First push to the new thing
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\chapter{Brownian Motion Monte Carlo}
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\section{Brownian Motion Preliminaries}
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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}.
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\begin{lemma}[Markov property of Brownian motions]
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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 $\mathcal{W}$.
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\end{lemma}
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\begin{proof}
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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.
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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.
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Note next that for $h\in \lp 0,\infty\rp$ it is the case that:
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\begin{align}
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\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 \\
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&= \E \lb \mathcal{W}_{s+t+h}-\mathcal{W}_{s+t}\rb -\E \lb \mathcal{W}_{s+h}-\mathcal{W}_s\rb \nonumber \\
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&=0-0 =0
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\end{align}
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We note finally that:
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\begin{align}
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\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 \\
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&= \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 \\
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&=h-0=h \nonumber
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\end{align}
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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.
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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.
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\end{proof}
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\begin{lemma}[Independence of Brownian Motion]\label{iobm}
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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.
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\end{lemma}
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\begin{proof}
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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.
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\end{proof}
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\begin{lemma}[Scaling Invariance]
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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.
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\end{lemma}
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\begin{proof}
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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.
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Note also for instance that $\mathcal{X}_0 = \frac{1}{a} \cdot \mathcal{W}_{a^2 \cdot 0} = 0$ with $\mathbb{P}$-a.s.
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Note that for all $h \in \lp 0,\infty\rp$, and $t\in \lb 0,T\rb$ it is the case that:
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\begin{align}
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\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 \\
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&=\frac{1}{a} \E \lb \mathcal{W}_{a^2\cdot \lp t+h\rp} - \mathcal{W}_{a^2\cdot t}\rb \nonumber \\
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&=0\nonumber
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\end{align}
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Note that for all $h \in \lp 0,\infty\rp$, and $t\in \lb 0,T\rb$ it is the case that:
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\begin{align}
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\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 \\
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&=\frac{1}{a^2}\var\lb\mathcal{W}_{a^2\cdot \lp t+h\rp} - \mathcal{W}_{a^2\cdot t}\rb \nonumber\\
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&= \frac{1}{\cancel{a^2}}\cancel{a^2} \lp \cancel{t}+h-\cancel{t}\rp \nonumber\\
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&=h
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\end{align}
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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 $\mathcal{X}_s$.
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\end{proof}
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\begin{lemma}[Summation of Brownian Motions]
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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.
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\end{lemma}
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\begin{proof}
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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.
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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.
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Note that for all $h \in \lp 0,\infty\rp$ and $t \in \lb t,T\rb$ it is the case that:
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\begin{align}
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\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 \\
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&= \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 \\
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&=0 \nonumber
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\end{align}
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Note that for all $h \in \lp 0,\infty\rp$, and $t\in \lb 0,T\rb$ it is the case that:
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\begin{align}
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\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 \\
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&=\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\\
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&= \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 \\
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&= h \nonumber
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\end{align}
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\end{proof}
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\begin{definition}[Of $\mathfrak{k}$]\label{def:1.17}
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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 $\lp \E \lb \lv \mathcal{X} - \E \lb \mathcal{X} \rb \rp^p \rb \rp ^{\frac{1}{p}} \leqslant c \lp \E \lb \lv \mathcal{X} \rv^p \rb \rp ^{\frac{1}{p}}.$
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\end{definition}
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\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:
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\begin{align}\label{(2.1.2)}
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\max\{|g(x)|\} \leqslant \mathfrak{L} \lp 1+\|x\|_E^p \rp
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\end{align}
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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:
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\begin{align}\label{(1.12)}
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u(t,x) &= \mathbb{E} \lb g \lp x+\mathcal{W}^0_{T-t} \rp \rb
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\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<T \\
|
||||
\sigma \left(\sigma \left(W^{d,0}_s: s\in [0,T]\right) \cup \{ A \in \mathcal{F}: \mathbb{P}(A)=0\}\right) & :t=T
|
||||
\end{cases}
|
||||
\end{align}
|
||||
Observe that (\ref{2.3.29}) guarantees that $\mathbb{F}^d_t \subseteq \mathcal{F}$, $d\in \N$, $t\in [0,T]$ satisfies that:
|
||||
\begin{enumerate}[label = (\Roman*)]
|
||||
\item it holds for all $d\in \N$ that $\{ A \in \mathcal{F}: \mathbb{P}(A) = 0 \} \subseteq \mathbb{F}^d_0$
|
||||
\item it holds for all $d \in \N$, $t\in [0,T]$, that $\mathbb{F}^d_t = \bigcap_{s \in (t,T]}\mathbb{F}^d_s$.
|
||||
\end{enumerate}
|
||||
Combining item (I), item (II), (\ref{2.3.29}) and \cite[Lemma 2.17]{hjw2020} assures us that for all $d \in \N$ it holds that $W^{d,0}:[0, T] \times \Omega \rightarrow \R^d$ is a standard $\left(\Omega, \mathcal{F}, \mathbb{P}, \left(\mathbb{F}^d_t\right)_{t\in [0, T]}\right)$-Brownian Brownian motion. In addition $(58)$ ensures that it is the case that for all $d\in N$, $x\in \R^d$ it holds that $[0,T] \times \Omega \ni (t,\omega) \mapsto x + W^{d,0}_t(\omega) \in \R^d$ is an $\left(\mathbb{F}^d_t\right)_{t\in [0,T]}/\mathcal{B}\left(\R^d\right)$-adapted stochastic process with continuous sample paths.
|
||||
\medskip
|
||||
|
||||
This and the fact that for all $d\in \N$, $t\in [0,T]$, $x\in \R^d$ it holds that $a_d(t,x) = 0$, and the fact that for all $d\in \N$, $t \in [0,T]$, $x$,$v\in \R^d$ it holds that $b_d(t,x)v = v$ yield that for all $d \in \N$, $x\in \R^d$ it holds that $[0,T] \times \Omega \ni (t,\omega) \mapsto x+W^{d,0}_t(\omega) \in \R^d$ satisfies for all $t\in [0,T]$ it holds $\mathbb{P}$-a.s. that:
|
||||
\begin{align}
|
||||
x+W^{d,0}_t = x + \int^t_0 0 ds + \int^t_0 dW^{d,0}_s = x + \int^t_0 a_d(s,x+W^{d,0}_s) ds + \int^t_0 b_d(s,x+W^{d,0}_s) dW^{d,0}_s
|
||||
\end{align}
|
||||
\medskip
|
||||
|
||||
This and \cite[Lemma 2.6]{hjw2020} (applied for every $d \in \N$, $x \in \R^d$ with $d \curvearrowleft d$, $m \curvearrowleft d$, $T \curvearrowleft T$, $C_1 \curvearrowleft d$, $ C_2 \curvearrowleft 0$, $\mathbb{F} \curvearrowleft \mathbb{F}^d$, $ \xi \curvearrowleft x, \mu \curvearrowleft a_d, \sigma \curvearrowleft b_d, W \curvearrowleft W^{d,0}, X \curvearrowleft \left(\left[0,T\right] \times \Omega \ni (t, \omega) \mapsto x+W^{d,0}_t(\omega) \in \R^d\right)$ in the notation of \cite[Lemma 2.6]{hjw2020} ensures that for all $r\in [0,\infty)$, $d\in \N$, $x\in \R^d$, $t \in [0,T]$ it holds that
|
||||
\begin{align}
|
||||
\E \left[\left\| x+W^{d,0}_t\right\|^r\right] \leqslant \max\{T,1\} \left(\left(1+\left\| x\right\|^2\right)^{\frac{r}{2}} + (r+1)d^{\frac{r}{2}}\right) \exp \left(\frac{r(r+3)T}{2}\right) < \infty
|
||||
\end{align}
|
||||
This, the triangle inequality, and the fact that for all $v$,$w\in [0,\infty)$, $r\in (0,1]$, it holds that $(v+w)^r \leqslant v^r + w^r$ assure that for all $\p \in [2,\infty)$, $d\in \N$, $x \in \R^d$ it holds that:
|
||||
\begin{align}
|
||||
&\sup_{s\in [0,T]} \left(\E \left[\left(1+ \left\| x+W^{d,0}_s \right\|_E^q\right)^\p\right]\right)^\frac{1}{\p} \nonumber
|
||||
\leqslant 1 + \sup_{s\in [0,T]} \left(\E \left[\left\| x+W^{d,0}_{s}\right\|_E^{q\p}\right]\right)^{\frac{1}{\p}} \nonumber \\
|
||||
&\leqslant 1 + \sup_{s \in [0,T]} \left(\max\{T,1\} \left(\left(1+\left\| x\right\|_E^2\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{q\p(q\p+3)T}{2}\right) \right)^\frac{1}{\p} \nonumber \\
|
||||
&\leqslant 1 + \max\{T^\frac{1}{\p},1\} \left(\left(1+\left\| x\right\|_E^2\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{q(q\p+3)T}{2}\right) \nonumber \\
|
||||
&\leqslant 2\left(\left(1+\left\| x\right\|_E^2\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{q(q\p+3)T}{2}+\frac{T}{\p}\right) \nonumber \\
|
||||
&\leqslant 2\left(\left(1+\left\| x\right\|_E^2\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right)
|
||||
\end{align}
|
||||
Given that for all $d\in \N$, $x \in [-L,L]^d$ it holds that $\left\| x \right\|_E \leqslant Ld^{\frac{1}{2}}$, this demonstrates for all $\p \in [2,\infty)$, $d\in \N$, it holds that:
|
||||
\begin{align}
|
||||
&L\left( \frac{ \mathfrak{m}_\p}{m^\frac{1}{2}}\right)\left(\sup_{x \in [-L,L]^d} \sup_{s \in [0,T]} \left( \E \left[\left(1+\left\| x + W^{d,0}_s \right\|_E ^q\right)^\p\right]\right)^{\frac{1}{\p}}\right) \nonumber\\
|
||||
&\leqslant L\left( \frac{ \mathfrak{m}_\p}{m^\frac{1}{2}}\right)\left(\sup_{x \in [-L,L]^d} \left[\left(\left(1+\left\| x\right\|_E^2\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right)\right] \right)\nonumber\\
|
||||
&\leqslant L\left( \frac{ \mathfrak{m}_\p}{m^\frac{1}{2}}\right)\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right)
|
||||
\end{align}
|
||||
|
||||
Combining this with Corollary \ref{cor:1.25.1} tells us that:
|
||||
\begin{align}\label{(2.3.33)}
|
||||
&\left( \E \left[ \left| u_d(t,x)-U^{d,0}_m\lp t,x \rp \right| ^\p \right] \right) ^{\frac{1}{\p}} \nonumber\\
|
||||
&\leqslant L\left(\frac{\mathfrak{m}_\p}{m^\frac{1}{2}}\right)\left(\sup_{x \in [-L,L]^d} \sup_{s \in [0,T]} \left( \E \left[\left(1+\left\| x + W^{d,0}_s \right\|_E^q\right)^\p\right]\right)^{\frac{1}{\p}}\right) \nonumber\\
|
||||
&\leqslant L\left( \frac{\mathfrak{m}_\p}{m^\frac{1}{2}}\right)\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right)
|
||||
\end{align}
|
||||
|
||||
This and the fact that for all $d \in \N$ and $\ve \in (0,\infty)$ and the fact that $\mathfrak{m}_\p \leqslant 2$, it holds that for fixed $L,q, \mathfrak{p}, d,T$ there exists an $\mathfrak{M}_{L,q,\mathfrak{p},d,T} \in \R$ such that $\mathfrak{N}_{d,\epsilon} \geqslant \mathfrak{M}_{L,q,\mathfrak{p},d,T}$ forces:
|
||||
\begin{align}\label{2.3.34}
|
||||
L\left[ \frac{\mathfrak{m}_\p}{\mathfrak{N}_{d,\epsilon}^\frac{1}{2}}\right]\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \leqslant \ve
|
||||
\end{align}
|
||||
|
||||
Thus (\ref{(2.3.33)}) and (\ref{2.3.34}) together proves (\ref{(2.48)}).
|
||||
|
||||
Note that $\mathfrak{C}_{d,\mathfrak{N}_{d,\epsilon},\mathfrak{N}_{d,\epsilon}}$ is 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 $U^{d,0}_{\mathfrak{N}_{d,\epsilon}}(T,0):\Omega \rightarrow \R$. Let $\widetilde{\mathfrak{N}_{d,\ve}}$ be the value of $\mathfrak{N}_{d,\ve}$ that causes equality in $(\ref{2.3.34})$. In such a situation the number of evaluations of $u_d(0,\cdot)$ do not exceed $\widetilde{\mathfrak{N}_{d,\ve}}$. Each evaluation of $u_d(0,\cdot)$ requires at most one realization of scalar random variables. Thus we do not exceed $2\widetilde{\mathfrak{N}_{d,\epsilon}}$. Thus note that:
|
||||
\begin{align}\label{(2.3.35)}
|
||||
\mathfrak{C}_{d,\mathfrak{N}_{d,\ve},\mathfrak{N}_{d,\ve}} \leqslant 2\lb L\mathfrak{m}_\p\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb \ve^{-1}
|
||||
\end{align}
|
||||
Note that other than $d$ and $\ve$ everything on the right-hand side is constant or fixed. Hence (\ref{(2.3.35)}) can be rendered as:
|
||||
\begin{align}
|
||||
\mathfrak{C}_{d,\mathfrak{N}_{d,\ve},\mathfrak{N}_{d,\ve}} \leq cd^k\ve^{-1}
|
||||
\end{align}
|
||||
Where both $c$ and $k$ are dependent on $L,\mathfrak{p,m},L$
|
||||
%Next observe that:
|
||||
%\begin{align}
|
||||
% \lb L\left( \frac{\mathfrak{m}_\p}{\mathfrak{N}_{d,\epsilon}^\frac{1}{2}}\right)\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb^{-1} \geqslant \frac{1}{\ve} \\
|
||||
% \mathfrak{N}_{d,\epsilon}\lb L\left(\mathfrak{m}_\p\right)\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb^{-1} \geqslant \frac{1}{\ve} \\
|
||||
%\end{align}
|
||||
%
|
||||
%Note that $\mathfrak{C}_{d,\mathfrak{N}_{d,\epsilon},\mathfrak{N}_{d,\epsilon}}$ is 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 $U^{d,0}_{\mathfrak{N}_{d,\epsilon}}(T,0):\Omega \rightarrow \R$. In such a situation the number of evaluations of $u_d(0,\cdot)$ do not exceed $\mathfrak{N}_{d,\ve}$. In other words, for a given precision $\ve$ in dimension $d$, the number computations required for evaluating $u_d(0,\cdot)$ is at most:
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||||
% \begin{align}
|
||||
% \lb L\mathfrak{m}_\p\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb \ve^{-1}
|
||||
% \end{align}
|
||||
% Furthermore note that for each evaluation of $u_d$ is associated with an evaluation of $\mathcal{W}_t^{d,(\theta, 0,-k)}$ which requires at most one realization of scalar random variables and thus in turn is also bounded by:
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% \begin{align}
|
||||
% \lb L\left( \frac{\mathfrak{m}_\p}{\mathfrak{N}_{d,\epsilon}^\frac{1}{2}}\right)\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb \ve^{-1}
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% \end{align}
|
||||
% And thus we have that:
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% \begin{align}
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% \mathfrak{C}_{d,\mathfrak{N}_{d,\ve},\mathfrak{N}_{d,\ve}} \leqslant 2 \lb L\left( \frac{\mathfrak{m}_\p}{\mathfrak{N}_{d,\epsilon}^\frac{1}{2}}\right)\left(\left(1+L^2d\right)^{\frac{q\p}{2}} + (q\p+1)d^{\frac{q\p}{2}}\right) \exp \left(\frac{\left[q(q\p+3)+1\right]T}{2}\right) \rb \ve^{-1}
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% \end{align}
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\end{proof}
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||||
|
||||
\bibitem[Bass, 2011]{bass_2011}
|
||||
Bass, R.~F. (2011).
|
||||
\newblock {\em Brownian motion}, page 6–12.
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||||
\newblock Cambridge Series in Statistical and Probabilistic Mathematics.
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||||
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|
||||
|
||||
\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.
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||||
|
||||
\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.
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\newblock {\em Stochastics and Dynamics}, 21(08).
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||||
\bibitem[Crandall et~al., 1992]{crandall_lions}
|
||||
Crandall, M.~G., Ishii, H., and Lions, P.-L. (1992).
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||||
\newblock User’s guide to viscosity solutions of second order partial
|
||||
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||||
\newblock {\em Bull. Amer. Math. Soc.}, 27(1):1--67.
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||||
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Da~Prato, G. and Zabczyk, J. (2002).
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\newblock {\em Second Order Partial Differential Equations in Hilbert Spaces}.
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||||
\newblock London Mathematical Society Lecture Note Series. Cambridge University
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||||
\bibitem[Durrett, 2019]{durrett2019probability}
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Durrett, R. (2019).
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\newblock Cambridge Series in Statistical and Probabilistic Mathematics.
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||||
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||||
\bibitem[Grohs et~al., 2020]{grohsetal}
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Grohs, P., Hornung, F., Jentzen, A., and von Wurstemberger, P. (2020).
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||||
\newblock A proof that artificial neural networks overcome the curse of
|
||||
dimensionality in the numerical approximation of {Black}-{Scholes} partial
|
||||
differential equations.
|
||||
|
||||
\bibitem[Gy{\"o}ngy and Krylov, 1996]{Gyngy1996ExistenceOS}
|
||||
Gy{\"o}ngy, I. and Krylov, N.~V. (1996).
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||||
\newblock Existence of strong solutions for {It\^o}'s stochastic equations via
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||||
approximations.
|
||||
\newblock {\em Probability Theory and Related Fields}, 105:143--158.
|
||||
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||||
\bibitem[Hutzenthaler et~al., 2020a]{hutzenthaler_overcoming_2020}
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||||
Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T.~A., and von Wurstemberger,
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||||
P. (2020a).
|
||||
\newblock Overcoming the curse of dimensionality in the numerical approximation
|
||||
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||||
\newblock {\em Proc. R. Soc. A.}, 476(2244):20190630.
|
||||
\newblock arXiv:1807.01212 [cs, math].
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||||
|
||||
\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, P. (2020b).
|
||||
\newblock Overcoming the curse of dimensionality in the approximative pricing
|
||||
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|
||||
\newblock page 73 p.
|
||||
\newblock Artwork Size: 73 p. Medium: application/pdf Publisher: ETH Zurich.
|
||||
|
||||
\bibitem[It\^o, 1942a]{Ito1942a}
|
||||
It\^o, K. (1942a).
|
||||
\newblock Differential equations determining {Markov} processes (original in
|
||||
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|
||||
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||||
|
||||
\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.
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||||
|
||||
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[]\OT1/cmr/m/it/10.95 A strong so-lu-tion to the stochas-tic dif-fer-en-tial eq
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[]\OT1/cmr/bx/n/10.95 Corollary 3.3.1.1. []\OT1/cmr/m/it/10.95 Let $\OML/cmm/m/
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[48] [49]
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l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0}
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\nonumber\\
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I've run across a `}' that doesn't seem to match anything.
|
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For example, `\def\a#1{...}' and `\a}' would produce
|
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this error. If you simply proceed now, the `\par' that
|
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I've just inserted will cause me to report a runaway
|
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argument that might be the root of the problem. But if
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your `}' was spurious, just type `2' and it will go away.
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Runaway argument?
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\| \mu (t,x) - \mu (t,y) \|_E+\|\sigma (t,x) - \sigma (t,y) \|_F &= \ETC.
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! Paragraph ended before \align was complete.
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<to be read again>
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\par
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l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0}
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\nonumber\\
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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.
|
||||
<inserted text>
|
||||
$
|
||||
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.
|
||||
<inserted text>
|
||||
}
|
||||
l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0}
|
||||
\nonumber\\
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> 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.
|
||||
<inserted text>
|
||||
\endgroup
|
||||
l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0}
|
||||
\nonumber\\
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> 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 $$.
|
||||
<to be read again>
|
||||
\@@par
|
||||
l.1754 ...\|_F &= \| \mu(t,x) - \mu(t,y) \|_E+0}
|
||||
\nonumber\\
|
||||
The `$' that I just saw supposedly matches a previous `$$'.
|
||||
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.
|
||||
<recently read> }
|
||||
|
||||
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 <return> for immediate help.
|
||||
...
|
||||
|
||||
l.1755 &
|
||||
= \|t\nabla_x\alpha(x) - t\nabla_x \alpha(y) \|_E} \nonumber\\
|
||||
Your command was ignored.
|
||||
Type I <command> <return> to replace it with another command,
|
||||
or <return> to continue without it.
|
||||
|
||||
! Misplaced alignment tab character &.
|
||||
<recently read> &
|
||||
|
||||
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.
|
||||
<inserted text>
|
||||
$
|
||||
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 &.
|
||||
<recently read> &
|
||||
|
||||
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 &.
|
||||
<recently read> &
|
||||
|
||||
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.
|
||||
<inserted text>
|
||||
$
|
||||
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.
|
||||
<inserted text>
|
||||
}
|
||||
l.1758 \end{align}
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> 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.
|
||||
<inserted text>
|
||||
$
|
||||
l.1758 \end{align}
|
||||
|
||||
I've inserted something that you may have forgotten.
|
||||
(See the <inserted text> 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 $$.
|
||||
<to be read again>
|
||||
\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
|
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Here is how much of TeX's memory you used:
|
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12969 strings out of 478994
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29889 multiletter control sequences out of 15000+600000
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421865 words of font info for 102 fonts, out of 8000000 for 9000
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1141 hyphenation exceptions out of 8191
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71i,18n,74p,917b,628s stack positions out of 5000i,500n,10000p,200000b,80000s
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||||
xxrl8a.pfb>
|
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\contentsline {chapter}{\numberline {1}Introduction}{4}{chapter.1}%
|
||||
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|
||||
\contentsline {chapter}{\numberline {2}Brownian Motion Monte Carlo}{10}{chapter.2}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|||
\chapter{Introduction.}
|
||||
\section{Motivation}
|
||||
|
||||
Artificial neural networks represent a sea change in computing. 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 $s <t$.
|
||||
\item The paths that $\mathcal{W}_t$ take are $\mathbb{P}$-a.s. continuous.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
\begin{definition}[$\lp \mathbb{F}_t \rp _{t\in [0,T]}$-adapted Stochastic Process]
|
||||
Let $T \in (0,\infty)$. Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F})$ be a filtered probability space with the filtration indexed over $[0,T]$. Let $(S,\Sigma)$ be a measurable space. Let $\mathcal{X}: [0,T] \times \Omega \rightarrow S$ be a stochastic process. We say that $\mathcal{X}$ is an $(\mathbb{F}_t)_{t\in [0,T]}$-adapted stochastic process if it is the case that $\mathcal{X}_t: \Omega \rightarrow S$ is $(\mathcal{F}_t, \Sigma)$ measurable for each $t \in [0,T]$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[$(\mathbb{F}_t)_{t\in[0,T]}$-adapted stopping time] Let $T \in (0,\infty)$, $\tau \in [0,T]$. Assume a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F})$. It is then the case that $\tau \in \R$ is a stopping time if the stochastic process $\mathcal{X} = (\mathcal{X}_t)_{t\in [0,T]}$ define as:
|
||||
\begin{align}
|
||||
\mathcal{X}_t := \begin{cases}
|
||||
1 : t < \tau \\
|
||||
0 : t \geqslant \tau
|
||||
\end{cases}
|
||||
\end{align}
|
||||
is adapted to the filtration $\mathbb{F}:= (\mathcal{F}_i )_{i \in [0,T]}$
|
||||
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Strong Solution of Stochastic Differential Equation]\label{1.9}
|
||||
Let $d,m \in \N$. Let $\mu: \R^d \rightarrow \R^d$, $\sigma: \R^d \rightarrow \R^{d \times m}$ be Borel-measurable. Let $(\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t \in [0,T]})$ be a stochastic basis, and let $\mathcal{W}: [0,T] \times \Omega \rightarrow \R^d$ be a standard $(\mathbb{F}_t)_{t\in [0,T]}$-Brownian motion. For all $t \in [0, T]$, $x \in \R^d$, let $\mathcal{X}^{t,x} = (\mathcal{X}^{t,x}_s)_{s\in [t, T]} \times \Omega \rightarrow \R^d$ be an $(\mathbb{F}_s)_{s\in [t, T]}$-adapted stochastic process with continuous sample paths satisfying that for all $t \in [0, T]$ we have $\mathbb{P}$-a.s. that:
|
||||
\begin{align}\label{1.5}
|
||||
\mathcal{X}^{t,x} = \mathcal{X}_0 + \int^t_0 \mu(r, \mathcal{X}^{t,x}_r)dr + \int^t_0 \sigma(r, \mathcal{X}^{t,x}_r) d\mathcal{W}_r
|
||||
\end{align}
|
||||
\medskip
|
||||
|
||||
A strong solution to the stochastic differential equation (\ref{1.5}) on probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t \in [0,T]})$, w.r.t Brownian motion $\mathcal{W}$, w.r.t to initial condition $\mathcal{X}_0 = 0$ is a stochastic process $(\mathcal{X}_t)_{t\in[0,\infty)}$ satisfying that:
|
||||
\begin{enumerate}[label = (\roman*)]
|
||||
\item $\mathcal{X}_t$ is adapted to the filtration $(\mathbb{F}_t)_{t \in [0,T]}$.
|
||||
\item $\mathbb{P}(\mathcal{X}_0 = 0) =1$.
|
||||
\item for all $t \in [0,T]$ it is the case that $\mathbb{P} \lp \int^t_0 \| \mu(r, \mathcal{X}^{t,x}_r) \|_E + \|\sigma(r, \mathcal{X}^{t,x}_r) \|_F d\mathcal{W}_r < \infty \rp =1$
|
||||
\item it holds with $\mathbb{P}$-a.s. that $\mathcal{X}$ satisfies the equation:
|
||||
\begin{align}
|
||||
\mathcal{X}^{t,x} = \mathcal{X}_0 + \int^t_0 \mu(r, \mathcal{X}^{t,x}_r)dr + \int^t_0 \sigma(r, \mathcal{X}^{t,x}_r) d\mathcal{W}_r
|
||||
\end{align}
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Strong Uniqueness Property for Solutions to Stochastic Differential Equations]
|
||||
Let it be the case that whenever we have two strong solutions $\mathcal{X}$ and $\widetilde{\mathcal{X}}$, w.r.t. process $\mathcal{W}$ and initial condition $\mathcal{X}_0 = 0$, as defined in Definition \ref{1.9}, it is also the case that $\mathbb{P}(\mathcal{X}_t = \widetilde{\mathcal{X}}_t) =1$ for all $t\in [0, T]$. We then say that the pair $(\mu, \sigma)$ exhibits a strong uniqueness property.
|
||||
\end{definition}
|
||||
\subsection{Lipschitz and Related Notions}
|
||||
\begin{definition}[Globally Lipschitz Function]\label{def:1.13}
|
||||
Let $d \in \N_0$. For every $d\in \N_0$, we say a function $f: \R^d \rightarrow \R^d$ is (globally) Lipschitz if there exists an $L \in (0,\infty)$ such that for all $x,y \in \R^d$ it is the case that :
|
||||
\begin{align}
|
||||
\left\| f(x)-f(y) \right\|_E \leqslant L \cdot \left\| x-y\right\|_E
|
||||
\end{align}
|
||||
The set of globally Lipschitz functions over set $X$ will be denoted $\lip_G(X)$
|
||||
\end{definition}
|
||||
|
||||
\begin{corollary}
|
||||
Let $d \in \N_0$. For every $d \in \N_0$, a continuous function $f \in C(\R^d,\R^d)$ over a compact set $\mathcal{K} \subsetneq \R^d$ is Lipschitz over that set.
|
||||
\end{corollary}
|
||||
\begin{proof}
|
||||
By Hiene-Cantor, $f$ is uniformly continuous over set $\mathcal{K}$. Fix an arbitrary $\epsilon$ and let $\delta$ be from the definition of uniform continuity. By compactness we have a finite cover of $\mathcal{K}$ by balls of radius $\delta$, centered around $x_i \in \mathcal{K}$:
|
||||
\begin{align}
|
||||
\mathcal{K} \subseteq \bigcup^N_{i=1} B_\delta(x_i)
|
||||
\end{align}
|
||||
Note that within a given ball, no point $x_j$ is such that $|x_i-x_j|> \delta$. 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 $g$. Thus, their product is open as well, proving the lemma.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
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|
|||
|
||||
\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"}
|
||||
|
||||
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|
||||
|
||||
\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"}
|
||||
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||||
|
||||
|
||||
|
||||
|
||||
|
|
@ -0,0 +1,60 @@
|
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\relax
|
||||
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|
||||
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|
||||
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|
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|
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||||
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\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}
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||||
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\chapter{Some categorical ideas about neural networks}
|
||||
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@ -0,0 +1,161 @@
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@ -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}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
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|
||||
\bibcite{bass_2011}{{1}{2011}{{Bass}}{{}}}
|
||||
\bibcite{Beck_2021}{{2}{2021a}{{Beck et~al.}}{{}}}
|
||||
\bibcite{BHJ21}{{3}{2021b}{{Beck et~al.}}{{}}}
|
||||
\bibcite{bhj20}{{4}{2021c}{{Beck et~al.}}{{}}}
|
||||
\bibcite{crandall_lions}{{5}{1992}{{Crandall et~al.}}{{}}}
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||||
\bibcite{da_prato_zabczyk_2002}{{6}{2002}{{Da~Prato and Zabczyk}}{{}}}
|
||||
\bibcite{durrett2019probability}{{7}{2019}{{Durrett}}{{}}}
|
||||
\bibcite{golub2013matrix}{{8}{2013}{{Golub and Van~Loan}}{{}}}
|
||||
\bibcite{grohsetal}{{9}{2018}{{Grohs et~al.}}{{}}}
|
||||
\bibcite{grohs2019spacetime}{{10}{2023}{{Grohs et~al.}}{{}}}
|
||||
\bibcite{Grohs_2022}{{11}{2022}{{Grohs et~al.}}{{}}}
|
||||
\bibcite{Gyngy1996ExistenceOS}{{12}{1996}{{Gy{\"o}ngy and Krylov}}{{}}}
|
||||
\bibcite{hutzenthaler_overcoming_2020}{{13}{2020a}{{Hutzenthaler et~al.}}{{}}}
|
||||
\bibcite{hutzenthaler_strong_2021}{{14}{2021}{{Hutzenthaler et~al.}}{{}}}
|
||||
\bibcite{hjw2020}{{15}{2020b}{{Hutzenthaler et~al.}}{{}}}
|
||||
\bibcite{Ito1942a}{{16}{1942a}{{It\^o}}{{}}}
|
||||
\bibcite{Ito1946}{{17}{1942b}{{It\^o}}{{}}}
|
||||
\bibcite{karatzas1991brownian}{{18}{1991}{{Karatzas and Shreve}}{{}}}
|
||||
\bibcite{rio_moment_2009}{{19}{2009}{{Rio}}{{}}}
|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
|
|
@ -0,0 +1,120 @@
|
|||
\begin{thebibliography}{}
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
It\^o, K. (1942a).
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||||
\newblock Differential equations determining {Markov} processes (original in
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||||
{Japanese}).
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||||
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||||
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||||
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||||
It\^o, K. (1942b).
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
|
||||
\end{thebibliography}
|
||||
|
|
@ -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}
|
||||
}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
@ -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
|
||||
You've used 19 entries,
|
||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
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|
|||
\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}
|
||||
|
|
@ -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}%
|
||||
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||||
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\chapter{A modified Multi-Level Picard and Associated Neural Network}
|
||||
We now look at neural networks in the context of multi-level Picard iterations.
|
||||
\begin{lemma}
|
||||
Let $\alpha, \beta, M \in \lb 0,\infty \rp$, $U_n \in \lb 0,\infty \rp$, for $n\in \N_0$ satisfy for all $n\in \N$ that:
|
||||
\begin{align}\label{7.0.1}
|
||||
U_n \les \alpha M^n + \sum^{n-1}_{i=0}M^{n-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
|
||||
\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}
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||||
\end{lemma}
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||||
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Binary file not shown.
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After Width: | Height: | Size: 73 KiB |
|
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@ -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)
|
||||
|
||||
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|
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@ -0,0 +1,383 @@
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<div class="quarto-title">
|
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<h1 class="title">Sharktooth Functions in 1-dimension</h1>
|
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</div>
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<div class="cell" data-execution_count="32">
|
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<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np </span>
|
||||
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||||
</div>
|
||||
<div class="cell" data-execution_count="33">
|
||||
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> max_conv_operator(samples, f_samples, <span class="bu">input</span>, L):</span>
|
||||
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> np.<span class="bu">max</span>(f_samples <span class="op">-</span> L <span class="op">*</span> np.<span class="bu">abs</span>(<span class="bu">input</span> <span class="op">-</span> samples))</span>
|
||||
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> sharktooth_function(function, x_start, x_stop, number_of_sharkteeth, L, plot_arg):</span>
|
||||
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a> samples <span class="op">=</span> np.random.uniform(x_start,x_stop, number_of_sharkteeth)</span>
|
||||
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a> f_samples <span class="op">=</span> function(samples)</span>
|
||||
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a> x <span class="op">=</span> np.linspace(x_start, x_stop, <span class="dv">10000</span>)</span>
|
||||
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a> approximate_y <span class="op">=</span> []</span>
|
||||
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(x)):</span>
|
||||
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> approximate_y.append(max_conv_operator(samples, f_samples, x[i], L))</span>
|
||||
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> error <span class="op">=</span> np.<span class="bu">max</span>(np.<span class="bu">abs</span>(f(x) <span class="op">-</span> approximate_y))</span>
|
||||
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> plot_arg <span class="op">==</span> <span class="dv">1</span>:</span>
|
||||
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a> plt.plot(x,approximate_y)</span>
|
||||
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a> <span class="cf">else</span>:</span>
|
||||
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> error</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||||
</div>
|
||||
<div class="cell" data-execution_count="34">
|
||||
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> f(x):</span>
|
||||
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> x<span class="op">/</span>(x<span class="op">+</span><span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||||
</div>
|
||||
<div class="cell" data-execution_count="35">
|
||||
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>L <span class="op">=</span> [<span class="dv">1</span>,<span class="dv">4</span>,<span class="dv">8</span>,<span class="dv">16</span>]</span>
|
||||
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>plt.figure(figsize<span class="op">=</span>(<span class="dv">17</span>,<span class="dv">5</span>))</span>
|
||||
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> L <span class="kw">in</span> L:</span>
|
||||
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> errors <span class="op">=</span> []</span>
|
||||
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">1</span>,<span class="dv">50</span>):</span>
|
||||
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> errors.append(sharktooth_function(f,<span class="dv">0</span>,<span class="dv">1</span>,i,L,<span class="dv">0</span>))</span>
|
||||
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> <span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
|
||||
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a> plt.plot(errors, label <span class="op">=</span>L)</span>
|
||||
<span id="cb4-9"><a href="#cb4-9" aria-hidden="true" tabindex="-1"></a></span>
|
||||
<span id="cb4-10"><a href="#cb4-10" aria-hidden="true" tabindex="-1"></a>plt.xlabel(<span class="st">"Number of teeth"</span>)</span>
|
||||
<span id="cb4-11"><a href="#cb4-11" aria-hidden="true" tabindex="-1"></a>plt.ylabel(<span class="vs">r"$\sup_{x\in x_i} | f(x)- \mathfrak</span><span class="sc">{R}</span><span class="vs">_{\mathfrak</span><span class="sc">{r}</span><span class="vs">} ( \mathsf</span><span class="sc">{P}</span><span class="vs">)|$"</span>)</span>
|
||||
<span id="cb4-12"><a href="#cb4-12" aria-hidden="true" tabindex="-1"></a>plt.legend()</span>
|
||||
<span id="cb4-13"><a href="#cb4-13" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">"Sup of deviance from f(x) as the number of teeth increase and as we get closer to L"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
|
||||
<div class="cell-output cell-output-display" data-execution_count="35">
|
||||
<pre><code>Text(0.5, 1.0, 'Sup of deviance from f(x) as the number of teeth increase and as we get closer to L')</code></pre>
|
||||
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|
||||
if (top !== null && height !== null && parent !== null) {
|
||||
// cook up a div (if necessary) and position it
|
||||
let div = window.document.getElementById("code-annotation-line-highlight");
|
||||
if (div === null) {
|
||||
div = window.document.createElement("div");
|
||||
div.setAttribute("id", "code-annotation-line-highlight");
|
||||
div.style.position = 'absolute';
|
||||
parent.appendChild(div);
|
||||
}
|
||||
div.style.top = top - 2 + "px";
|
||||
div.style.height = height + 4 + "px";
|
||||
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
|
||||
if (gutterDiv === null) {
|
||||
gutterDiv = window.document.createElement("div");
|
||||
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
|
||||
gutterDiv.style.position = 'absolute';
|
||||
const codeCell = window.document.getElementById(targetCell);
|
||||
const gutter = codeCell.querySelector('.code-annotation-gutter');
|
||||
gutter.appendChild(gutterDiv);
|
||||
}
|
||||
gutterDiv.style.top = top - 2 + "px";
|
||||
gutterDiv.style.height = height + 4 + "px";
|
||||
}
|
||||
selectedAnnoteEl = annoteEl;
|
||||
}
|
||||
};
|
||||
const unselectCodeLines = () => {
|
||||
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
|
||||
elementsIds.forEach((elId) => {
|
||||
const div = window.document.getElementById(elId);
|
||||
if (div) {
|
||||
div.remove();
|
||||
}
|
||||
});
|
||||
selectedAnnoteEl = undefined;
|
||||
};
|
||||
// Attach click handler to the DT
|
||||
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
|
||||
for (const annoteDlNode of annoteDls) {
|
||||
annoteDlNode.addEventListener('click', (event) => {
|
||||
const clickedEl = event.target;
|
||||
if (clickedEl !== selectedAnnoteEl) {
|
||||
unselectCodeLines();
|
||||
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
|
||||
if (activeEl) {
|
||||
activeEl.classList.remove('code-annotation-active');
|
||||
}
|
||||
selectCodeLines(clickedEl);
|
||||
clickedEl.classList.add('code-annotation-active');
|
||||
} else {
|
||||
// Unselect the line
|
||||
unselectCodeLines();
|
||||
clickedEl.classList.remove('code-annotation-active');
|
||||
}
|
||||
});
|
||||
}
|
||||
const findCites = (el) => {
|
||||
const parentEl = el.parentElement;
|
||||
if (parentEl) {
|
||||
const cites = parentEl.dataset.cites;
|
||||
if (cites) {
|
||||
return {
|
||||
el,
|
||||
cites: cites.split(' ')
|
||||
};
|
||||
} else {
|
||||
return findCites(el.parentElement)
|
||||
}
|
||||
} else {
|
||||
return undefined;
|
||||
}
|
||||
};
|
||||
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
|
||||
for (var i=0; i<bibliorefs.length; i++) {
|
||||
const ref = bibliorefs[i];
|
||||
const citeInfo = findCites(ref);
|
||||
if (citeInfo) {
|
||||
tippyHover(citeInfo.el, function() {
|
||||
var popup = window.document.createElement('div');
|
||||
citeInfo.cites.forEach(function(cite) {
|
||||
var citeDiv = window.document.createElement('div');
|
||||
citeDiv.classList.add('hanging-indent');
|
||||
citeDiv.classList.add('csl-entry');
|
||||
var biblioDiv = window.document.getElementById('ref-' + cite);
|
||||
if (biblioDiv) {
|
||||
citeDiv.innerHTML = biblioDiv.innerHTML;
|
||||
}
|
||||
popup.appendChild(citeDiv);
|
||||
});
|
||||
return popup.innerHTML;
|
||||
});
|
||||
}
|
||||
}
|
||||
});
|
||||
</script>
|
||||
</div> <!-- /content -->
|
||||
|
||||
|
||||
|
||||
</body></html>
|
||||
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|
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|
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2018
Dissertation/sharktooth_functions_files/libs/bootstrap/bootstrap-icons.css
vendored
Normal file
2018
Dissertation/sharktooth_functions_files/libs/bootstrap/bootstrap-icons.css
vendored
Normal file
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Load Diff
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|
|
@ -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: #5E5E5E;
|
||||
--quarto-hl-cv-color: #5E5E5E;
|
||||
--quarto-hl-cn-color: #8f5902;
|
||||
--quarto-hl-sc-color: #5E5E5E;
|
||||
--quarto-hl-dv-color: #AD0000;
|
||||
--quarto-hl-kw-color: #003B4F;
|
||||
}
|
||||
|
||||
/* other quarto variables */
|
||||
:root {
|
||||
--quarto-font-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace;
|
||||
}
|
||||
|
||||
pre > code.sourceCode > span {
|
||||
color: #003B4F;
|
||||
}
|
||||
|
||||
code span {
|
||||
color: #003B4F;
|
||||
}
|
||||
|
||||
code.sourceCode > span {
|
||||
color: #003B4F;
|
||||
}
|
||||
|
||||
div.sourceCode,
|
||||
div.sourceCode pre.sourceCode {
|
||||
color: #003B4F;
|
||||
}
|
||||
|
||||
code span.ot {
|
||||
color: #003B4F;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.at {
|
||||
color: #657422;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.ss {
|
||||
color: #20794D;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.an {
|
||||
color: #5E5E5E;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.fu {
|
||||
color: #4758AB;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.st {
|
||||
color: #20794D;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.cf {
|
||||
color: #003B4F;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.op {
|
||||
color: #5E5E5E;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.er {
|
||||
color: #AD0000;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.bn {
|
||||
color: #AD0000;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.al {
|
||||
color: #AD0000;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.va {
|
||||
color: #111111;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.bu {
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.ex {
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.pp {
|
||||
color: #AD0000;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.in {
|
||||
color: #5E5E5E;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.vs {
|
||||
color: #20794D;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.wa {
|
||||
color: #5E5E5E;
|
||||
font-style: italic;
|
||||
}
|
||||
|
||||
code span.do {
|
||||
color: #5E5E5E;
|
||||
font-style: italic;
|
||||
}
|
||||
|
||||
code span.im {
|
||||
color: #00769E;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.ch {
|
||||
color: #20794D;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.dt {
|
||||
color: #AD0000;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.fl {
|
||||
color: #AD0000;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
code span.co {
|
||||
color: #5E5E5E;
|
||||
font-style: inherit;
|
||||
}
|
||||
|
||||
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: "</";
|
||||
}
|
||||
|
||||
/*# sourceMappingURL=debc5d5d77c3f9108843748ff7464032.css.map */
|
||||
|
|
@ -0,0 +1,902 @@
|
|||
const sectionChanged = new CustomEvent("quarto-sectionChanged", {
|
||||
detail: {},
|
||||
bubbles: true,
|
||||
cancelable: false,
|
||||
composed: false,
|
||||
});
|
||||
|
||||
const layoutMarginEls = () => {
|
||||
// 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) {
|
||||
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|
||||
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|
||||
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|
||||
|
||||
// Throttle the scroll event and walk peridiocally
|
||||
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|
||||
"scroll",
|
||||
throttle(() => {
|
||||
if (tocEl) {
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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||||
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|
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
document.querySelectorAll(`div[data-group] a[id^='tabset-']`)
|
||||
);
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
}
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
Object.entries(tabSetsByValue).forEach(([value, items]) => {
|
||||
items.forEach((item) => {
|
||||
item.addEventListener("click", (_event) => {
|
||||
setTabState(group, value);
|
||||
toggleAll(value, selectorsToSync[group]);
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
return selectorsToSync;
|
||||
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|
||||
|
||||
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|
||||
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);
|
||||
}
|
||||
|
|
@ -0,0 +1 @@
|
|||
.tippy-box[data-animation=fade][data-state=hidden]{opacity:0}[data-tippy-root]{max-width:calc(100vw - 10px)}.tippy-box{position:relative;background-color:#333;color:#fff;border-radius:4px;font-size:14px;line-height:1.4;white-space:normal;outline:0;transition-property:transform,visibility,opacity}.tippy-box[data-placement^=top]>.tippy-arrow{bottom:0}.tippy-box[data-placement^=top]>.tippy-arrow:before{bottom:-7px;left:0;border-width:8px 8px 0;border-top-color:initial;transform-origin:center top}.tippy-box[data-placement^=bottom]>.tippy-arrow{top:0}.tippy-box[data-placement^=bottom]>.tippy-arrow:before{top:-7px;left:0;border-width:0 8px 8px;border-bottom-color:initial;transform-origin:center bottom}.tippy-box[data-placement^=left]>.tippy-arrow{right:0}.tippy-box[data-placement^=left]>.tippy-arrow:before{border-width:8px 0 8px 8px;border-left-color:initial;right:-7px;transform-origin:center left}.tippy-box[data-placement^=right]>.tippy-arrow{left:0}.tippy-box[data-placement^=right]>.tippy-arrow:before{left:-7px;border-width:8px 8px 8px 0;border-right-color:initial;transform-origin:center right}.tippy-box[data-inertia][data-state=visible]{transition-timing-function:cubic-bezier(.54,1.5,.38,1.11)}.tippy-arrow{width:16px;height:16px;color:#333}.tippy-arrow:before{content:"";position:absolute;border-color:transparent;border-style:solid}.tippy-content{position:relative;padding:5px 9px;z-index:1}
|
||||
File diff suppressed because one or more lines are too long
File diff suppressed because one or more lines are too long
Binary file not shown.
|
|
@ -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)
|
||||
|
|
@ -0,0 +1,44 @@
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|
||||
File diff suppressed because one or more lines are too long
|
After Width: | Height: | Size: 11 KiB |
|
|
@ -0,0 +1,199 @@
|
|||
\relax
|
||||
\providecommand\hyper@newdestlabel[2]{}
|
||||
\citation{crandall_lions}
|
||||
\citation{Beck_2021}
|
||||
\citation{Ito1942a}
|
||||
\citation{Ito1946}
|
||||
\citation{Beck_2021}
|
||||
\citation{BHJ21}
|
||||
\@writefile{toc}{\contentsline {chapter}{\numberline {3}That $u$ is a viscosity solution}{30}{chapter.3}\protected@file@percent }
|
||||
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||||
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||||
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|
||||
\newlabel{lemma:2.7}{{3.1.1}{30}{}{theorem.3.1.1}{}}
|
||||
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|
||||
\citation{karatzas1991brownian}
|
||||
\citation{karatzas1991brownian}
|
||||
\newlabel{lem:3.4}{{3.1.2}{32}{}{theorem.3.1.2}{}}
|
||||
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|
||||
\newlabel{2.13}{{3.1.7}{32}{}{equation.3.1.7}{}}
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\citation{karatzas1991brownian}
|
||||
\newlabel{(2.19)}{{3.1.10}{33}{Some Preliminaries}{equation.3.1.10}{}}
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||||
\newlabel{(2.19)@cref}{{[equation][10][3,1]3.1.10}{[1][33][]33}}
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||||
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\newlabel{sumofusc}{{3.2.3.1}{34}{}{corollary.3.2.3.1}{}}
|
||||
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|
||||
\newlabel{neglsc}{{3.2.3.2}{35}{}{corollary.3.2.3.2}{}}
|
||||
\newlabel{neglsc@cref}{{[corollary][2][]3.2.3.2}{[1][35][]35}}
|
||||
\newlabel{negdegel}{{3.2.6}{36}{}{theorem.3.2.6}{}}
|
||||
\newlabel{negdegel@cref}{{[lemma][6][3,2]3.2.6}{[1][36][]36}}
|
||||
\newlabel{def:viscsubsolution}{{3.2.7}{36}{Viscosity subsolutions}{theorem.3.2.7}{}}
|
||||
\newlabel{def:viscsubsolution@cref}{{[definition][7][3,2]3.2.7}{[1][36][]36}}
|
||||
\newlabel{def:viscsupsolution}{{3.2.8}{36}{Viscosity supersolutions}{theorem.3.2.8}{}}
|
||||
\newlabel{def:viscsupsolution@cref}{{[definition][8][3,2]3.2.8}{[1][36][]36}}
|
||||
\newlabel{def:viscsolution}{{3.2.9}{37}{Viscosity solution}{theorem.3.2.9}{}}
|
||||
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|
||||
\newlabel{maxviscosity}{{3.2.10}{37}{}{theorem.3.2.10}{}}
|
||||
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||||
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||||
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||||
\newlabel{limitofun@cref}{{[equation][13][3,2]3.2.13}{[1][38][]38}}
|
||||
\newlabel{hessungeq0}{{3.2.15}{38}{}{equation.3.2.15}{}}
|
||||
\newlabel{hessungeq0@cref}{{[equation][15][3,2]3.2.15}{[1][38][]38}}
|
||||
\newlabel{phieps}{{3.2.16}{38}{Viscosity Solutions}{equation.3.2.16}{}}
|
||||
\newlabel{phieps@cref}{{[equation][16][3,2]3.2.16}{[1][38][]38}}
|
||||
\newlabel{maxphiu}{{3.2.20}{39}{Viscosity Solutions}{equation.3.2.20}{}}
|
||||
\newlabel{maxphiu@cref}{{[equation][20][3,2]3.2.20}{[1][39][]39}}
|
||||
\newlabel{unleq0}{{3.2.11.1}{40}{}{corollary.3.2.11.1}{}}
|
||||
\newlabel{unleq0@cref}{{[corollary][1][]3.2.11.1}{[1][40][]40}}
|
||||
\newlabel{limitofun}{{3.2.27}{41}{}{equation.3.2.27}{}}
|
||||
\newlabel{limitofun@cref}{{[equation][27][3,2]3.2.27}{[1][41][]41}}
|
||||
\newlabel{viscsolutionvn}{{3.2.28}{41}{}{equation.3.2.28}{}}
|
||||
\newlabel{viscsolutionvn@cref}{{[equation][28][3,2]3.2.28}{[1][41][]41}}
|
||||
\newlabel{hessungeq0}{{3.2.29}{41}{}{equation.3.2.29}{}}
|
||||
\newlabel{hessungeq0@cref}{{[equation][29][3,2]3.2.29}{[1][41][]41}}
|
||||
\newlabel{hgeq0}{{3.2.30}{41}{Viscosity Solutions}{equation.3.2.30}{}}
|
||||
\newlabel{hgeq0@cref}{{[equation][30][3,2]3.2.30}{[1][41][]41}}
|
||||
\newlabel{unneq0}{{3.2.11.2}{42}{}{corollary.3.2.11.2}{}}
|
||||
\newlabel{unneq0@cref}{{[corollary][2][]3.2.11.2}{[1][42][]42}}
|
||||
\newlabel{absq}{{3.2.12}{42}{}{theorem.3.2.12}{}}
|
||||
\newlabel{absq@cref}{{[lemma][12][3,2]3.2.12}{[1][42][]42}}
|
||||
\citation{karatzas1991brownian}
|
||||
\newlabel{ugoesto0}{{3.2.13}{43}{}{theorem.3.2.13}{}}
|
||||
\newlabel{ugoesto0@cref}{{[lemma][13][3,2]3.2.13}{[1][43][]43}}
|
||||
\newlabel{limsupis0}{{3.2.39}{43}{}{equation.3.2.39}{}}
|
||||
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|
||||
\newlabel{xnasintuvxn}{{3.2.40}{43}{}{equation.3.2.40}{}}
|
||||
\newlabel{xnasintuvxn@cref}{{[equation][40][3,2]3.2.40}{[1][43][]43}}
|
||||
\newlabel{lipformun}{{3.2.42}{43}{Viscosity Solutions}{equation.3.2.42}{}}
|
||||
\newlabel{lipformun@cref}{{[equation][42][3,2]3.2.42}{[1][43][]43}}
|
||||
\newlabel{expofxisbounded}{{3.2.43}{43}{Viscosity Solutions}{equation.3.2.43}{}}
|
||||
\newlabel{expofxisbounded@cref}{{[equation][43][3,2]3.2.43}{[1][43][]43}}
|
||||
\newlabel{mathcalxn-mathclx0}{{3.2.44}{44}{Viscosity Solutions}{equation.3.2.44}{}}
|
||||
\newlabel{mathcalxn-mathclx0@cref}{{[equation][44][3,2]3.2.44}{[1][43][]44}}
|
||||
\newlabel{2.19}{{3.2.14}{45}{}{theorem.3.2.14}{}}
|
||||
\newlabel{2.19@cref}{{[lemma][14][3,2]3.2.14}{[1][45][]45}}
|
||||
\newlabel{2.59}{{3.2.50}{46}{}{equation.3.2.50}{}}
|
||||
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|
||||
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|
||||
\newlabel{2.60@cref}{{[equation][51][3,2]3.2.51}{[1][46][]46}}
|
||||
\newlabel{2.62}{{3.2.53}{46}{Viscosity Solutions}{equation.3.2.53}{}}
|
||||
\newlabel{2.62@cref}{{[equation][53][3,2]3.2.53}{[1][46][]46}}
|
||||
\newlabel{2.63}{{3.2.54}{46}{Viscosity Solutions}{equation.3.2.54}{}}
|
||||
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|
||||
\newlabel{2.64}{{3.2.55}{46}{Viscosity Solutions}{equation.3.2.55}{}}
|
||||
\newlabel{2.64@cref}{{[equation][55][3,2]3.2.55}{[1][46][]46}}
|
||||
\citation{BHJ21}
|
||||
\citation{Beck_2021}
|
||||
\newlabel{2.65}{{3.2.56}{47}{Viscosity Solutions}{equation.3.2.56}{}}
|
||||
\newlabel{2.65@cref}{{[equation][56][3,2]3.2.56}{[1][46][]47}}
|
||||
\newlabel{2.66}{{3.2.57}{47}{Viscosity Solutions}{equation.3.2.57}{}}
|
||||
\newlabel{2.66@cref}{{[equation][57][3,2]3.2.57}{[1][47][]47}}
|
||||
\newlabel{ungn}{{3.2.58}{47}{Viscosity Solutions}{equation.3.2.58}{}}
|
||||
\newlabel{ungn@cref}{{[equation][58][3,2]3.2.58}{[1][47][]47}}
|
||||
\newlabel{u0gn}{{3.2.59}{47}{Viscosity Solutions}{equation.3.2.59}{}}
|
||||
\newlabel{u0gn@cref}{{[equation][59][3,2]3.2.59}{[1][47][]47}}
|
||||
\newlabel{2.70}{{3.2.61}{47}{Viscosity Solutions}{equation.3.2.61}{}}
|
||||
\newlabel{2.70@cref}{{[equation][61][3,2]3.2.61}{[1][47][]47}}
|
||||
\newlabel{unu0}{{3.2.62}{48}{Viscosity Solutions}{equation.3.2.62}{}}
|
||||
\newlabel{unu0@cref}{{[equation][62][3,2]3.2.62}{[1][48][]48}}
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\documentclass{article}
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\setlength\parindent{0pt}
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\title{Numerical Analysis of DiffEq HW 1}
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\author{Shakil Rafi}
|
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\begin{document}
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||||
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\maketitle
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||||
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||||
\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}
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\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}}) \\
|
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\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}
|
||||
|
|
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\global\let\oldnewlabel\newlabel
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||||
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
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\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
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\AtEndDocument{\ifx\hyper@anchor\@undefined
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\fi}
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|
||||
%
|
||||
|
||||
\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}
|
||||
|
|
@ -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}%
|
||||
|
|
@ -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}
|
||||
}
|
||||
|
|
@ -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}}
|
||||
\newcommand{\cD}{\mathcal{D}}
|
||||
\newcommand{\cE}{\mathcal{E}}
|
||||
\newcommand{\cF}{\mathcal{F}}
|
||||
\newcommand{\cG}{\mathcal{G}}
|
||||
\newcommand{\cH}{\mathcal{H}}
|
||||
\newcommand{\cI}{\mathcal{I}}
|
||||
\newcommand{\cJ}{\mathcal{J}}
|
||||
\newcommand{\cK}{\mathcal{K}}
|
||||
\newcommand{\cL}{\mathcal{L}}
|
||||
\newcommand{\cM}{\mathcal{M}}
|
||||
\newcommand{\cN}{\mathcal{N}}
|
||||
\newcommand{\cO}{\mathcal{O}}
|
||||
\newcommand{\cP}{\mathcal{P}}
|
||||
\newcommand{\cQ}{\mathcal{Q}}
|
||||
\newcommand{\cR}{\mathcal{R}}
|
||||
\newcommand{\cS}{\mathcal{S}}
|
||||
\newcommand{\cT}{\mathcal{T}}
|
||||
\newcommand{\cU}{\mathcal{U}}
|
||||
\newcommand{\cV}{\mathcal{V}}
|
||||
\newcommand{\cW}{\mathcal{W}}
|
||||
\newcommand{\cX}{\mathcal{X}}
|
||||
\newcommand{\cY}{\mathcal{Y}}
|
||||
\newcommand{\cZ}{\mathcal{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}{{\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
|
||||
Some files were not shown because too many files have changed in this diff Show More
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Reference in New Issue