical solution of stochastic differential equations}, + Author = {Kloeden, Peter E. and Platen, Eckhard}, + Publisher = {Springer-Verlag}, + Year = {1992}, + + Address = {Berlin}, + Series = {Applications of Mathematics (New York)}, + Volume = {23}, + + ISBN = {3-540-54062-8}, + Mrclass = {60H10 (34A50 34F05 65L99 65P05)}, + Mrnumber = {MR1214374 (94b:60069)}, + Mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n}, + Pages = {xxxvi+632} +} + +@Book{Kloeden1994, + author = {Peter E. Kloeden and Eckhard Platen and Henri Schurz}, + publisher = {Springer-Verlag}, + title = {Numerical solution of {SDE} through computer experiments}, + year = {1994}, + address = {Berlin}, + isbn = {3-540-57074-8}, + note = {With 1 IBM-PC floppy disk (3.5 inch; HD)}, + series = {Universitext}, + groups = {sdiyora:6}, + mrclass = {60H10 (34-04 34F05 65U05)}, + mrnumber = {MR1260431 (95a:60084)}, + mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n}, + owner = {sdiyora}, + pages = {xiv+292}, + timestamp = {2016.02.02}, +} + +@Article{Kloeden2001, + author = {P. E. Kloeden and S. Shott}, + journal = {J. Appl. Math. Stochastic Anal.}, + title = {Linear-implicit strong schemes for {I}t{\^o}-{G}alerkin approximations of stochastic {PDE}s}, + year = {2001}, + issn = {1048-9533}, + number = {1}, + pages = {47--53}, + volume = {14}, + fjournal = {Journal of Applied Mathematics and Stochastic Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (35A35 35R60 60H35 65C30)}, + mrnumber = {MR1825100 (2002b:60113)}, + mrreviewer = {Wilfried Grecksch}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{ks01, + Title = {Linear-implicit strong schemes for {I}t\^o-{G}alerkin + approximations of stochastic {PDE}s}, + Author = {Kloeden, P. E. and Shott, S.}, + Journal = {J. Appl. Math. Stochastic Anal.}, + Year = {2001}, + Note = {Special issue: Advances in applied stochastics}, + Number = {1}, + Pages = {47--53}, + Volume = {14}, + + Fjournal = {Journal of Applied Mathematics and Stochastic Analysis}, + ISSN = {1048-9533}, + Mrclass = {60H15 (35A35 35R60 60H35 65C30)}, + Mrnumber = {MR1825100 (2002b:60113)}, + Mrreviewer = {Wilfried Grecksch} +} + +@InProceedings{Kohatsu-Higa2012, + author = {Kohatsu-Higa, Arturo and Lejay, Antoine and Yasuda, Kazuhiro}, + booktitle = {Mathematical Economics}, + title = {On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient}, + year = {2012}, + pages = {94--106}, + volume = {1788}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kolmogoroff1941, + author = {Kolmogoroff, A.}, + journal = {C. R. (Doklady) Acad. Sci. URSS (N.S.)}, + title = {The local structure of turbulence in incompressible viscous fluid for very large {R}eynold's numbers}, + year = {1941}, + pages = {301--305}, + volume = {30}, + groups = {sdiyora:6}, + mrclass = {76.1X}, + mrnumber = {0004146 (2,327d)}, + mrreviewer = {W. R. Sears}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kolmogorov1991, + author = {Kolmogorov, A. N.}, + journal = {Proc. Roy. Soc. London Ser. A}, + title = {The local structure of turbulence in incompressible viscous fluid for very large {R}eynolds numbers}, + year = {1991}, + issn = {0962-8444}, + note = {Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov's ideas 50 years on}, + number = {1890}, + pages = {9--13}, + volume = {434}, + coden = {PRLAAZ}, + doi = {10.1098/rspa.1991.0075}, + fjournal = {Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences}, + groups = {sdiyora:6}, + mrclass = {76F05}, + mrnumber = {1124922 (92h:76049)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1098/rspa.1991.0075}, +} + +@Article{Kolmogorov1962, + author = {Kolmogorov, A. N.}, + journal = {J. Fluid Mech.}, + title = {A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high {R}eynolds number}, + year = {1962}, + issn = {0022-1120}, + pages = {82--85}, + volume = {13}, + fjournal = {Journal of Fluid Mechanics}, + groups = {sdiyora:6}, + mrclass = {76.60}, + mrnumber = {0139329 (25 \#2763)}, + mrreviewer = {A. A. Townsend}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kolmogorov1931, + author = {Kolmogorov, Andrey N.}, + journal = {Math. Ann.}, + title = {\"{U}ber die analytischen {M}ethoden in der {W}ahrscheinlichkeitsrechnung (in german)}, + year = {1931}, + pages = {415--458}, + volume = {104}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kopec2014, + author = {Marie Kopec}, + journal = {Ph{D} thesis, ENS Rennes}, + title = {Quelques contributions \`{a} l'analyse num\'{e}rique d'\'{e}quations stochastiques}, + year = {2014}, + pages = {viii+189}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kossioris2009, + Title = {{Fully-Discrete Finite Element Approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case}}, + Author = {{Kossioris}, G.~T. and {Zouraris}, G.~E.}, + Journal = {arXiv:0906.1828v1}, + Year = {2009}, + + Archiveprefix = {arXiv}, + Eprint = {0906.1828}, + Keywords = {Mathematics - Numerical Analysis, 65M60, 65M15, 65C20}, + Primaryclass = {math.NA} +} + +@Article{kz09, + Title = {Fully-Discrete Finite Element Approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case}, + Author = {Kossioris, G. T. and Zouraris, G. E.}, + Journal = {arXiv:0906.1828v1}, + Year = {2009}, + Pages = {23} +} + +@Article{Kotelenez1982/83, + author = {Kotelenez, Peter}, + journal = {Stochastics}, + title = {A submartingale type inequality with applications to stochastic evolution equations}, + year = {1982/83}, + issn = {0090-9491}, + number = {2}, + pages = {139--151}, + volume = {8}, + coden = {STOCB2}, + fjournal = {Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H15 (60H05)}, + mrnumber = {686575 (84h:60115)}, + mrreviewer = {Maurizio Pratelli}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Kotelenez2008, + author = {Peter Kotelenez}, + publisher = {Springer-Verlag}, + title = {Stochastic Ordinary and Stochastic Partial Differential Equations}, + year = {2008}, + address = {Berlin}, + isbn = {978-0-387-74316-5}, + series = {Stochastic Modelling and Applied Probability}, + volume = {58}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {460}, + timestamp = {2016.02.02}, +} + +@Article{Kotelenez1987, + author = {Kotelenez, Peter}, + journal = {Stochastics}, + title = {A maximal inequality for stochastic convolution integrals on {H}ilbert spaces and space-time regularity of linear stochastic partial differential equations}, + year = {1987}, + issn = {0090-9491}, + number = {4}, + pages = {345--358}, + volume = {21}, + coden = {STOCB2}, + doi = {10.1080/17442508708833463}, + fjournal = {Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H05 (35R60 46G99 47D05 60H15)}, + mrnumber = {905052 (89h:60087)}, + mrreviewer = {Ya. {\=I}. B{\={\i}}lopol{\cprime}s{\cprime}ka}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1080/17442508708833463}, +} + +@Article{Kotelenez1984, + author = {Kotelenez, Peter}, + journal = {Stochastic Anal. Appl.}, + title = {A stopped {D}oob inequality for stochastic convolution integrals and stochastic evolution equations}, + year = {1984}, + issn = {0736-2994}, + number = {3}, + pages = {245--265}, + volume = {2}, + coden = {SAAPDA}, + doi = {10.1080/07362998408809036}, + fjournal = {Stochastic Analysis and Applications}, + groups = {sdiyora:6}, + mrclass = {60H05 (60G44 60H15)}, + mrnumber = {757338 (86k:60096)}, + mrreviewer = {Ali S{\"u}leyman {\"U}st{\"u}nel}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1080/07362998408809036}, +} + +@Article{Kouritzin2002, + author = {Kouritzin, Michael A. and Long, Hongwei}, + journal = {Ann. Appl. Probab.}, + title = {Convergence of {M}arkov chain approximations to stochastic reaction-diffusion equations}, + year = {2002}, + issn = {1050-5164}, + number = {3}, + pages = {1039--1070}, + volume = {12}, + doi = {10.1214/aoap/1031863180}, + fjournal = {The Annals of Applied Probability}, + groups = {sdiyora:6}, + mrclass = {60H15 (60F15 60G55 60J27 60K35)}, + mrnumber = {1925451 (2003h:60097)}, + mrreviewer = {Min Kang}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/aoap/1031863180}, +} + +@Article{Kouritzin2003, + author = {Kouritzin, Michael A. and Long, Hongwei and Sun, Wei}, + journal = {Stochastic Anal. Appl.}, + title = {On {M}arkov chain approximations to semilinear partial differential equations driven by {P}oisson measure noise}, + year = {2003}, + issn = {0736-2994}, + number = {2}, + pages = {419--441}, + volume = {21}, + coden = {SAAPDA}, + doi = {10.1081/SAP-120019293}, + fjournal = {Stochastic Analysis and Applications}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K57 35R60 60K37)}, + mrnumber = {1967721 (2004c:60185)}, + mrreviewer = {Vivek S. Borkar}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1081/SAP-120019293}, +} + +@article {KovacsLarssonLindgren2018, + AUTHOR = {Kov\'{a}cs, Mih\'{a}ly and Larsson, Stig and Lindgren, Fredrik}, + TITLE = {On the discretisation in time of the stochastic {A}llen-{C}ahn + equation}, + JOURNAL = {Math. Nachr.}, + FJOURNAL = {Mathematische Nachrichten}, + VOLUME = {291}, + YEAR = {2018}, + NUMBER = {5-6}, + PAGES = {966--995}, + ISSN = {0025-584X}, + MRCLASS = {60H15 (60H35 65M75)}, + MRNUMBER = {3795566}, + DOI = {10.1002/mana.201600283}, + URL = {https://doi.org/10.1002/mana.201600283}, +} + +@Article{kll2015, + Title = {{On the discretization in time of the stochastic Allen-Cahn equation}}, + Author = {{Kov{\'a}cs}, M. and {Larsson}, S. and {Lindgren}, F.}, + Journal = {arXiv:1510.03684}, + Year = {2015}, + Pages = {34 pages} +} + +@Article{Kovacs2015, + author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, Fredrik}, + journal = {J. Appl. Probab.}, + title = {On the backward {E}uler approximation of the stochastic {A}llen-{C}ahn equation}, + year = {2015}, + issn = {0021-9002}, + number = {2}, + pages = {323--338}, + volume = {52}, + doi = {10.1239/jap/1437658601}, + fjournal = {Journal of Applied Probability}, + groups = {sdiyora:6}, + mrclass = {60H15 (60H35 65C30)}, + mrnumber = {3372078}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1239/jap/1437658601}, +} + +@Article{Kovacs2013, + author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, Fredrik}, + journal = {arXiv:1311.2067v3}, + title = {On the backward {E}uler approximation of the stochastic {Allen-Cahn} equation}, + year = {2013}, + pages = {16 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kovacs2013a, + author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, Fredrik}, + journal = {BIT}, + title = {Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise {II}. {F}ully discrete schemes}, + year = {2013}, + issn = {0006-3835}, + number = {2}, + pages = {497--525}, + volume = {53}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {65M60 (60H15 60H35 65M75)}, + mrnumber = {3123856}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kovacs2012a, + author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, Fredrik}, + journal = {BIT}, + title = {Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise}, + year = {2012}, + issn = {0006-3835}, + number = {1}, + pages = {85--108}, + volume = {52}, + doi = {10.1007/s10543-011-0344-2}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {65M60 (35R60 60H15 60H35 65C30)}, + mrnumber = {2891655 (2012m:65336)}, + mrreviewer = {B{\"u}lent Karas{\"o}zen}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s10543-011-0344-2}, +} + +@Article{kll08, + Title = {Strong convergence of the finite element method with truncated + noise for semilinear parabolic stochastic equations with + additive noise}, + Author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, + Fredrik}, + Journal = {Numer. Algorithms}, + Year = {2010}, + Number = {2-3}, + Pages = {309--320}, + Volume = {53}, + + Doi = {10.1007/s11075-009-9281-4}, + Fjournal = {Numerical Algorithms}, + ISSN = {1017-1398}, + Mrclass = {65C30 (35K58 35R60 60H15 65N75)}, + Mrnumber = {2600932 (2011c:65015)}, + Mrreviewer = {Marius Ghergu} +} + +@Article{Kovacs2010, + author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Lindgren, Fredrik}, + journal = {Numer. Algorithms}, + title = {Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise}, + year = {2010}, + issn = {1017-1398}, + number = {2-3}, + pages = {309--320}, + volume = {53}, + doi = {10.1007/s11075-009-9281-4}, + fjournal = {Numerical Algorithms}, + groups = {sdiyora:6}, + mrclass = {65C30 (35K58 35R60 60H15 65N75)}, + mrnumber = {2600932}, + mrreviewer = {Marius Ghergu}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s11075-009-9281-4}, +} + +@Article{Kovacs2011a, + author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Mesforush, Ali}, + journal = {SIAM J. Numer. Anal.}, + title = {Finite element approximation of the {C}ahn-{H}illiard-{C}ook equation}, + year = {2011}, + issn = {0036-1429}, + number = {6}, + pages = {2407--2429}, + volume = {49}, + coden = {SJNAAM}, + doi = {10.1137/110828150}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {65M75 (35R60 60H15 60H35 65M60)}, + mrnumber = {2854602 (2012j:65357)}, + mrreviewer = {Bruno Scheurer}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/110828150}, +} + +@Article{Kovacs2010a, + author = {Mih{\'a}ly Kov{\'a}cs and Stig Larsson and Fardin Saedpanah}, + journal = {SIAM J. Numer. Anal.}, + title = {Finite element approximation of the linear stochastic wave equation with additive noise}, + year = {2010}, + issn = {0036-1429}, + number = {2}, + pages = {408--427}, + volume = {48}, + doi = {10.1137/090772241}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {65M60 (60H15 65C30)}, + mrnumber = {2646102}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/090772241}, +} + +@Article{kss10, + Title = {Finite element approximation of the linear stochastic wave + equation with additive noise}, + Author = {Kov{\'a}cs, Mih{\'a}ly and Larsson, Stig and Saedpanah, + Fardin}, + Journal = {SIAM J. Numer. Anal.}, + Year = {2010}, + Number = {2}, + Pages = {408--427}, + Volume = {48}, + + Doi = {10.1137/090772241}, + Fjournal = {SIAM Journal on Numerical Analysis}, + ISSN = {0036-1429}, + Mrclass = {65M60 (60H35 65C30)}, + Mrnumber = {2646102 (2011c:65215)}, + Mrreviewer = {Gon{\c{c}}alo Dos Reis}, + Url = {http://dx.doi.org/10.1137/090772241} +} + +@Book{Kruse2014book, + author = {Kruse, Raphael}, + publisher = {Springer, Cham}, + title = {Strong and weak approximation of semilinear stochastic evolution equations}, + year = {2014}, + isbn = {978-3-319-02230-7; 978-3-319-02231-4}, + series = {Lecture Notes in Mathematics}, + volume = {2093}, + doi = {10.1007/978-3-319-02231-4}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R60 60H07 65-02 65Cxx)}, + mrnumber = {3154916}, + owner = {sdiyora}, + pages = {xiv+177}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-3-319-02231-4}, +} + +@Article{rk13, + Title = {{Consistency and Stability of a Milstein-Galerkin Finite Element Scheme for Semilinear SPDE}}, + Author = {{Kruse}, R.}, + Journal = {arXiv:1307.4120v1}, + Year = {2013}, + Pages = {38 pages}, + + Keywords = {Mathematics - Numerical Analysis, Mathematics - Probability, 60H15, 65C30, 65M60, 65M70} +} + +@Article{k11, + Title = {{Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise}}, + Author = {{Kruse}, R.}, + Journal = {arXiv:1103.4504v1}, + Year = {2011}, + + Archiveprefix = {arXiv}, + Eprint = {1103.4504}, + Keywords = {Mathematics - Numerical Analysis, Mathematics - Analysis of PDEs, Mathematics - Probability, 60H15, 65C30, 65M60, 65M70}, + Primaryclass = {math.NA} +} + +@Article{Kruse2010, + author = {Raphael Kruse}, + journal = {BIT}, + title = {Characterization of bistability for stochastic multistep methods}, + year = {2010}, + number = {2}, + pages = {389--407}, + volume = {52}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{k08, + Title = {Diskrete Approximation + gew{\"o}hnlicher und partieller + stochastischer Differentialgleichungen}, + Author = {Kruse, R.}, + Publisher = {Univerist{\"a}t Bielefeld}, + Year = {2008}, + + Address = {Bielefeld, Germany}, + Note = {Diplomarbeit} +} + +@Book{Kruse2008, + author = {R. Kruse}, + publisher = {Univerist{\"a}t Bielefeld}, + title = {Diskrete Approximation gew{\"o}hnlicher und partieller stochastischer Differentialgleichungen}, + year = {2008}, + address = {Bielefeld, Germany}, + note = {Diplomarbeit}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{kl10, + Title = {Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise}, + Author = {Raphael Kruse and Stig Larsson}, + Journal = {Electron. J. Probab.}, + Year = {2012}, + Pages = {no. 65, 1-19}, + Volume = {17}, + + Fjournal = {Electronic Journal of Probability} +} + +@Article{Kruse2011a, + author = {Raphael Kruse and Stig Larsson}, + journal = {arXiv:1109.6487v1}, + title = {Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise}, + year = {2011}, + pages = {19 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Kruse2010a, + author = {Raphael Kruse and Stig Larsson}, + journal = {http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb10093.pdf}, + title = {Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise}, + year = {2010}, + pages = {17 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{kr02, + Title = {Some new results in the theory of {SPDE}s in {S}obolev spaces}, + Author = {Krylov, N. V.}, + Booktitle = {Stochastic partial differential equations and applications + ({T}rento, 2002)}, + Publisher = {Dekker}, + Year = {2002}, + + Address = {New York}, + Pages = {325--336}, + Series = {Lecture Notes in Pure and Appl. Math.}, + Volume = {227}, + + Mrclass = {60H15 (35R60)}, + Mrnumber = {1919516 (2003g:60108)} +} + +@InCollection{Krylov1999, + author = {Krylov, N. V.}, + booktitle = {Stochastic {PDE}'s and {K}olmogorov equations in infinite dimensions ({C}etraro, 1998)}, + publisher = {Springer}, + title = {On {K}olmogorov's equations for finite-dimensional diffusions}, + year = {1999}, + address = {Berlin}, + pages = {1--63}, + series = {Lecture Notes in Math.}, + volume = {1715}, + doi = {10.1007/BFb0092417}, + groups = {sdiyora:6}, + mrclass = {60J60 (35R60 58J65 60J65)}, + mrnumber = {1731794 (2000k:60155)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BFb0092417}, +} + +@Article{k94, + Title = {A {$W^n_2$}-theory of the {D}irichlet problem for + {SPDE}s in general smooth domains}, + Author = {Krylov, N. V.}, + Journal = {Probab. Theory Related Fields}, + Year = {1994}, + Number = {3}, + Pages = {389--421}, + Volume = {98}, + + Coden = {PTRFEU}, + Doi = {10.1007/BF01192260}, + Fjournal = {Probability Theory and Related Fields}, + ISSN = {0178-8051}, + Mrclass = {60H15 (35R60)}, + Mrnumber = {MR1262972 (95e:60060)}, + Mrreviewer = {Krystyna Twardowska} +} + +@Article{Krylov1994, + author = {N. V. Krylov}, + journal = {Probab. Theory Related Fields}, + title = {A {$W^n\_2$}-theory of the {D}irichlet problem for {SPDE}s in general smooth domains}, + year = {1994}, + issn = {0178-8051}, + number = {3}, + pages = {389--421}, + volume = {98}, + coden = {PTRFEU}, + doi = {10.1007/BF01192260}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R60)}, + mrnumber = {MR1262972 (95e:60060)}, + mrreviewer = {Krystyna Twardowska}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF01192260}, +} + +@Article{Krylov1991, + author = {Krylov, N. V.}, + journal = {Theory Probab. Appl.}, + title = {A simple proof of the existence of a solution to the {I}t\^o equation with monotone coefficients}, + year = {1991}, + issn = {0040-361X}, + number = {3}, + pages = {583--587}, + volume = {35}, + doi = {10.1137/1135082}, + fjournal = {Theory of Probability and its Applications}, + groups = {sdiyora:6}, + mrclass = {60H10}, + mrnumber = {1091217 (92m:60047)}, + mrreviewer = {I. Gy{\"o}ngy}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/1135082}, +} + +@Book{Krylov1999a, + author = {Krylov, N. V. and R{\"o}ckner, M. and Zabczyk, J.}, + publisher = {Springer-Verlag}, + title = {Stochastic {PDE}'s and {K}olmogorov equations in infinite dimensions}, + year = {1999}, + address = {Berlin}, + isbn = {3-540-66545-5}, + note = {Lectures given at the 2nd C.I.M.E. Session held in Cetraro, August 24--September 1, 1998, Edited by G. Da Prato, Fondazione C.I.M.E.. [C.I.M.E. Foundation]}, + series = {Lecture Notes in Mathematics}, + volume = {1715}, + groups = {sdiyora:6}, + mrclass = {60-06 (00B25 35-06)}, + mrnumber = {1730228 (2000g:60004)}, + owner = {sdiyora}, + pages = {viii+213}, + timestamp = {2016.02.02}, +} + +@InCollection{kr79, + Title = {Stochastic evolution equations}, + Author = {Krylov, N. V. and Rozovski{\u\i}, B. L.}, + Booktitle = {Current problems in mathematics, {V}ol. 14 ({R}ussian)}, + Publisher = {Akad. Nauk SSSR}, + Year = {1979}, + + Address = {Vsesoyuz. Inst. Nauchn. i Tekhn. 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An International Journal of Probability and Stochastic Processes}, + groups = {sdiyora:6}, + mrclass = {65C30}, + mrnumber = {2677551}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.tandfonline.com/doi/abs/10.1080/17442508.2011.618884}, +} + +@Article{clp09, + Title = {Almost sure convergence for a semidiscrete {M}ilstein scheme + for {SPDE}s of {Z}akai type}, + Author = {Lang, Annika and Chow, Pao-Liu and Potthoff, J{\"u}rgen}, + Journal = {Stochastics}, + Year = {2010}, + Number = {1-3}, + Pages = {315--326}, + Volume = {82}, + + Doi = {10.1080/17442501003653497}, + Fjournal = {Stochastics. 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Comput. Math.}, + Year = {2007}, + Number = {2}, + Pages = {135--181}, + Volume = {7}, + + Fjournal = {Foundations of Computational Mathematics. The Journal of the + Society for the Foundations of Computational Mathematics}, + ISSN = {1615-3375}, + Mrclass = {65C30}, + Mrnumber = {MR2324415 (2008b:65014)}, + Mrreviewer = {A. D. Booth} +} + +@Article{mr07b, + Title = {An implicit {E}uler scheme with non-uniform time + discretization for heat equations with multiplicative noise}, + Author = {M{\"u}ller-Gronbach, Thomas and Ritter, Klaus}, + Journal = {BIT}, + Year = {2007}, + Number = {2}, + Pages = {393--418}, + Volume = {47}, + + Fjournal = {BIT. Numerical Mathematics}, + ISSN = {0006-3835}, + Mrclass = {60H35 (60H15 65C30)}, + Mrnumber = {MR2334046 (2008m:60132)}, + Mrreviewer = {St{\'e}phane Menozzi} +} + +@Article{Mueller-Gronbach2007, + author = {Thomas M{\"u}ller-Gronbach and Klaus Ritter}, + journal = {Found. Comput. Math.}, + title = {Lower bounds and nonuniform time discretization for approximation of stochastic heat equations}, + year = {2007}, + issn = {1615-3375}, + number = {2}, + pages = {135--181}, + volume = {7}, + fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics}, + groups = {sdiyora:6}, + mrclass = {65C30}, + mrnumber = {MR2324415 (2008b:65014)}, + mrreviewer = {A. D. Booth}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Mueller-Gronbach2007a, + author = {Thomas M{\"u}ller-Gronbach and Klaus Ritter}, + journal = {BIT}, + title = {An implicit {E}uler scheme with non-uniform time discretization for heat equations with multiplicative noise}, + year = {2007}, + issn = {0006-3835}, + number = {2}, + pages = {393--418}, + volume = {47}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H15 65C30)}, + mrnumber = {MR2334046 (2008m:60132)}, + mrreviewer = {St{\'e}phane Menozzi}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{mrw08, + Title = {Optimal pointwise approximation of infinite-dimensional + {O}rnstein-{U}hlenbeck processes}, + Author = {M{\"u}ller-Gronbach, Thomas and Ritter, Klaus and Wagner, Tim}, + Journal = {Stoch. Dyn.}, + Year = {2008}, + Number = {3}, + Pages = {519--541}, + Volume = {8}, + + Fjournal = {Stochastics and Dynamics}, + ISSN = {0219-4937}, + Mrclass = {60H15 (60H35 65C30)}, + Mrnumber = {MR2444516} +} + +@Article{Mueller-Gronbach2008a, + author = {Thomas M{\"u}ller-Gronbach and Klaus Ritter and Tim Wagner}, + journal = {Stoch. Dyn.}, + title = {Optimal pointwise approximation of infinite-dimensional {O}rnstein-{U}hlenbeck processes}, + year = {2008}, + issn = {0219-4937}, + number = {3}, + pages = {519--541}, + volume = {8}, + fjournal = {Stochastics and Dynamics}, + groups = {sdiyora:6}, + mrclass = {60H15 (60H35 65C30)}, + mrnumber = {MR2444516}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{mrw07, + Title = {Optimal pointwise approximation of a linear stochastic heat + equation with additive space-time white noise}, + Author = {M{\"u}ller-Gronbach, Thomas and Ritter, Klaus and Wagner, Tim}, + Booktitle = {Monte {C}arlo and quasi-{M}onte {C}arlo methods 2006}, + Publisher = {Springer}, + Year = {2007}, + + Address = {Berlin}, + Pages = {577--589}, + + Mrclass = {65C30 (60H15 60H35)}, + Mrnumber = {MR2479247} +} + +@InCollection{Mueller-Gronbach2007b, + author = {Thomas M{\"u}ller-Gronbach and Klaus Ritter and Tim Wagner}, + booktitle = {Monte {C}arlo and quasi-{M}onte {C}arlo methods 2006}, + publisher = {Springer}, + title = {Optimal pointwise approximation of a linear stochastic heat equation with additive space-time white noise}, + year = {2007}, + address = {Berlin}, + pages = {577--589}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H15 60H35)}, + mrnumber = {MR2479247}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Ma1994, + author = {Ma, Jin and Protter, Philip and Yong, Jiongmin}, + journal = {Probability Theory and Related Fields}, + title = {Solving forward-backward stochastic differential equations explicitly -- a four step scheme}, + year = {1994}, + number = {3}, + pages = {339--359}, + volume = {98}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Springer}, + timestamp = {2016.02.02}, +} + +@Book{Ma1999, + author = {Ma, Jin and Yong, Jiongmin}, + publisher = {Springer-Verlag}, + title = {Forward-backward stochastic differential equations and their applications}, + year = {1999}, + address = {Berlin}, + isbn = {3-540-65960-9}, + series = {Lecture Notes in Mathematics}, + volume = {1702}, + groups = {sdiyora:6}, + mrclass = {60H10}, + mrnumber = {1704232 (2000k:60118)}, + mrreviewer = {Fabio Antonelli}, + owner = {sdiyora}, + pages = {xiv+270}, + timestamp = {2016.02.02}, +} + +@Article{Ma2002, + author = {Ma, Jin and Zhang, Jianfeng and others}, + journal = {The Annals of Applied Probability}, + title = {Representation theorems for backward stochastic differential equations}, + year = {2002}, + number = {4}, + pages = {1390--1418}, + volume = {12}, + groups = {sdiyora:6}, + owner = {martin}, + publisher = {Institute of Mathematical Statistics}, + timestamp = {2014.07.25}, +} + +@Book{Ma1992a, + author = {Ma, Zhi Ming and R{\"o}ckner, Michael}, + publisher = {Springer-Verlag, Berlin}, + title = {Introduction to the theory of (nonsymmetric) {D}irichlet forms}, + year = {1992}, + isbn = {3-540-55848-9}, + series = {Universitext}, + doi = {10.1007/978-3-642-77739-4}, + groups = {sdiyora:6}, + mrclass = {60J40 (31C25 60J45)}, + mrnumber = {1214375 (94d:60119)}, + mrreviewer = {Tu Sheng Zhang}, + owner = {sdiyora}, + pages = {vi+209}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-3-642-77739-4}, +} + +@Book{Ma1992, + author = {Zhi-Ming Ma and Michael R{\"o}ckner}, + publisher = {Springer-Verlag}, + title = {Introduction to the Theory of (Non Symmetric) Dirichlet Forms}, + year = {1992}, + address = {Berlin}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Malliavin1995, + author = {Malliavin, Paul}, + publisher = {Springer-Verlag}, + title = {Integration and probability}, + year = {1995}, + address = {New York}, + isbn = {0-387-94409-5}, + note = {With the collaboration of H{\'e}l{\`e}ne Airault, Leslie Kay and G{\'e}rard Letac, Edited and translated from the French by Kay, With a foreword by Mark Pinsky}, + series = {Graduate Texts in Mathematics}, + volume = {157}, + doi = {10.1007/978-1-4612-4202-4}, + groups = {sdiyora:6}, + mrclass = {28-01 (42-01 46-01 60-01)}, + mrnumber = {1335234 (97f:28001a)}, + mrreviewer = {R. 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Math.}, + title = {Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally {L}ipschitz continuous coefficients}, + year = {2013}, + issn = {0377-0427}, + pages = {14--28}, + volume = {238}, + coden = {JCAMDI}, + doi = {10.1016/j.cam.2012.08.015}, + fjournal = {Journal of Computational and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {65C60}, + mrnumber = {2972586}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.cam.2012.08.015}, +} + +@Article{Mao2013a, + author = {Mao, Xuerong and Szpruch, Lukasz}, + journal = {Stochastics}, + title = {Strong convergence rates for backward {E}uler-{M}aruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients}, + year = {2013}, + issn = {1744-2508}, + number = {1}, + pages = {144--171}, + volume = {85}, + doi = {10.1080/17442508.2011.651213}, + fjournal = {Stochastics. 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Anal.}, + title = {Balanced implicit methods for stiff stochastic systems}, + year = {1998}, + issn = {0036-1429}, + number = {3}, + pages = {1010--1019 (electronic)}, + volume = {35}, + doi = {10.1137/S0036142994273525}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {65C20 (60H10)}, + mrnumber = {1619926 (99a:65019)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/S0036142994273525}, +} + +@Article{Milstein2013, + author = {Milstein, Grigori N and Schoenmakers, John}, + journal = {arXiv:1312.0876}, + title = {Uniform approximation of the Cox-Ingersoll-Ross process}, + year = {2013}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Milstein2013a, + author = {Milstein, G. N. and Tretyakov, M. V.}, + journal = {Adv. Appl. Prob.}, + title = {Probabilistic methods for the incompressible {N}avier-{S}tokes equations with space periodic conditions}, + year = {2013}, + number = {to appear}, + volume = {45}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Milstein2009, + author = {Milstein, G. N. and Tretyakov, M. V.}, + journal = {Math. Comp.}, + title = {Solving parabolic stochastic partial differential equations via averaging over characteristics}, + year = {2009}, + issn = {0025-5718}, + number = {268}, + pages = {2075--2106}, + volume = {78}, + coden = {MCMPAF}, + doi = {10.1090/S0025-5718-09-02250-9}, + fjournal = {Mathematics of Computation}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H15 60H35)}, + mrnumber = {2521279 (2010k:65014)}, + mrreviewer = {Carlos M. Mora Gonz{\'a}lez}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1090/S0025-5718-09-02250-9}, +} + +@Article{Milstein2005, + author = {G. N. Milstein and M. V. Tretyakov}, + journal = {SIAM J. Numer. Anal.}, + title = {Numerical integration of stochastic differential equations with nonglobally {L}ipschitz coefficients}, + year = {2005}, + issn = {0036-1429}, + number = {3}, + pages = {1139--1154 (electronic)}, + volume = {43}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H10 65C30)}, + mrnumber = {MR2177799}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Milstein2004, + author = {G. N. Milstein and M. V. Tretyakov}, + publisher = {Springer-Verlag}, + title = {Stochastic numerics for mathematical physics}, + year = {2004}, + address = {Berlin}, + isbn = {3-540-21110-1}, + series = {Scientific Computation}, + groups = {sdiyora:6}, + mrclass = {60-02 (35R60 37M15 60H10 60H15 60H35 65C30)}, + mrnumber = {2069903 (2005f:60004)}, + mrreviewer = {Peter E. Kloeden}, + owner = {sdiyora}, + pages = {xx+594}, + timestamp = {2016.02.02}, +} + +@Article{Minty1963, + author = {Minty, George J.}, + journal = {Proc. Nat. Acad. Sci. U.S.A.}, + title = {on a ``monotonicity'' method for the solution of non-linear equations in {B}anach spaces}, + year = {1963}, + issn = {0027-8424}, + pages = {1038--1041}, + volume = {50}, + fjournal = {Proceedings of the National Academy of Sciences of the United States of America}, + groups = {sdiyora:6}, + mrclass = {47.80}, + mrnumber = {0162159 (28 \$5358)}, + mrreviewer = {J. Peetre}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Minty1962, + author = {Minty, George J.}, + journal = {Duke Math. J.}, + title = {Monotone (nonlinear) operators in {H}ilbert space}, + year = {1962}, + issn = {0012-7094}, + pages = {341--346}, + volume = {29}, + fjournal = {Duke Mathematical Journal}, + groups = {sdiyora:6}, + mrclass = {47.80}, + mrnumber = {0169064 (29 \$6319)}, + mrreviewer = {I. {\v{S}}ragin}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Mishura2006, + Title = {Approximation schemes for stochastic differential equations in + a {H}ilbert space}, + Author = {Mishura, Yu. S. and Shevchenko, G. M.}, + Journal = {Teor. Veroyatn. Primen.}, + Year = {2006}, + Number = {3}, + Pages = {476--495}, + Volume = {51}, + + Doi = {10.1137/S0040585X97982487}, + Fjournal = {Rossi\u\i skaya Akademiya Nauk. Teoriya Veroyatnoste\u\i\ i ee + Primeneniya}, + ISSN = {0040-361X}, + Mrclass = {60H35 (60H20 65C30)}, + Mrnumber = {MR2325541 (2008h:60282)}, + Mrreviewer = {Andrzej Kozlowski} +} + +@Article{Mishura2006a, + author = {Yu. S. Mishura and G. M. Shevchenko}, + journal = {Teor. Veroyatn. Primen.}, + title = {Approximation schemes for stochastic differential equations in a {H}ilbert space}, + year = {2006}, + issn = {0040-361X}, + number = {3}, + pages = {476--495}, + volume = {51}, + doi = {10.1137/S0040585X97982487}, + fjournal = {Rossi\u\i skaya Akademiya Nauk. Teoriya Veroyatnoste\u\i\ i ee Primeneniya}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H20 65C30)}, + mrnumber = {MR2325541 (2008h:60282)}, + mrreviewer = {Andrzej Kozlowski}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/S0040585X97982487}, +} + +@Article{ms06, + Title = {Approximation schemes for stochastic differential equations in + a {H}ilbert space}, + Author = {Mishura, Yu. S. and Shevchenko, G. M.}, + Journal = {Teor. Veroyatn. Primen.}, + Year = {2006}, + Number = {3}, + Pages = {476--495}, + Volume = {51}, + + Doi = {10.1137/S0040585X97982487}, + Fjournal = {Rossi\u\i skaya Akademiya Nauk. Teoriya Veroyatnoste\u\i\ i ee + Primeneniya}, + ISSN = {0040-361X}, + Mrclass = {60H35 (60H20 65C30)}, + Mrnumber = {2325541 (2008h:60282)}, + Mrreviewer = {Andrzej Kozlowski} +} + +@Article{Mittmann2003, + author = {Mittmann, Katrin and Steinwart, Ingo}, + journal = {Georgian Math. J.}, + title = {On the existence of continuous modifications of vector-valued random fields}, + year = {2003}, + issn = {1072-947X}, + note = {Dedicated to the memory of Professor Revaz Chitashvili}, + number = {2}, + pages = {311--317}, + volume = {10}, + fjournal = {Georgian Mathematical Journal}, + groups = {sdiyora:6}, + mrclass = {60G60 (46N40)}, + mrnumber = {2009978 (2004i:60071)}, + mrreviewer = {Randall James Swift}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Mohammed2003, + author = {Mohammed, Salah-Eldin A. and Scheutzow, Michael K. R.}, + journal = {J. Funct. Anal.}, + title = {The stable manifold theorem for non-linear stochastic systems with memory. {I}. {E}xistence of the semiflow}, + year = {2003}, + issn = {0022-1236}, + number = {2}, + pages = {271--305}, + volume = {205}, + coden = {JFUAAW}, + doi = {10.1016/j.jfa.2002.04.001}, + fjournal = {Journal of Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H20 (34F05 34K50 37H05)}, + mrnumber = {2017689 (2004i:60097)}, + mrreviewer = {Vivek S. Borkar}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.jfa.2002.04.001}, +} + + + + +@article {MatthiasMorzfeld2011, + AUTHOR = {Morzfeld, Matthias and Tu, Xuemin and Atkins, Ethan and + Chorin, Alexandre J.}, + TITLE = {A random map implementation of implicit filters}, + JOURNAL = {J. Comput. Phys.}, + FJOURNAL = {Journal of Computational Physics}, + VOLUME = {231}, + YEAR = {2012}, + NUMBER = {4}, + PAGES = {2049--2066}, + ISSN = {0021-9991}, + MRCLASS = {62M20 (60J60 65C05)}, + MRNUMBER = {2876600}, + DOI = {10.1016/j.jcp.2011.11.022}, + URL = {https://doi.org/10.1016/j.jcp.2011.11.022}, +} + + +@Article{mtac12, + Title = {{A random map implementation of implicit filters}}, + Author = {{Morzfeld}, M. and {Tu}, X. and {Atkins}, E. and {Chorin}, A.~J. }, + Journal = {Journal of Computational Physics}, + Year = {2012}, + + Month = feb, + Pages = {2049-2066}, + Volume = {231}, + + Archiveprefix = {arXiv}, + Eprint = {1102.4375}, + Primaryclass = {math.NA} +} + +@Article{Mueller1994, + author = {Mueller, Carl and Tribe, Roger}, + journal = {Probab. Theory Related Fields}, + title = {A phase transition for a stochastic {PDE} related to the contact process}, + year = {1994}, + issn = {0178-8051}, + number = {2}, + pages = {131--156}, + volume = {100}, + coden = {PTRFEU}, + doi = {10.1007/BF01199262}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K55 35R60 60K35)}, + mrnumber = {1296425 (95k:60151)}, + mrreviewer = {Ya. {\=I}. B{\={\i}}lopol{\cprime}s{\cprime}ka}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF01199262}, +} + +@Article{Naurois2015, + author = {Ladislas Jacobe de Naurois and Arnulf Jentzen and Timo Welti}, + journal = {arXiv:1508.05168}, + title = {Weak convergence rates for spatial spectral {G}alerkin approximations of semilinear stochastic wave equations with multiplicative noise}, + year = {2015}, + pages = {1--27}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Neerven1992, + author = {Neerven, J.}, + publisher = {Springer-Verlag Berlin and New York}, + title = {The adjoint of a semigroup of linear operators}, + year = {1992}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.getcited.org/pub/103006106}, + urldate = {2012-12-06}, +} + +@Article{Neerven2012, + author = {van Neerven, Jan and Veraar, Mark and Weis, Lutz}, + journal = {SIAM J. Math. Anal.}, + title = {Maximal {$L^p$}-regularity for stochastic evolution equations}, + year = {2012}, + issn = {0036-1410}, + number = {3}, + pages = {1372--1414}, + volume = {44}, + coden = {SJMAAH}, + doi = {10.1137/110832525}, + fjournal = {SIAM Journal on Mathematical Analysis}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R60 46B09 47D06)}, + mrnumber = {2982717}, + mrreviewer = {Elisa Al{\`o}s}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/110832525}, +} + +@Article{Neerven2008, + author = {van Neerven, J. M. A. M. and Veraar, M. C. and Weis, L.}, + journal = {J. Funct. Anal.}, + title = {Stochastic evolution equations in {UMD} {B}anach spaces}, + year = {2008}, + issn = {0022-1236}, + number = {4}, + pages = {940--993}, + volume = {255}, + coden = {JFUAAW}, + doi = {10.1016/j.jfa.2008.03.015}, + fjournal = {Journal of Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {35R60 (35B65 35K55 46N30 47D06 60H15 60H20)}, + mrnumber = {2433958 (2009h:35465)}, + mrreviewer = {Krystyna Twardowska}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.jfa.2008.03.015}, +} + +@Article{Neerven2007, + author = {van Neerven, J. M. A. M. and Veraar, M. C. and Weis, L.}, + journal = {Ann. Probab.}, + title = {Stochastic integration in {UMD} {B}anach spaces}, + year = {2007}, + issn = {0091-1798}, + number = {4}, + pages = {1438--1478}, + volume = {35}, + coden = {APBYAE}, + doi = {10.1214/009117906000001006}, + fjournal = {The Annals of Probability}, + groups = {sdiyora:6}, + mrclass = {60H05 (28C20 60B11)}, + mrnumber = {2330977 (2008g:60170)}, + mrreviewer = {Martin Ondrej{\'a}t}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/009117906000001006}, +} + +@Article{Neuenkirch2014, + author = {Neuenkirch, Andreas and Szpruch, Lukasz}, + journal = {Numerische Mathematik}, + title = {First order strong approximations of scalar {SDEs} defined in a domain}, + year = {2014}, + pages = {1--34}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Springer}, + timestamp = {2016.02.02}, +} + +@Article{Neuenkirch2012, + author = {Andreas Neuenkirch and Lukasz Szpruch}, + journal = {arXiv:1209.0390}, + title = {First order strong approximations of scalar {SDE}s with values in a domain}, + year = {2012}, + pages = {27 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Newey1989, + author = {Newey, Whitney K. and Princeton University.}, + publisher = {Econometric Research Program, Princeton University Princeton, N.J}, + title = {Uniform convergence in probability and stochastic equicontinuity / Whitney K. 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Werschulz}, + owner = {sdiyora}, + pages = {xii+384}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.4171/026}, +} + +@Book{Nualart1995, + author = {Nualart, David}, + publisher = {Springer-Verlag}, + title = {The {M}alliavin calculus and related topics}, + year = {1995}, + address = {New York}, + isbn = {0-387-94432-X}, + series = {Probability and its Applications (New York)}, + groups = {sdiyora:6}, + mrclass = {60H07 (60-02)}, + mrnumber = {1344217 (96k:60130)}, + mrreviewer = {Daniel Ocone}, + owner = {sdiyora}, + pages = {xii+266}, + timestamp = {2016.02.02}, +} + +@Article{Nualart1988, + author = {Nualart, D. and Pardoux, {\'E}.}, + journal = {Probab. Theory Related Fields}, + title = {Stochastic calculus with anticipating integrands}, + year = {1988}, + issn = {0178-8051}, + number = {4}, + pages = {535--581}, + volume = {78}, + coden = {PTRFEU}, + doi = {10.1007/BF00353876}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H05 (60H07)}, + mrnumber = {950346 (89h:60089)}, + mrreviewer = {Halim Doss}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF00353876}, +} + +@Article{Ocone1984, + author = {Ocone, Daniel}, + journal = {Stochastics}, + title = {Malliavin's calculus and stochastic integral representations of functionals of diffusion processes}, + year = {1984}, + issn = {0090-9491}, + number = {3-4}, + pages = {161--185}, + volume = {12}, + coden = {STOCB2}, + doi = {10.1080/17442508408833299}, + fjournal = {Stochastics}, + groups = {sdiyora:6}, + mrclass = {60H07 (58G32 60H05)}, + mrnumber = {749372 (85m:60101)}, + mrreviewer = {L. C. G. Rogers}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1080/17442508408833299}, +} + +@Article{Ocone1989, + author = {Ocone, Daniel and Pardoux, {\'E}tienne}, + journal = {Ann. Inst. H. Poincar\'e Probab. Statist.}, + title = {A generalized {I}t\^o-{V}entzell formula. {A}pplication to a class of anticipating stochastic differential equations}, + year = {1989}, + issn = {0246-0203}, + number = {1}, + pages = {39--71}, + volume = {25}, + coden = {AHPBAR}, + fjournal = {Annales de l'Institut Henri Poincar\'e. Probabilit\'es et Statistique}, + groups = {sdiyora:6}, + mrclass = {60H10 (60H05)}, + mrnumber = {995291 (90g:60055)}, + mrreviewer = {David Nualart}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.numdam.org/item?id=AIHPB_1989__25_1_39_0}, +} + +@Article{Ocone1991, + author = {Ocone, Daniel L. and Karatzas, Ioannis}, + journal = {Stochastics Stochastics Rep.}, + title = {A generalized {C}lark representation formula, with application to optimal portfolios}, + year = {1991}, + issn = {1045-1129}, + number = {3-4}, + pages = {187--220}, + volume = {34}, + coden = {STOCBS}, + fjournal = {Stochastics and Stochastics Reports}, + groups = {sdiyora:6}, + mrclass = {60G44 (60H05 90A09)}, + mrnumber = {1124835 (93b:60098)}, + mrreviewer = {John P. Lehoczky}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Oeksendal2000, + author = {Bernt Oeksendal}, + publisher = {Springer-Verlag}, + title = {Stochastic Differential Equations: An Introduction with Applications}, + year = {2000}, + address = {Berlin}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Palmeri2000, + author = {Palmeri, Maria Carla}, + journal = {J. Convex Anal.}, + title = {Homographic approximation for some nonlinear parabolic unilateral problems}, + year = {2000}, + issn = {0944-6532}, + number = {2}, + pages = {353--373}, + volume = {7}, + fjournal = {Journal of Convex Analysis}, + groups = {sdiyora:6}, + mrclass = {35K85 (35K35 49J40)}, + mrnumber = {1811685 (2002e:35146)}, + mrreviewer = {Rodolfo Salvi}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Panteleev2006, + author = {Panteleev, Mikhail A and Ovanesov, Mikhail V and Kireev, Dmitrii A and Shibeko, Aleksei M and Sinauridze, Elena I and Ananyeva, Natalya M and Butylin, Andrey A and Saenko, Evgueni L and Ataullakhanov, Fazoil I}, + journal = {Biophysical journal}, + title = {Spatial propagation and localization of blood coagulation are regulated by intrinsic and protein {C} pathways, respectively}, + year = {2006}, + number = {5}, + pages = {1489--1500}, + volume = {90}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Elsevier}, + timestamp = {2016.02.02}, +} + +@InCollection{Pardoux1987, + author = {{\'E}. Pardoux}, + booktitle = {Stochastic partial differential equations and applications ({T}rento, 1985)}, + publisher = {Springer}, + title = {Two-sided stochastic calculus for {SPDE}s}, + year = {1987}, + address = {Berlin}, + pages = {200--207}, + series = {Lecture Notes in Math.}, + volume = {1236}, + doi = {10.1007/BFb0072891}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R60 60H05 60H10)}, + mrnumber = {899304 (88j:60110)}, + mrreviewer = {Philip Protter}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BFb0072891}, +} + +@InCollection{Pardoux1975, + author = {{\'E}tienne Pardoux}, + booktitle = {S\'eminaire sur les \'{E}quations aux {D}\'eriv\'ees {P}artielles (1974--1975), {III}, {E}xp. {N}o. 2}, + publisher = {Coll\`ege de France}, + title = {\'{E}quations aux d\'eriv\'ees partielles stochastiques de type monotone}, + year = {1975}, + address = {Paris}, + pages = {10}, + groups = {sdiyora:6}, + mrclass = {60H15 (35R99)}, + mrnumber = {0651582 (58 \$31406)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Pardoux1975a, + author = {{\'E}tienne Pardoux}, + publisher = {Sci. Math. Univ. Paris Sud.}, + title = {\'{E}quations aux d\'{e}riv\'{e}es partielles stochastiques non lineaires monotones. Etude de solutions fortes de type It\^{o}}, + year = {1975}, + address = {Paris, France}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Pardoux1972, + author = {{\'E}tienne Pardoux}, + journal = {C. R. Acad. Sci. Paris S{\'e}r. A-B}, + title = {Sur des {\'e}quations aux d{\'e}riv{\'e}es partielles stochastiques monotones}, + year = {1972}, + pages = {A101--A103}, + volume = {275}, + groups = {sdiyora:6}, + mrclass = {60H15}, + mrnumber = {0312572 (47 \#1129)}, + mrreviewer = {R. Temam}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Pardoux1992, + author = {Pardoux, {\'E}. and Peng, S.}, + booktitle = {Stochastic partial differential equations and their applications ({C}harlotte, {NC}, 1991)}, + publisher = {Springer}, + title = {Backward stochastic differential equations and quasilinear parabolic partial differential equations}, + year = {1992}, + address = {Berlin}, + pages = {200--217}, + series = {Lecture Notes in Control and Inform. Sci.}, + volume = {176}, + doi = {10.1007/BFb0007334}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K55 35R60)}, + mrnumber = {1176785 (93k:60157)}, + mrreviewer = {Ali S{\"u}leyman {\"U}st{\"u}nel}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BFb0007334}, +} + +@Article{Pardoux1990, + author = {Pardoux, {\'E}. and Peng, S. G.}, + journal = {Systems Control Lett.}, + title = {Adapted solution of a backward stochastic differential equation}, + year = {1990}, + issn = {0167-6911}, + number = {1}, + pages = {55--61}, + volume = {14}, + coden = {SCLEDC}, + doi = {10.1016/0167-6911(90)90082-6}, + fjournal = {Systems \& Control Letters}, + groups = {sdiyora:6}, + mrclass = {60H10 (60H20 93E03)}, + mrnumber = {1037747 (91e:60171)}, + mrreviewer = {Kiyomasa Narita}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/0167-6911(90)90082-6}, +} + +@Article{Pardoux1999, + author = {Pardoux, Etienne and Tang, Shanjian}, + journal = {Probab. Theory Related Fields}, + title = {Forward-backward stochastic differential equations and quasilinear parabolic {PDE}s}, + year = {1999}, + issn = {0178-8051}, + number = {2}, + pages = {123--150}, + volume = {114}, + coden = {PTRFEU}, + doi = {10.1007/s004409970001}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H10 (35K55 35R60 60G44)}, + mrnumber = {1701517 (2000f:60088)}, + mrreviewer = {Rainer Buckdahn}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s004409970001}, +} + +@Article{Parisi1981, + author = {Parisi, G. and Wu, Y.S.}, + journal = {Sci.Sinica}, + title = {Perturbation theory without gauge fixing}, + year = {1981}, + number = {4}, + pages = {483-496}, + volume = {24}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/euclid.aop/1068646370}, + urldate = {2012-11-12}, +} + +@Book{p83, + Title = {Semigroups of linear operators and applications to partial + differential equations}, + Author = {Pazy, A.}, + Publisher = {Springer-Verlag}, + Year = {1983}, + + Address = {New York}, + Series = {Applied Mathematical Sciences}, + Volume = {44}, + + ISBN = {0-387-90845-5}, + Mrclass = {47D05 (34Gxx 35Fxx 35Gxx 47H20)}, + Mrnumber = {MR710486 (85g:47061)}, + Mrreviewer = {H. O. Fattorini}, + Pages = {viii+279} +} + +@Book{Pazy1983, + author = {A. Pazy}, + publisher = {Springer-Verlag}, + title = {Semigroups of linear operators and applications to partial differential equations}, + year = {1983}, + address = {New York}, + isbn = {0-387-90845-5}, + series = {Applied Mathematical Sciences}, + volume = {44}, + groups = {sdiyora:6}, + mrclass = {47D05 (34Gxx 35Fxx 35Gxx 47H20)}, + mrnumber = {MR710486 (85g:47061)}, + mrreviewer = {H. O. Fattorini}, + owner = {sdiyora}, + pages = {viii+279}, + timestamp = {2016.02.02}, +} + +@Article{Peng2010, + author = {Shige Peng}, + journal = {arXiv:1002.4546v1}, + title = {Nonlinear Expectations and Stochastic Calculus under Uncertainty}, + year = {2010}, + pages = {149 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Peng1993, + author = {Peng, Shige}, + journal = {Appl. Math. Optim.}, + title = {Backward stochastic differential equations and applications to optimal control}, + year = {1993}, + issn = {0095-4616}, + number = {2}, + pages = {125--144}, + volume = {27}, + coden = {AMOMBN}, + doi = {10.1007/BF01195978}, + fjournal = {Applied Mathematics and Optimization. An International Journal with Applications to Stochastics}, + groups = {sdiyora:6}, + mrclass = {49K45 (60H10 93E20)}, + mrnumber = {1202528 (94c:49035)}, + mrreviewer = {N. U. Ahmed}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF01195978}, +} + +@Article{Peng1991, + author = {Peng, Shi Ge}, + journal = {Stochastics Stochastics Rep.}, + title = {Probabilistic interpretation for systems of quasilinear parabolic partial differential equations}, + year = {1991}, + issn = {1045-1129}, + number = {1-2}, + pages = {61--74}, + volume = {37}, + coden = {STOCBS}, + fjournal = {Stochastics and Stochastics Reports}, + groups = {sdiyora:6}, + mrclass = {35R60 (35K50 60H30 93E20)}, + mrnumber = {1149116 (93a:35159)}, + mrreviewer = {Krystyna Twardowska}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Peng1991a, + author = {Peng, Shi Ge}, + booktitle = {Control theory, stochastic analysis and applications ({H}angzhou, 1991)}, + publisher = {World Sci. Publ., River Edge, NJ}, + title = {A nonlinear {F}eynman-{K}ac formula and applications}, + year = {1991}, + pages = {173--184}, + groups = {sdiyora:6}, + mrclass = {60H30 (35R60 60H10)}, + mrnumber = {1171673 (93g:60134)}, + mrreviewer = {Vo-Khac Khoan}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Peskin1995, + author = {Peskin, Michael E. and Schroeder, Daniel V.}, + publisher = {Addison-Wesley Publishing Company Advanced Book Program}, + title = {An introduction to quantum field theory}, + year = {1995}, + address = {Reading, MA}, + isbn = {0-201-50397-2}, + note = {Edited and with a foreword by David Pines}, + groups = {sdiyora:6}, + mrclass = {81-01 (81Txx)}, + mrnumber = {1402248 (97j:81001)}, + owner = {sdiyora}, + pages = {xxii+842}, + timestamp = {2016.02.02}, +} + +@Book{pz07, + Title = {Stochastic partial differential equations with {L}\'evy noise}, + Author = {Peszat, S. and Zabczyk, J.}, + Publisher = {Cambridge University Press}, + Year = {2007}, + + Address = {Cambridge}, + Note = {An evolution equation approach}, + Series = {Encyclopedia of Mathematics and its Applications}, + Volume = {113}, + + Doi = {10.1017/CBO9780511721373}, + ISBN = {978-0-521-87989-7}, + Mrclass = {60H15 (35R60 60-02 60G51)}, + Mrnumber = {2356959 (2009b:60200)}, + Mrreviewer = {Sandra Cerrai}, + Pages = {xii+419} +} + +@Article{Peters2004, + author = {Baron Peters and Andreas Heyden and Alexis T. Bell and Arup Chakraborty}, + journal = {J. Chem. Phys.}, + title = {A growing stsring method for determining transition states: Comparison to the nudged elastic band and string methods}, + year = {2004}, + number = {17}, + pages = {7877--7885}, + volume = {120}, + fjournal = {Journal of Chemical Physics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Petersdorff2010, + author = {von Petersdorff, Tobias and Schwab, Christoph}, + journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, + title = {Numerical solution of parabolic equations in high dimensions}, + year = {2010}, + month = {3}, + number = {1}, + pages = {93-127}, + volume = {38}, + groups = {sdiyora:6}, + keywords = {Discontinuous Galerkin method; sparse grid; wavelets.; discontinuous Galerkin method; wavelets; parabolic partial differential equation; high dimension; finite elements; GMRES iterations; preconditioner; error estimates; algorithm; convergence; numerical results; heat equation}, + language = {eng}, + owner = {sdiyora}, + publisher = {EDP Sciences}, + timestamp = {2016.02.02}, + url = {http://eudml.org/doc/194210}, +} + +@Article{Petersdorff2004, + author = {Petersdorff, Tobias Von and Schwab, Christoph}, + journal = {ESAIM: Mathematical Modelling and Numerical Analysis-Mod{\'e}lisation Math{\'e}matique et Analyse Num{\'e}rique}, + title = {Numerical solution of parabolic equations in high dimensions}, + year = {2004}, + number = {1}, + pages = {93--127}, + volume = {38}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Pettersson2005, + author = {Roger Pettersson and Mikael Signahl}, + journal = {Potential Anal.}, + title = {Numerical approximation for a white noise driven {SPDE} with locally bounded drift}, + year = {2005}, + issn = {0926-2601}, + number = {4}, + pages = {375--393}, + volume = {22}, + coden = {POANE2}, + fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H15 60H40)}, + mrnumber = {MR2135265 (2005k:60215)}, + mrreviewer = {Isamu D{\^o}ku}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{ps05, + Title = {Numerical approximation for a white noise driven {SPDE} with + locally bounded drift}, + Author = {Pettersson, Roger and Signahl, Mikael}, + Journal = {Potential Anal.}, + Year = {2005}, + Number = {4}, + Pages = {375--393}, + Volume = {22}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International Journal Devoted to the + Interactions between Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H35 (60H15 60H40)}, + Mrnumber = {MR2135265 (2005k:60215)}, + Mrreviewer = {Isamu D{\^o}ku} +} + +@Book{Platen2010, + author = {Platen, Eckhard and Bruti-Liberati, Nicola}, + publisher = {Springer-Verlag, Berlin}, + title = {Numerical solution of stochastic differential equations with jumps in finance}, + year = {2010}, + isbn = {978-3-642-12057-2}, + note = {856 pages}, + series = {Stochastic Modelling and Applied Probability}, + volume = {64}, + doi = {10.1007/978-3-642-13694-8}, + groups = {sdiyora:6}, + mrclass = {60-02 (60H10 60H35 60J75 65C30 91G60 91G70)}, + mrnumber = {2723480 (2012b:60001)}, + mrreviewer = {Igor Cialenco}, + owner = {sdiyora}, + pages = {xxviii+856}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-3-642-13694-8}, +} + +@Article{Pol1926, + author = {van der Pol, Balth.}, + journal = {Philosophical Magazine Series 7}, + title = {LXXXVIII. {O}n relaxation-oscillations}, + year = {1926}, + number = {11}, + pages = {978-992}, + volume = {2}, + doi = {10.1080/14786442608564127}, + eprint = {http://www.tandfonline.com/doi/pdf/10.1080/14786442608564127}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.tandfonline.com/doi/abs/10.1080/14786442608564127}, +} + +@Book{pr07, + Title = {{A {C}oncise {C}ourse on {S}tochastic {P}artial {D}ifferential {E}quations}}, + Author = {Pr{\'e}v{\^o}t, Claudia and R{\"o}ckner, Michael}, + Publisher = {Springer}, + Year = {2007}, + + Address = {Berlin}, + Series = {Lecture Notes in Mathematics}, + Volume = {1905}, + + ISBN = {978-3-540-70780-6; 3-540-70780-8}, + Mrclass = {60H15 (60-01)}, + Mrnumber = {MR2329435 (2009a:60069)}, + Mrreviewer = {Sandra Cerrai}, + Pages = {vi+144} +} + +@Article{Prigogine1968, + author = {Prigogine, I. and Lefever, R.}, + journal = {J. Chem. Phys}, + title = {Symmetry breaking instabilities in dissipative systems II}, + year = {1968}, + pages = {1695--1700}, + volume = {48}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{p01, + Title = {On the discretization in time of parabolic stochastic partial + differential equations}, + Author = {Printems, Jacques}, + Journal = {M2AN Math. Model. Numer. Anal.}, + Year = {2001}, + Number = {6}, + Pages = {1055--1078}, + Volume = {35}, + + Fjournal = {M2AN. Mathematical Modelling and Numerical Analysis}, + ISSN = {0764-583X}, + Mrclass = {60H15 (34G20 35R60 65C30)}, + Mrnumber = {MR1873517 (2002j:60116)}, + Mrreviewer = {Stanis{\l}aw W{\polhk{e}}drychowicz} +} + +@Article{Priola2006, + author = {Priola, E.}, + journal = {Journal of Evolution Equations}, + title = {Formulae for the derivatives of degenerate diffusion semigroups}, + year = {2006}, + number = {4}, + pages = {577--600}, + volume = {6}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {Springer}, + timestamp = {2016.02.02}, +} + +@Article{Priola2004, + author = {Priola, Enrico and Zabczyk, Jerzy}, + journal = {J. Funct. Anal.}, + title = {Liouville theorems for non-local operators}, + year = {2004}, + issn = {0022-1236}, + number = {2}, + pages = {455--490}, + volume = {216}, + coden = {JFUAAW}, + doi = {10.1016/j.jfa.2004.04.001}, + fjournal = {Journal of Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {35S05 (31B05 47D07 60H10 60H20 60H30 60J75)}, + mrnumber = {2095690 (2005g:35315)}, + mrreviewer = {Niels Jacob}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.jfa.2004.04.001}, +} + +@Article{Protter1997, + author = {Protter, Philip and Talay, Denis}, + journal = {Ann. Probab.}, + title = {The {E}uler scheme for {L}\'evy driven stochastic differential equations}, + year = {1997}, + issn = {0091-1798}, + number = {1}, + pages = {393--423}, + volume = {25}, + coden = {APBYAE}, + doi = {10.1214/aop/1024404293}, + fjournal = {The Annals of Probability}, + groups = {sdiyora:6}, + mrclass = {60H10 (60E07 65C05 65R20 65U05)}, + mrnumber = {1428514 (98c:60063)}, + mrreviewer = {Eckhard Platen}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/aop/1024404293}, +} + +@Book{Protter2004, + author = {Philipp E. Protter}, + publisher = {Springer-Verlag}, + title = {Stochastic integration and differential equations (Second edition)}, + year = {2004}, + address = {Berlin}, + isbn = {3-540-00313-4}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {xiv+415}, + timestamp = {2016.02.02}, +} + +@Article{Przybylowicz2013, + author = {Przybylowicz, Pawel}, + journal = {International Journal of Computer Mathematics}, + title = {Optimality of {E}uler-type algorithms for approximation of stochastic differential equations with discontinuous coefficients}, + year = {2013}, + pages = {1-19}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.tandfonline.com/doi/abs/10.1080/00207160.2013.844336}, +} + +@Article{qs06, + Title = {Space semi-discretisations + for a stochastic wave equation}, + Author = {Quer-Sardanyons, Llu{\'{\i}}s and Sanz-Sol{\'e}, Marta}, + Journal = {Potential Anal.}, + Year = {2006}, + Number = {4}, + Pages = {303--332}, + Volume = {24}, + + Coden = {POANE2}, + Fjournal = {Potential Analysis. An International + Journal Devoted to the + Interactions between + Potential Theory, Probability Theory, + Geometry and Functional Analysis}, + ISSN = {0926-2601}, + Mrclass = {60H35 (60H15)}, + Mrnumber = {MR2224753 (2007b:60170)}, + Mrreviewer = {Roger Pettersson} +} + +@Article{Quer-Sardanyons2006, + author = {Llu{\'i}s Quer-Sardanyons and Marta Sanz-Sol{\'e}}, + journal = {Potential Anal.}, + title = {Space semi-discretisations for a stochastic wave equation}, + year = {2006}, + issn = {0926-2601}, + number = {4}, + pages = {303--332}, + volume = {24}, + coden = {POANE2}, + fjournal = {Potential Analysis. An International Journal Devoted to the Interactions between Potential Theory, Probability Theory, Geometry and Functional Analysis}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H15)}, + mrnumber = {MR2224753 (2007b:60170)}, + mrreviewer = {Roger Pettersson}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Roesler2009, + author = {Andreas R{\"o}{\ss}ler}, + journal = {SIAM J. Numer. Anal.}, + title = {Second order {R}unge-{K}utta methods for {I}t{\^o} stochastic differential equations}, + year = {2009}, + issn = {0036-1429}, + number = {3}, + pages = {1713--1738}, + volume = {47}, + doi = {10.1137/060673308}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H10 65C30)}, + mrnumber = {2505871 (2010f:60193)}, + mrreviewer = {Angel Tocino}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/060673308}, +} + +@Article{r06a, + Title = {Rooted tree analysis for order conditions of stochastic + {R}unge-{K}utta methods for the weak approximation of + stochastic differential equations}, + Author = {R{\"o}{\ss}ler, Andreas}, + Journal = {Stoch. Anal. Appl.}, + Year = {2006}, + Number = {1}, + Pages = {97--134}, + Volume = {24}, + + Fjournal = {Stochastic Analysis and Applications}, + ISSN = {0736-2994}, + Mrclass = {65C30 (34F05 37H10 60H10 60H35)}, + Mrnumber = {MR2198539 (2006k:65016)}, + Mrreviewer = {Henri Schurz} +} + +@Article{Roesler2006, + author = {Andreas R{\"o}{\ss}ler}, + journal = {Stochastic Process. Appl.}, + title = {Infinite dimensional stochastic differential equations of Ornstein–Uhlenbeck tipe}, + year = {2006}, + number = {3}, + pages = {381--406}, + volume = {116}, + fjournal = {Stochastic Processes and their Applications}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Roesler2006a, + author = {Andreas R{\"o}{\ss}ler}, + journal = {Stoch. Anal. Appl.}, + title = {Rooted tree analysis for order conditions of stochastic {R}unge-{K}utta methods for the weak approximation of stochastic differential equations}, + year = {2006}, + issn = {0736-2994}, + number = {1}, + pages = {97--134}, + volume = {24}, + doi = {10.1080/07362990500397699}, + fjournal = {Stochastic Analysis and Applications}, + groups = {sdiyora:6}, + mrclass = {65C30 (34F05 37H10 60H10 60H35)}, + mrnumber = {2198539 (2006k:65016)}, + mrreviewer = {Henri Schurz}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1080/07362990500397699}, +} + +@Article{r04, + Title = {Stochastic {T}aylor expansions for the expectation of + functionals of diffusion processes}, + Author = {R{\"o}{\ss}ler, Andreas}, + Journal = {Stochastic Anal. Appl.}, + Year = {2004}, + Number = {6}, + Pages = {1553--1576}, + Volume = {22}, + + Coden = {SAAPDA}, + Fjournal = {Stochastic Analysis and Applications}, + ISSN = {0736-2994}, + Mrclass = {60H35 (41A58 60H10 60J60 65C30)}, + Mrnumber = {MR2095070 (2005m:60151)}, + Mrreviewer = {G. Leha} +} + +@Article{Roesler2004, + author = {Andreas R{\"o}{\ss}ler}, + journal = {Stochastic Anal. Appl.}, + title = {Stochastic {T}aylor expansions for the expectation of functionals of diffusion processes}, + year = {2004}, + issn = {0736-2994}, + number = {6}, + pages = {1553--1576}, + volume = {22}, + coden = {SAAPDA}, + doi = {10.1081/SAP-200029495}, + fjournal = {Stochastic Analysis and Applications}, + groups = {sdiyora:6}, + mrclass = {60H35 (41A58 60H10 60J60 65C30)}, + mrnumber = {2095070 (2005m:60151)}, + mrreviewer = {G. Leha}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1081/SAP-200029495}, +} + +@Book{r03, + Title = {Runge-Kutta Methods for + the Numerical Solution of + Stochastic Differential Equations}, + Author = {R{\"o}{\ss}ler, Andreas}, + Publisher = {Shaker Verlag}, + Year = {2003}, + + Address = {Aachen}, + Note = {Dissertation, Technische + Universit{\"a}t Darmstadt, + Darmstadt, Germany} +} + +@Book{Roesler2003, + author = {Andreas R{\"o}{\ss}ler}, + publisher = {Shaker Verlag}, + title = {Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations}, + year = {2003}, + address = {Aachen}, + note = {Dissertation, Technische Universit{\"a}t Darmstadt, Darmstadt, Germany}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Roeckner1999, + author = {Michael R{\"o}ckner}, + booktitle = {Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions}, + publisher = {Springer}, + title = {${L}^p$-analysis of finite and infinite dimensional diffusions}, + year = {1999}, + address = {Berlin}, + pages = {65--116}, + series = {Lecture Notes in Math.}, + volume = {1715}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Roeckner2006, + author = {R{\"o}ckner, Michael and Sobol, Zeev}, + journal = {Ann. Probab.}, + title = {Kolmogorov equations in infinite dimensions: well-posedness and regularity of solutions, with applications to stochastic generalized {B}urgers equations}, + year = {2006}, + issn = {0091-1798}, + number = {2}, + pages = {663--727}, + volume = {34}, + coden = {APBYAE}, + doi = {10.1214/009117905000000666}, + fjournal = {The Annals of Probability}, + groups = {sdiyora:6}, + mrclass = {35R15 (35B30 35Q53 35R60 37L55 60H15 76D06)}, + mrnumber = {2223955 (2007b:35323)}, + mrreviewer = {Alp O. Eden}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1214/009117905000000666}, +} + +@Article{Roessler2007, + author = {Andreas R{\"o}ssler}, + journal = {BIT}, + title = {Second order {R}unge-{K}utta methods for {S}tratonovich stochastic differential equations}, + year = {2007}, + issn = {0006-3835}, + number = {3}, + pages = {657--680}, + volume = {47}, + doi = {10.1007/s10543-007-0130-3}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H10 65C30)}, + mrnumber = {2338537 (2009b:60214)}, + mrreviewer = {Evelyn Buckwar}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/s10543-007-0130-3}, +} + +@Article{Ruemelin1982, + author = {R{\"u}melin, W.}, + journal = {SIAM J. Numer. Anal.}, + title = {Numerical treatment of stochastic differential equations}, + year = {1982}, + issn = {0036-1429}, + number = {3}, + pages = {604--613}, + volume = {19}, + coden = {SJNAAM}, + doi = {10.1137/0719041}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {60H10 (65L05)}, + mrnumber = {656474 (83i:60075)}, + mrreviewer = {G. N. Mil{\cprime}shte{\u\i}n}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1137/0719041}, +} + +@Book{Rebonato1997, + author = {Rebonato, R.}, + publisher = {Wiley}, + title = {Interest rate option models}, + year = {1997}, + address = {Chichester}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Reed1972, + author = {Reed, Michael and Simon, Barry}, + publisher = {Academic Press}, + title = {Methods of modern mathematical physics. {I}. {F}unctional analysis}, + year = {1972}, + address = {New York}, + groups = {sdiyora:6}, + mrclass = {47-02 (81.47)}, + mrnumber = {0493419 (58 \#12429a)}, + mrreviewer = {P. R. Chernoff}, + owner = {sdiyora}, + pages = {xvii+325}, + timestamp = {2016.02.02}, +} + +@Article{Ren2007, + author = {Jiagang Ren and Michael R{\"o}ckner and Feng-Yu Wang}, + journal = {J. Differential Equations}, + title = {Stochastic generalized porous media and fast diffusion equations}, + year = {2007}, + issn = {0022-0396}, + number = {1}, + pages = {118--152}, + volume = {238}, + coden = {JDEQAK}, + doi = {10.1016/j.jde.2007.03.027}, + fjournal = {Journal of Differential Equations}, + groups = {sdiyora:6}, + mrclass = {76S05 (35K57 35R60 47N20 60H15)}, + mrnumber = {2334594 (2008e:76170)}, + mrreviewer = {Jorge A. Le{\'o}n}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.jde.2007.03.027}, +} + +@Article{Ren2003a, + author = {Ren, Jiagang and Zhang, Xicheng}, + journal = {Bull. Sci. Math.}, + title = {Stochastic flow for {SDE}s with non-{L}ipschitz coefficients}, + year = {2003}, + issn = {0007-4497}, + number = {8}, + pages = {739--754}, + volume = {127}, + coden = {BSMQA9}, + doi = {10.1016/S0007-4497(03)00062-9}, + fjournal = {Bulletin des Sciences Math\'ematiques}, + groups = {sdiyora:6}, + mrclass = {60H10 (34A12 34F05)}, + mrnumber = {2014757 (2004h:60091)}, + mrreviewer = {B. G. Pachpatte}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/S0007-4497(03)00062-9}, +} + +@Article{Ren2003, + author = {Wei-qing Ren}, + journal = {Commun. Math. Sci.}, + title = {Higher order string method for finding minimum energy paths}, + year = {2003}, + issn = {1539-6746}, + number = {2}, + pages = {377--384}, + volume = {1}, + fjournal = {Communications in Mathematical Sciences}, + groups = {sdiyora:6}, + mrclass = {65N06 (74N99)}, + mrnumber = {1980482 (2004d:65122)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/getRecord?id=euclid.cms/1118152077}, +} + +@Book{rr04, + Title = {An introduction to partial differential equations}, + Author = {Renardy, Michael and Rogers, Robert C.}, + Publisher = {Springer-Verlag}, + Year = {2004}, + + Address = {New York}, + Edition = {Second}, + Series = {Texts in Applied Mathematics}, + Volume = {13}, + + ISBN = {0-387-00444-0}, + Mrclass = {35-01 (46N20 47F05 47N20)}, + Mrnumber = {2028503 (2004j:35001)}, + Pages = {xiv+434} +} + +@book {RenardyRogers1993, + AUTHOR = {Renardy, Michael and Rogers, Robert C.}, + TITLE = {An introduction to partial differential equations}, + SERIES = {Texts in Applied Mathematics}, + VOLUME = {13}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1993}, + PAGES = {xiv+428}, + ISBN = {0-387-97952-2}, + MRCLASS = {35-01 (00A05 46-01 47-01 47F05)}, + MRNUMBER = {1211418}, +MRREVIEWER = {P. Szeptycki}, +} + +@Book{Revuz1994, + author = {Revuz, Daniel and Yor, Marc}, + publisher = {Springer-Verlag}, + title = {Continuous martingales and {B}rownian motion}, + year = {1994}, + address = {Berlin}, + edition = {Second}, + isbn = {3-540-57622-3}, + series = {Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]}, + volume = {293}, + groups = {sdiyora:6}, + mrclass = {60G44 (60G07 60H05)}, + mrnumber = {1303781 (95h:60072)}, + owner = {sdiyora}, + pages = {xii+560}, + timestamp = {2016.02.02}, +} + +@Article{Reznikoff2005, + author = {Maria G. Reznikoff and Eric Vanden-Eijnden}, + journal = {C. R. Math. Acad. Sci. Paris}, + title = {Invariant measures of stochastic partial differential equations and conditioned diffusions}, + year = {2005}, + issn = {1631-073X}, + number = {4}, + pages = {305--308}, + volume = {340}, + doi = {10.1016/j.crma.2004.12.025}, + fjournal = {Comptes Rendus Math{\'e}matique. Acad{\'e}mie des Sciences. Paris}, + groups = {sdiyora:6}, + mrclass = {60H15 (37L40 37L55)}, + mrnumber = {2121896}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.crma.2004.12.025}, +} + +@Article{Ritter1990, + author = {Klaus Ritter}, + journal = {J. Complexity}, + title = {Approximation and optimization on the {W}iener space}, + year = {1990}, + number = {6}, + pages = {337--364}, + volume = {6}, + fjournal = {Journal of Complexity}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Roberts1996, + author = {Gareth O. Roberts and Richard L. Tweedie}, + journal = {Bernoulli}, + title = {Exponential convergence of {L}angevin distributions and their discrete approximations}, + year = {1996}, + issn = {1350-7265}, + number = {4}, + pages = {341--363}, + volume = {2}, + doi = {10.2307/3318418}, + fjournal = {Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability}, + groups = {sdiyora:6}, + mrclass = {62E25 (65C05)}, + mrnumber = {MR1440273 (98j:62014)}, + mrreviewer = {Arnoldo Frigessi}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.2307/3318418}, +} + +@Book{r01, + Title = {Infinite-dimensional dynamical systems}, + Author = {Robinson, James C.}, + Publisher = {Cambridge University Press}, + Year = {2001}, + + Address = {Cambridge}, + Note = {An introduction to dissipative parabolic PDEs and the theory + of global attractors}, + Series = {Cambridge Texts in Applied Mathematics}, + + ISBN = {0-521-63204-8}, + Mrclass = {37-02 (34G10 34L30 35B41 35K57 35Q30 37Lxx)}, + Mrnumber = {MR1881888 (2003f:37001a)}, + Mrreviewer = {Ricardo M. S. Rosa}, + Pages = {xviii+461} +} + +@Book{Rogers2000, + author = {Rogers, L. C. G. and Williams, David}, + publisher = {Cambridge University Press}, + title = {Diffusions, {M}arkov processes and martingales. {V}ol. 1}, + year = {2000}, + address = {Cambridge}, + isbn = {0-521-77594-9}, + note = {Foundations, Reprint of the second (1994) edition}, + series = {Cambridge Mathematical Library}, + groups = {sdiyora:6}, + mrclass = {60J60 (60G07 60H05 60J25)}, + mrnumber = {MR1796539 (2001g:60188)}, + owner = {sdiyora}, + pages = {xx+386}, + timestamp = {2016.02.02}, +} + +@Article{r09b, + Title = {On numerical solutions of stochastic differential equations}, + Author = {Roman, L. J.}, + Journal = {PhD Thesis}, + Year = {2009}, + + Address = {Minnesota}, + Publisher = {University of Minnesota} +} + +@Article{r07, + Title = {Weak approximations of solutions of a first order hyperbolic + stochastic partial differential equation}, + Author = {Roth, Christian}, + Journal = {Monte Carlo Methods Appl.}, + Year = {2007}, + Number = {2}, + Pages = {117--133}, + Volume = {13}, + + Fjournal = {Monte Carlo Methods and Applications}, + ISSN = {0929-9629}, + Mrclass = {65C30 (60H15 60H35)}, + Mrnumber = {MR2338085 (2008i:65013)}, + Mrreviewer = {Albert Gardo{\'n}} +} + +@Article{Roth2007, + author = {Christian Roth}, + journal = {Monte Carlo Methods Appl.}, + title = {Weak approximations of solutions of a first order hyperbolic stochastic partial differential equation}, + year = {2007}, + issn = {0929-9629}, + number = {2}, + pages = {117--133}, + volume = {13}, + fjournal = {Monte Carlo Methods and Applications}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H15 60H35)}, + mrnumber = {MR2338085 (2008i:65013)}, + mrreviewer = {Albert Gardo{\'n}}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{r06b, + Title = {A combination of finite difference and {W}ong-{Z}akai methods + for hyperbolic stochastic partial differential equations}, + Author = {Roth, Christian}, + Journal = {Stoch. 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Ordnung}, + Author = {Roth, Christian}, + Publisher = {Martin-Luther-Universit{\"a}t + Halle-Wittenberg}, + Year = {2002}, + + Address = {Halle-Wittenberg, Germany}, + Note = {Dissertation} +} + +@Article{r02b, + Title = {Difference methods for stochastic partial differential + equations}, + Author = {Roth, Ch.}, + Journal = {ZAMM Z. Angew. Math. Mech.}, + Year = {2002}, + Note = {4th GAMM-Workshop ``Stochastic Models and Control Theory'' + (Lutherstadt Wittenberg, 2001)}, + Number = {11-12}, + Pages = {821--830}, + Volume = {82}, + + Fjournal = {ZAMM. Zeitschrift f\"ur Angewandte Mathematik und Mechanik. + Journal of Applied Mathematics and Mechanics}, + ISSN = {0044-2267}, + Mrclass = {60H15 (65C30)}, + Mrnumber = {MR1944425 (2004b:60160)}, + Mrreviewer = {Frederi G. Viens} +} + +@Book{Roth2002, + author = {Christian Roth}, + publisher = {Martin-Luther-Universit{\"a}t Halle-Wittenberg}, + title = {Stochastische partielle Differentialgleichungen 1. Ordnung}, + year = {2002}, + address = {Halle-Wittenberg, Germany}, + note = {Dissertation}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{r90, + Title = {Stochastic evolution systems}, + Author = {Rozovski{\u\i}, B. L.}, + Publisher = {Kluwer Academic Publishers Group}, + Year = {1990}, + + Address = {Dordrecht}, + Note = {Linear theory and applications to nonlinear filtering, + Translated from the Russian by A. 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Szeptycki}, + Pages = {x+547} +} + +@Book{Runst1996, + author = {Thomas Runst and Winfried Sickel}, + publisher = {Walter de Gruyter \& Co.}, + title = {Sobolev spaces of fractional order, {N}emytskij operators, and nonlinear partial differential equations}, + year = {1996}, + address = {Berlin}, + isbn = {3-11-015113-8}, + series = {de Gruyter Series in Nonlinear Analysis and Applications}, + volume = {3}, + groups = {sdiyora:6}, + mrclass = {47H30 (35J65 46E35 46F10 47N20)}, + mrnumber = {MR1419319 (98a:47071)}, + mrreviewer = {P. Szeptycki}, + owner = {sdiyora}, + pages = {x+547}, + timestamp = {2016.02.02}, +} + +@Article{rw01, + Title = {On the simulation of iterated {I}t\^o integrals}, + Author = {Ryd{\'e}n, Tobias and Wiktorsson, Magnus}, + Journal = {Stochastic Process. Appl.}, + Year = {2001}, + Number = {1}, + Pages = {151--168}, + Volume = {91}, + + Coden = {STOPB7}, + Fjournal = {Stochastic Processes and their Applications}, + ISSN = {0304-4149}, + Mrclass = {60H10 (60H05)}, + Mrnumber = {MR1807367 (2001j:60115)}, + Mrreviewer = {Eckhard Platen} +} + +@Article{Sabanis2013E, + Title = {{Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients}}, + Author = {{Sabanis}, S.}, + Journal = {arXiv:1308.1796}, + Year = {2015}, + Pages = {24 pages} +} + +@article {Sabanis2013, + AUTHOR = {Sabanis, Sotirios}, + TITLE = {A note on tamed {E}uler approximations}, + JOURNAL = {Electron. Commun. Probab.}, + FJOURNAL = {Electronic Communications in Probability}, + VOLUME = {18}, + YEAR = {2013}, + PAGES = {no. 47, 10}, + ISSN = {1083-589X}, + MRCLASS = {60H35}, + MRNUMBER = {3070913}, + DOI = {10.1214/ECP.v18-2824}, + URL = {http://dx.doi.org/10.1214/ECP.v18-2824}, +} + +@Book{Sanz-Sole1970, + author = {Marta Sanz-Sol{\'e}}, + publisher = {EFPL Press}, + title = {{M}alliavin {C}alculus with {A}pplications to {S}tochastic {P}artial {D}ifferential {E}quations}, + year = {1970}, + address = {Lausanne}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {150}, + timestamp = {2016.02.02}, +} + +@Book{Sato2013, + author = {Sato, Ken-iti}, + publisher = {Cambridge University Press, Cambridge}, + title = {L\'evy processes and infinitely divisible distributions}, + year = {2013}, + isbn = {978-1-107-65649-9}, + note = {Translated from the 1990 Japanese original, Revised edition of the 1999 English translation}, + series = {Cambridge Studies in Advanced Mathematics}, + volume = {68}, + groups = {sdiyora:6}, + mrclass = {60G51 (60E07 60G18 60G52 60J45)}, + mrnumber = {3185174}, + owner = {sdiyora}, + pages = {xiv+521}, + timestamp = {2016.02.02}, +} + +@Article{Sauer2014, + author = {Martin Sauer and Wilhelm Stannat}, + journal = {arXiv:1402.4791}, + title = {Analysis and Approximation of Stochastic Nerve Axon Equations}, + year = {2014}, + pages = {25 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Sauer2013, + author = {Martin Sauer and Wilhelm Stannat}, + journal = {arXiv:1301.6350}, + title = {Lattice Approximation for Stochastic Reaction Diffusion Equations with One-Sided {L}ipschitz Condition}, + year = {2013}, + pages = {22 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schaedle2006, + author = {Sch{\"a}dle, A. and L{\'o}pez-Fern{\'a}ndez, M. and Lubich, Ch.}, + journal = {SIAM J. Sci. Comput.}, + title = {Fast and oblivious convolution quadrature}, + year = {2006}, + number = {2}, + pages = {421--438}, + volume = {28}, + fjournal = {SIAM Journal on Scientific Computing}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schoener1986, + author = {Sch\"{o}ner, G. and Haken, H. and Kelso, J. A. S.}, + journal = {Biol. Cybernet.}, + title = {A stochastic theory of phase transitions in human hand movements}, + year = {1986}, + pages = {247--257}, + volume = {53}, + fjournal = {Biological Cybernetics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Schenk-Hoppe2001, + author = {Schenk-Hopp{\'e}, Klaus Reiner}, + booktitle = {I{UTAM} {S}ymposium on {N}onlinearity and {S}tochastic {S}tructural {D}ynamics ({M}adras, 1999)}, + publisher = {Kluwer Acad. Publ.}, + title = {Bounds on sample paths of stochastic nonlinear systems---a {L}yapunov function approach}, + year = {2001}, + address = {Dordrecht}, + pages = {249--260}, + series = {Solid Mech. Appl.}, + volume = {85}, + doi = {10.1007/978-94-010-0886-0_20}, + groups = {sdiyora:6}, + mrclass = {37H99 (34D45 34F05 60G10)}, + mrnumber = {1896845 (2003k:37079)}, + mrreviewer = {Henri Schurz}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-94-010-0886-0_20}, +} + +@Article{Schenk-Hoppe1996, + author = {Schenk-Hopp{\'e}, Klaus Reiner}, + journal = {Nonlinear Dynam.}, + title = {Bifurcation scenarios of the noisy {D}uffing-van der {P}ol oscillator}, + year = {1996}, + issn = {0924-090X}, + number = {3}, + pages = {255--274}, + volume = {11}, + coden = {NODYES}, + doi = {10.1007/BF00120720}, + fjournal = {Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems}, + groups = {sdiyora:6}, + mrclass = {70L05 (34C23 70K50)}, + mrnumber = {1423974 (97g:70028)}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF00120720}, +} + +@Article{Schenk-Hoppe1996a, + author = {Schenk-Hopp{\'e}, Klaus Reiner}, + journal = {Z. Angew. Math. Phys.}, + title = {Deterministic and stochastic {D}uffing-van der {P}ol oscillators are non-explosive}, + year = {1996}, + issn = {0044-2275}, + number = {5}, + pages = {740--759}, + volume = {47}, + coden = {AZMPA8}, + doi = {10.1007/BF00915273}, + fjournal = {Zeitschrift f\"ur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Math\'ematiques et de Physique Appliqu\'ees}, + groups = {sdiyora:6}, + mrclass = {34F05 (34C15 60H10 93E03)}, + mrnumber = {1420853 (97i:34082)}, + mrreviewer = {Wolfgang Kliemann}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF00915273}, +} + +@Article{Scheutzow2013, + author = {Michael Scheutzow}, + journal = {arXiv:1304.5424}, + title = {A stochastic {G}ronwall lemma}, + year = {2013}, + pages = {4 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Scheutzow1986, + author = {Scheutzow, Michael}, + journal = {Probab. Theory Relat. Fields}, + title = {Periodic behavior of the stochastic {B}russelator in the mean-field limit}, + year = {1986}, + issn = {0178-8051}, + number = {3}, + pages = {425--462}, + volume = {72}, + coden = {PTRFEU}, + doi = {10.1007/BF00334195}, + fjournal = {Probability Theory and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H10 (60K99 92A40)}, + mrnumber = {843504 (88d:60163)}, + mrreviewer = {Toshio Yamada}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/BF00334195}, +} + +@Article{Schmalfus1997, + author = {Schmalfu\ss, Bj\"orn}, + journal = {Zeitschrift f\"ur Angewandte Mathematik und Physik (ZAMP)}, + title = {The random attractor of the stochastic {L}orenz system}, + year = {1997}, + pages = {951-975}, + volume = {48}, + groups = {sdiyora:6}, + issue = {6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schmidli2002, + author = {Schmidli, Hanspeter}, + journal = {Annals of Applied Probability}, + title = {On minimizing the ruin probability by investment and reinsurance}, + year = {2002}, + pages = {890--907}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {JSTOR}, + timestamp = {2016.02.02}, +} + +@InCollection{Schnaubelt2002, + author = {Schnaubelt, Roland}, + booktitle = {Evolution equations, semigroups and functional analysis ({M}ilano, 2000)}, + publisher = {Birkh\"auser, Basel}, + title = {Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations}, + year = {2002}, + pages = {311--338}, + series = {Progr. 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Parthasarathy}, + owner = {sdiyora}, + pages = {xiv+163}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-1-4612-1170-9}, +} + +@Article{s08, + Title = {Analysis and discretization of semi-linear stochastic wave + equations with cubic nonlinearity and additive space-time + noise}, + Author = {Schurz, Henri}, + Journal = {Discrete Contin. Dyn. Syst. Ser. S}, + Year = {2008}, + Number = {2}, + Pages = {353--363}, + Volume = {1}, + + Fjournal = {Discrete and Continuous Dynamical Systems. Series S}, + ISSN = {1937-1632}, + Mrclass = {60H15 (35L70 35R60 65C30)}, + Mrnumber = {MR2379913 (2009b:60201)}, + Mrreviewer = {Hisao Watanabe} +} + +@Article{Schurz2008, + author = {Henri Schurz}, + journal = {Discrete Contin. Dyn. Syst. Ser. S}, + title = {Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise}, + year = {2008}, + issn = {1937-1632}, + number = {2}, + pages = {353--363}, + volume = {1}, + fjournal = {Discrete and Continuous Dynamical Systems. Series S}, + groups = {sdiyora:6}, + mrclass = {60H15 (35L70 35R60 65C30)}, + mrnumber = {MR2379913 (2009b:60201)}, + mrreviewer = {Hisao Watanabe}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{s07, + Title = {A numerical method for nonlinear stochastic wave equations in + {$\Bbb R\sp 1$}}, + Author = {Schurz, Henri}, + Journal = {Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.}, + Year = {2007}, + Number = {Advances in Dynamical Systems, suppl. S2}, + Pages = {74--78}, + Volume = {14}, + + Coden = {DCDIS}, + Fjournal = {Dynamics of Continuous, Discrete \& Impulsive Systems. Series + A. Mathematical Analysis}, + ISSN = {1201-3390}, + Mrclass = {65C30 (34F05 35R60 60H15 60H35)}, + Mrnumber = {MR2384109 (2009d:65014)}, + Mrreviewer = {Dror Givon} +} + +@Article{Schurz2007, + author = {Henri Schurz}, + journal = {Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.}, + title = {A numerical method for nonlinear stochastic wave equations in {$\Bbb R\sp 1$}}, + year = {2007}, + issn = {1201-3390}, + number = {Advances in Dynamical Systems, suppl. S2}, + pages = {74--78}, + volume = {14}, + coden = {DCDIS}, + fjournal = {Dynamics of Continuous, Discrete \& Impulsive Systems. Series A. Mathematical Analysis}, + groups = {sdiyora:6}, + mrclass = {65C30 (34F05 35R60 60H15 60H35)}, + mrnumber = {MR2384109 (2009d:65014)}, + mrreviewer = {Dror Givon}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{s06, + Title = {Dynamics of linearly partial-implicit midpoint methods for numerical integration of some infinite systems of ODEs with cubic-type nonlinearity and Q-regular additive noise}, + Author = {Schurz, Henri}, + Journal = {Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.}, + Year = {2006}, + Note = {accepted}, + + Fjournal = {Dynamics of Continuous, Discrete \& Impulsive Systems. Series + A. Mathematical Analysis} +} + +@Article{Schurz2006a, + author = {Schurz, Henri}, + journal = {Int. J. Numer. Anal. Model.}, + title = {An axiomatic approach to numerical approximations of stochastic processes}, + year = {2006}, + issn = {1705-5105}, + number = {4}, + pages = {459--480}, + volume = {3}, + fjournal = {International Journal of Numerical Analysis and Modeling}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H10)}, + mrnumber = {2238168 (2007e:65010)}, + mrreviewer = {David J. Galvin}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schurz2005, + author = {Schurz, Henri}, + journal = {Int. J. Numer. Anal. Model.}, + title = {Convergence and stability of balanced implicit methods for systems of {SDE}s}, + year = {2005}, + issn = {1705-5105}, + number = {2}, + pages = {197--220}, + volume = {2}, + fjournal = {International Journal of Numerical Analysis and Modeling}, + groups = {sdiyora:6}, + mrclass = {60H10 (65C30)}, + mrnumber = {2111748 (2005k:60184)}, + mrreviewer = {St{\'e}phane Menozzi}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schurz2003, + author = {Schurz, Henri}, + journal = {Electron. Trans. Numer. Anal.}, + title = {General theorems for numerical approximation of stochastic processes on the {H}ilbert space {$H_2([0,T],\mu,\mathbb{R}^d)$}}, + year = {2003}, + issn = {1068-9613}, + pages = {50--69 (electronic)}, + volume = {16}, + fjournal = {Electronic Transactions on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {65C30 (60G07 60H10)}, + mrnumber = {1988720 (2004e:65014)}, + mrreviewer = {Suresh V. Lawande}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@InCollection{Schurz2002, + author = {Schurz, H.}, + booktitle = {Handbook of stochastic analysis and applications}, + publisher = {Dekker}, + title = {Numerical analysis of stochastic differential equations without tears}, + year = {2002}, + address = {New York}, + pages = {237--359}, + series = {Statist. Textbooks Monogr.}, + volume = {163}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H10)}, + mrnumber = {1882714}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schuss1980, + author = {Schuss, Zeev}, + journal = {SIAM Review}, + title = {Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics}, + year = {1980}, + issn = {00361445}, + number = {2}, + pages = {119--155}, + volume = {22}, + abstract = {Stochastic differential equations are used as models for various physical phenomena, such as chemical reactions, atomic migration in crystals, thermal fluctuations in electrical networks, noisy signals in radio transmission, etc. First passage times of solutions of such equations from certain domains and the distribution of the exit points are computed from the solutions of singularly perturbed elliptic boundary value problems. Physical interpretation of these quantities is given. Application in communication theory and in reliability of structures are shown.}, + copyright = {Copyright © 1980 Society for Industrial and Applied Mathematics}, + groups = {sdiyora:6}, + jstor_articletype = {primary_article}, + jstor_formatteddate = {Apr., 1980}, + owner = {sdiyora}, + publisher = {Society for Industrial and Applied Mathematics}, + timestamp = {2016.02.02}, + url = {http://www.jstor.org/stable/2029958}, +} + +@Article{Schwab2011, + author = {Christoph Schwab and Endre Sueli}, + journal = {Report 2011-69, SAM, ETH Zurich http://www.sam.math.ethz.ch/reports/2011/69}, + title = {Adaptive {G}alerkin approximation algorithms for partial differential equations in infinite dimensions}, + year = {2011}, + pages = {27 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Schwiening2012, + author = {Christof J. Schwiening}, + journal = {The Journal of Physiology}, + title = {A brief historical perspective: Hodgkin and Huxley}, + year = {2012}, + pages = {2571--2575}, + volume = {590}, + doi = {10.1113/jphysiol.2012.230458}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{SellYou2002, + Title = {Dynamics of evolutionary equations}, + Author = {Sell, George R. and You, Yuncheng}, + Publisher = {Springer-Verlag}, + Year = {2002}, + + Address = {New York}, + Series = {Applied Mathematical Sciences}, + Volume = {143}, + + ISBN = {0-387-98347-3}, + Mrclass = {37-02 (34G20 35K55 35L70 35Q30 37L30 37Lxx)}, + Mrnumber = {MR1873467 (2003f:37001b)}, + Mrreviewer = {Ricardo M. S. Rosa}, + Pages = {xiv+670} +} + +@Article{s05, + Title = {Numerical simulation of stochastic {PDE}s for excitable media}, + Author = {Shardlow, Tony}, + Journal = {J. Comput. Appl. Math.}, + Year = {2005}, + Number = {2}, + Pages = {429--446}, + Volume = {175}, + + Coden = {JCAMDI}, + Fjournal = {Journal of Computational and Applied Mathematics}, + ISSN = {0377-0427}, + Mrclass = {65C30 (35K57 35R60 60H15 60H35 92C30)}, + Mrnumber = {MR2108585 (2005k:65017)}, + Mrreviewer = {Mikhail V. Tretyakov} +} + +@Article{s03, + Title = {Weak convergence of a numerical method for a stochastic heat + equation}, + Author = {Shardlow, Tony}, + Journal = {BIT}, + Year = {2003}, + Number = {1}, + Pages = {179--193}, + Volume = {43}, + + Fjournal = {BIT. Numerical Mathematics}, + ISSN = {0006-3835}, + Mrclass = {65C30 (60H15)}, + Mrnumber = {MR1981647 (2004d:65018)} +} + +@Article{s99, + Title = {Numerical methods for stochastic parabolic {PDE}s}, + Author = {Shardlow, Tony}, + Journal = {Numer. Funct. Anal. Optim.}, + Year = {1999}, + Number = {1-2}, + Pages = {121--145}, + Volume = {20}, + + Coden = {NFADOL}, + Fjournal = {Numerical Functional Analysis and Optimization. An + International Journal}, + ISSN = {0163-0563}, + Mrclass = {65C30 (60H35 65M06)}, + Mrnumber = {MR1683281 (2000g:65004)}, + Mrreviewer = {Marianne Huebner} +} + +@Article{Shen2013, + author = {Hao Shen}, + journal = {arXiv:1311.2305}, + title = {A renormalization group method by harmonic extensions and the classical dipole gas}, + year = {2013}, + pages = {43 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Shen2013a, + author = {Hao Shen}, + journal = {Ph{D} thesis, Princeton University}, + title = {Renormalization theory in statistical physics and stochastic analysis}, + year = {2013}, + pages = {155 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Simon1979, + author = {Simon, B.}, + publisher = {Academic Press}, + title = {Functional integration and quantum physics}, + year = {1979}, + volume = {86}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://books.google.com/books?hl=zh-CN\&lr=\&id=0xenVeK2rf8C\&oi=fnd\&pg=PP2\&dq=barry+simon+\&ots=m8RTq5CzFj\&sig=2Sx8J7yNBaX7dgOVwSes_s4GYc0}, + urldate = {2012-12-06}, +} + +@Article{Sirimungkala1999, + author = {Sirimungkala, Atchara and F{\"o}rsterling, Horst-Dieter and Dlask, Vladimir and Field, Richard J}, + journal = {The Journal of Physical Chemistry A}, + title = {Bromination reactions important in the mechanism of the {B}elousov-{Z}habotinsky system}, + year = {1999}, + number = {8}, + pages = {1038--1043}, + volume = {103}, + groups = {sdiyora:6}, + owner = {sdiyora}, + publisher = {ACS Publications}, + timestamp = {2016.02.02}, +} + +@Book{Situ2005, + author = {Situ, Rong}, + publisher = {Springer, New York}, + title = {Theory of stochastic differential equations with jumps and applications}, + year = {2005}, + isbn = {978-0387-25083-0; 0-387-25083-2}, + note = {Mathematical and analytical techniques with applications to engineering}, + series = {Mathematical and Analytical Techniques with Applications to Engineering}, + groups = {sdiyora:6}, + mrclass = {60-02 (34-02 34F05 60H10 60H30 91B28 93E03)}, + mrnumber = {2160585 (2006e:60004)}, + mrreviewer = {Nicolas Fournier}, + owner = {sdiyora}, + pages = {xx+434}, + timestamp = {2016.02.02}, +} + +@Book{Skorokhod1982, + author = {Skorokhod, A. V.}, + publisher = {Dover}, + title = {Studies in the theory of random processes}, + year = {1982}, + address = {New York}, + groups = {sdiyora:6}, + owner = {sdiyora}, + pages = {199}, + timestamp = {2016.02.02}, +} + +@InProceedings{Smolyak1963, + author = {Smolyak, Sergey A}, + booktitle = {Dokl. Akad. Nauk SSSR}, + title = {Quadrature and interpolation formulas for tensor products of certain classes of functions}, + year = {1963}, + pages = {1042--1045}, + volume = {148}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{SongLuLiu2015, + Title = {{Convergence of the tamed Euler scheme for stochastic differential equations with piecewise continuous arguments under non-Lipschitz continuous coefficients}}, + Author = {{Song}, M.~H. and {Lu}, Y.~L. and {Liu}, M.~Z.}, + Journal = {arXiv:1502.00058}, + Year = {2015}, + Pages = {16 pages} +} + +@Article{Song2013, + author = {Qingshuo Song and George Yin and Qing Zhang}, + journal = {arXiv:1302.4278}, + title = {Weak Convergence Methods for Approximation of Path-dependent Functionals}, + year = {2013}, + pages = {25 pages}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Stannat2013, + author = {Stannat, Wilhelm}, + journal = {arXiv:1301:6350}, + title = {Lattice approximation for stochastic reaction diffusion equations with one-sided {L}ipschitz condition}, + year = {2013}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Statistic2015, + author = {Statistica}, + journal = {http://www.statista.com/statistics/272832/largest-international-futures-exchanges-by-number-of-contracts-traded/}, + title = {European {R}esearch {C}ouncil {F}unded {P}rojects}, + year = {2015}, + note = {access date November 14th, 2015}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Stein1970, + author = {Stein, Elias M.}, + publisher = {Princeton University Press}, + title = {Singular integrals and differentiability properties of functions}, + year = {1970}, + address = {Princeton, N.J.}, + series = {Princeton Mathematical Series, No. 30}, + groups = {sdiyora:6}, + mrclass = {46.38 (26.00)}, + mrnumber = {0290095 (44 \#7280)}, + mrreviewer = {R. E. Edwards}, + owner = {sdiyora}, + pages = {xiv+290}, + timestamp = {2016.02.02}, +} + +@Book{Stoer2002, + author = {Stoer, J. and Bulirsch, R.}, + publisher = {Springer-Verlag, New York}, + title = {Introduction to numerical analysis}, + year = {2002}, + edition = {Third}, + isbn = {0-387-95452-X}, + note = {Translated from the German by R. Bartels, W. Gautschi and C. Witzgall}, + series = {Texts in Applied Mathematics}, + volume = {12}, + doi = {10.1007/978-0-387-21738-3}, + groups = {sdiyora:6}, + mrclass = {65-01}, + mrnumber = {1923481 (2003d:65001)}, + owner = {sdiyora}, + pages = {xvi+744}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1007/978-0-387-21738-3}, +} + +@Book{Strauss1989, + author = {Strauss, Walter A.}, + publisher = {Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI}, + title = {Nonlinear wave equations}, + year = {1989}, + isbn = {0-8218-0725-0}, + series = {CBMS Regional Conference Series in Mathematics}, + volume = {73}, + groups = {sdiyora:6}, + mrclass = {35-02 (35L70 35Q55 58D25 81T13 82D10)}, + mrnumber = {1032250 (91g:35002)}, + mrreviewer = {John Albert}, + owner = {sdiyora}, + pages = {x+91}, + timestamp = {2016.02.02}, +} + +@Article{Stroock1972, + author = {Stroock, D. and Varadhan, S. R. S.}, + journal = {Comm. Pure Appl. Math.}, + title = {On degenerate elliptic-parabolic operators of second order and their associated diffusions}, + year = {1972}, + issn = {0010-3640}, + pages = {651--713}, + volume = {25}, + fjournal = {Communications on Pure and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {35J70 (35K10 60J45)}, + mrnumber = {0387812 (52 \$8651)}, + mrreviewer = {M. Biroli}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Stroock2010, + author = {Stroock, Daniel W.}, + publisher = {Cambridge University Press}, + title = {Probability Theory}, + year = {2010}, + edition = {Second}, + isbn = {9780511974243}, + note = {Cambridge Books Online}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1017/CBO9780511974243}, +} + +@Book{Stroock1993, + author = {Daniel W. Stroock}, + publisher = {Cambridge University Press}, + title = {Probability theory, an analytic view}, + year = {1993}, + address = {Cambridge}, + isbn = {0-521-43123-9}, + groups = {sdiyora:6}, + mrclass = {60-02 (60F05 60G44 60J30 60J35)}, + mrnumber = {MR1267569 (95f:60003)}, + mrreviewer = {F. B. Knight}, + owner = {sdiyora}, + pages = {xvi+512}, + timestamp = {2016.02.02}, +} + +@Article{Stuart2004, + author = {Andrew M. Stuart and Jochen Voss and Petter Wiberg}, + journal = {Commun. Math. Sci.}, + title = {Conditional path sampling of {SDE}s and the {L}angevin {MCMC} method}, + year = {2004}, + issn = {1539-6746}, + number = {4}, + pages = {685--697}, + volume = {2}, + fjournal = {Communications in Mathematical Sciences}, + groups = {sdiyora:6}, + mrclass = {60H35 (60H15 65C30)}, + mrnumber = {2119934}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/getRecord?id=euclid.cms/1109885503}, +} + +@Article{Sz.-Nagy1954, + author = {Sz.-Nagy, B{\'e}la}, + journal = {Acta Sci. Math. Szeged}, + title = {Transformations de l'espace de {H}ilbert, fonctions de type positif sur un groupe}, + year = {1954}, + issn = {0001-6969}, + pages = {104--114}, + volume = {15}, + fjournal = {Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum}, + groups = {sdiyora:6}, + mrclass = {46.3X}, + mrnumber = {0060740 (15,719h)}, + mrreviewer = {E. Hille}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Sz.-Nagy1953, + author = {Sz.-Nagy, B{\'e}la}, + journal = {Acta Sci. Math. Szeged}, + title = {Sur les contractions de l'espace de {H}ilbert}, + year = {1953}, + issn = {0001-6969}, + pages = {87--92}, + volume = {15}, + fjournal = {Acta Universitatis Szegediensis. Acta Scientiarum Mathematicarum}, + groups = {sdiyora:6}, + mrclass = {46.3X}, + mrnumber = {0058128 (15,326d)}, + mrreviewer = {N. Dunford}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Sz.-Nagy1970, + author = {Sz.-Nagy, B{\'e}la and Foia{\lfhook{s}}, Ciprian}, + publisher = {North-Holland Publishing Co.}, + title = {Harmonic analysis of operators on {H}ilbert space}, + year = {1970}, + address = {Amsterdam}, + series = {Translated from the French and revised}, + groups = {sdiyora:6}, + mrclass = {47.00}, + mrnumber = {0275190 (43 \#947)}, + owner = {sdiyora}, + pages = {xiii+389}, + timestamp = {2016.02.02}, +} + +@Article{Szepessy2001, + author = {Szepessy, Anders and Tempone, Ra{\'u}l and Zouraris, Georgios E.}, + journal = {Comm. Pure Appl. Math.}, + title = {Adaptive weak approximation of stochastic differential equations}, + year = {2001}, + issn = {0010-3640}, + number = {10}, + pages = {1169--1214}, + volume = {54}, + coden = {CPAMA}, + doi = {10.1002/cpa.10000}, + fjournal = {Communications on Pure and Applied Mathematics}, + groups = {sdiyora:6}, + mrclass = {60H35 (65C30)}, + mrnumber = {1843985 (2002h:60138)}, + mrreviewer = {Denis Talay}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1002/cpa.10000}, +} + +@Article{Szpruch2013, + author = {Szpruch, Lukasz}, + journal = {arXiv:1310.0785}, + title = {V-stable tamed {E}uler schemes}, + year = {2013}, + pages = {1--15}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Book{Szpruch2010, + author = {Lukasz Szpruch}, + publisher = {University of Strathclyde}, + title = {Numerical Approximations of Nonlinear Stochastic Systems}, + year = {2010}, + address = {Glasgow, UK}, + note = {Dissertation}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Szpruch2011, + author = {Lukasz Szpruch and Xuerong Mao and Desmond J. Higham and Jiazhu Pan}, + journal = {BIT}, + title = {Numerical simulation of a strongly nonlinear {A}it-{S}ahalia-type interest rate model}, + year = {2011}, + number = {2}, + pages = {405--425}, + volume = {51}, + doi = {10.1007/s10543-010-0288-y}, + fjournal = {BIT. Numerical Mathematics}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://www.springerlink.com/content/424tt0076k38685u/}, +} + +@Article{SzpruchZhang2013, + Title = {{$V$-integrability, asymptotic stability and comparison theorem of explicit numerical schemes for SDEs}}, + Author = {{Szpruch}, L. and {Zh{\= a}ng}, X.}, + Journal = {arXiv:1310.0785}, + Year = {2015}, + Pages = {33 pages} +} + +@Article{Tadmor2012, + author = {Tadmor, Eitan}, + journal = {Bull. Amer. Math. Soc. (N.S.)}, + title = {A review of numerical methods for nonlinear partial differential equations}, + year = {2012}, + issn = {0273-0979}, + number = {4}, + pages = {507--554}, + volume = {49}, + coden = {BAMOAD}, + doi = {10.1090/S0273-0979-2012-01379-4}, + fjournal = {American Mathematical Society. Bulletin. New Series}, + groups = {sdiyora:6}, + mrclass = {65-02 (35J60 35K55 35L65 35L70 65Mxx 65Nxx)}, + mrnumber = {2958929}, + mrreviewer = {Paul Andrew Martin}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1090/S0273-0979-2012-01379-4}, +} + +@Article{Talay2002, + author = {D. Talay}, + journal = {Markov Process. Related Fields}, + title = {Stochastic {H}amiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit {E}uler scheme}, + year = {2002}, + issn = {1024-2953}, + note = {Inhomogeneous random systems (Cergy-Pontoise, 2001)}, + number = {2}, + pages = {163--198}, + volume = {8}, + fjournal = {Markov Processes and Related Fields}, + groups = {sdiyora:6}, + mrclass = {60H10 (37J99 60H35)}, + mrnumber = {MR1924934 (2003e:60129)}, + mrreviewer = {Jan I. 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Anal.}, + title = {Galerkin finite element methods for stochastic parabolic partial differential equations}, + year = {2005}, + issn = {0036-1429}, + number = {4}, + pages = {1363--1384 (electronic)}, + volume = {43}, + fjournal = {SIAM Journal on Numerical Analysis}, + groups = {sdiyora:6}, + mrclass = {65C30 (60H15)}, + mrnumber = {MR2182132 (2007a:65013)}, + mrreviewer = {Fr{\'e}d{\'e}ric Bernardin}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{y04, + Title = {Semidiscrete {G}alerkin approximation for a linear stochastic + parabolic partial differential equation driven by an additive + noise}, + Author = {Yan, Yubin}, + Journal = {BIT}, + Year = {2004}, + Number = {4}, + Pages = {829--847}, + Volume = {44}, + + Fjournal = {BIT. Numerical Mathematics}, + ISSN = {0006-3835}, + Mrclass = {60H35 (60H15 65C30 65M60)}, + Mrnumber = {MR2211047 (2007c:60065)}, + Mrreviewer = {Roger Pettersson} +} + +@Article{Y2004, + author = {Yubin Yan}, + journal = {BIT}, + title = {Semidiscrete {G}alerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise}, + year = {2004}, + issn = {0006-3835}, + number = {4}, + pages = {829--847}, + volume = {44}, + fjournal = {BIT. 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Notes Pure Appl. Math.}, + Volume = {245}, + + Mrclass = {60H15 (35K55 35R60 60H07)}, + Mrnumber = {MR2227240 (2007g:60072)}, + Mrreviewer = {Jan I. Seidler} +} + +@InCollection{Zambotti2006, + author = {Lorenzo Zambotti}, + booktitle = {Stochastic partial differential equations and applications---{VII}}, + publisher = {Chapman \& Hall/CRC, Boca Raton, FL}, + title = {It{\^o}-{T}anaka's formula for stochastic partial differential equations driven by additive space-time white noise}, + year = {2006}, + pages = {337--347}, + series = {Lect. Notes Pure Appl. Math.}, + volume = {245}, + groups = {sdiyora:6}, + mrclass = {60H15 (35K55 35R60 60H07)}, + mrnumber = {MR2227240 (2007g:60072)}, + mrreviewer = {Jan I. 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Comput.}, + title = {Convergence of the {E}uler scheme for stochastic functional partial differential equations}, + year = {2004}, + issn = {0096-3003}, + number = {2}, + pages = {479--492}, + volume = {155}, + coden = {AMHCBQ}, + fjournal = {Applied Mathematics and Computation}, + groups = {sdiyora:6}, + mrclass = {60H15 (65C30)}, + mrnumber = {MR2077063 (2005h:60190)}, + mrreviewer = {Carlos M. Mora Gonz{\'a}lez}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{zzn04, + Title = {Convergence of the {E}uler scheme for stochastic functional + partial differential equations}, + Author = {Zhang, Qi-Min and Zhang, Wei-Guo and Nie, Zan-Kan}, + Journal = {Appl. Math. Comput.}, + Year = {2004}, + Number = {2}, + Pages = {479--492}, + Volume = {155}, + + Coden = {AMHCBQ}, + Fjournal = {Applied Mathematics and Computation}, + ISSN = {0096-3003}, + Mrclass = {60H15 (65C30)}, + Mrnumber = {MR2077063 (2005h:60190)}, + Mrreviewer = {Carlos M. Mora Gonz{\'a}lez} +} + +@Article{Zhang2010, + author = {Zhang, Xicheng}, + journal = {Stochastic Process. Appl.}, + title = {Stochastic flows and {B}ismut formulas for stochastic {H}amiltonian systems}, + year = {2010}, + issn = {0304-4149}, + number = {10}, + pages = {1929--1949}, + volume = {120}, + coden = {STOPB7}, + doi = {10.1016/j.spa.2010.05.015}, + fjournal = {Stochastic Processes and their Applications}, + groups = {sdiyora:6}, + mrclass = {60H10 (37H99 60H07 60H20)}, + mrnumber = {2673982 (2012c:60160)}, + mrreviewer = {Shi Zan Fang}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.spa.2010.05.015}, +} + +@Article{z07, + Title = {Regularities for semilinear stochastic partial differential + equations}, + Author = {Zhang, Xicheng}, + Journal = {J. Funct. 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Levakov}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://dx.doi.org/10.1016/j.spa.2004.10.003}, +} + +@Article{Zhang2014, + Title = {{New explicit balanced schemes for SDEs with locally Lipschitz coefficients}}, + Author = {{Zhang}, Z.}, + Journal = {arXiv:1402.3708}, + Year = {2014}, + Pages = {15 pages} +} + +@Article{Zhang2014a, + author = {Zhongqiang Zhang}, + journal = {arXiv:1402.3708}, + title = {New explicit balanced schemes for {SDEs} with locally {L}ipschitz coefficients}, + year = {2014}, + pages = {1--15}, + groups = {sdiyora:6}, + owner = {sdiyora}, + timestamp = {2016.02.02}, +} + +@Article{Zhou2010, + author = {Zhou, Xiang and E, Weinan}, + journal = {Commun. Math. Sci.}, + title = {Study of noise-induced transitions in the {L}orenz system using the minimum action method}, + year = {2010}, + issn = {1539-6746}, + number = {2}, + pages = {341--355}, + volume = {8}, + fjournal = {Communications in Mathematical Sciences}, + groups = {sdiyora:6}, + mrclass = {37D45 (34C23 34C37 37N20)}, + mrnumber = {2664454 (2011j:37062)}, + mrreviewer = {Bernd Krauskopf}, + owner = {sdiyora}, + timestamp = {2016.02.02}, + url = {http://projecteuclid.org/getRecord?id=euclid.cms/1274816885}, +} + +@Article{ZongWuHuang2014, + Title = {{Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients}}, + Author = {Xiaofeng Zong and Fuke Wu and Chengming Huang}, + Journal = {Appl. Math. Comput.}, + Year = {2014}, + Pages = {240 - 250}, + Volume = {228}, + + Doi = {http://dx.doi.org/10.1016/j.amc.2013.11.100}, + ISSN = {0096-3003}, + Url = {http://www.sciencedirect.com/science/article/pii/S0096300313012691} +} + +@book{liu2015stochastic, + title={Stochastic partial differential equations: an introduction}, + author={Liu, Wei and R{\"o}ckner, Michael}, + year={2015}, + publisher={Springer} +} + +@Book{Carmona1999, + editor = {Rene A. Carmona and Boris Rozovskii}, + publisher = {American Mathematical Society}, + title = {Stochastic partial differential equations: six perspectives}, + year = {1999}, + address = {Providence, RI}, + isbn = {0-8218-0806-0}, + series = {Mathematical Surveys and Monographs}, + volume = {64}, + groups = {sdiyora:6}, + mrclass = {60-06 (35-06 35R60 60H15)}, + mrnumber = {1661761 (99g:60005)}, + owner = {sdiyora}, + pages = {xii+334}, + timestamp = {2016.02.02}, +} + +@Comment{jabref-meta: databaseType:bibtex;} + +@Comment{jabref-meta: grouping: +0 AllEntriesGroup:; +1 StaticGroup:Markings\;2\;1\;\;\;\;; +2 StaticGroup:sdiyora:6\;2\;1\;\;\;\;; +}

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