408 lines
9.4 KiB
TeX
408 lines
9.4 KiB
TeX
|
\documentclass[12pt]{article}
|
||
|
|
|
||
|
|
\usepackage{amsmath,
|
||
|
|
mleftright,
|
||
|
|
amssymb,
|
||
|
|
amsthm,
|
||
|
|
nicefrac,
|
||
|
|
etoolbox,
|
||
|
|
xparse,
|
||
|
|
geometry,
|
||
|
|
enumitem,
|
||
|
|
mathtools,
|
||
|
|
bbm
|
||
|
|
}
|
||
|
|
|
||
|
|
\mleftright
|
||
|
|
|
||
|
|
\usepackage[colorlinks=true]{hyperref}
|
||
|
|
|
||
|
|
\geometry{margin=1in}
|
||
|
|
|
||
|
|
\usepackage[sort,capitalize]{cleveref}
|
||
|
|
\newcommand{\creflastconjunction}{, and\nobreakspace}
|
||
|
|
\crefformat{equation}{(#2#1#3)}
|
||
|
|
\crefname{enumi}{item}{items}
|
||
|
|
\crefname{equation}{}{}
|
||
|
|
|
||
|
|
\theoremstyle{plain}
|
||
|
|
\newtheorem{theorem}{Theorem}[section]
|
||
|
|
\newtheorem{lemma}[theorem]{Lemma}
|
||
|
|
\newtheorem{proposition}[theorem]{Proposition}
|
||
|
|
\newtheorem{corollary}[theorem]{Corollary}
|
||
|
|
\newtheorem{example}[theorem]{Example}
|
||
|
|
\newtheorem{setting}[theorem]{Setting}
|
||
|
|
\newtheorem{conjecture}[theorem]{Conjecture}
|
||
|
|
|
||
|
|
\theoremstyle{remark}
|
||
|
|
\newtheorem{remark}[theorem]{Remark}
|
||
|
|
|
||
|
|
\theoremstyle{definition}
|
||
|
|
\newtheorem{definition}[theorem]{Definition}
|
||
|
|
|
||
|
|
\numberwithin{equation}{section}
|
||
|
|
|
||
|
|
%%%
|
||
|
|
|
||
|
|
\input{commands.tex}
|
||
|
|
|
||
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||
|
|
|
||
|
|
\begin{document}
|
||
|
|
|
||
|
|
%
|
||
|
|
%
|
||
|
|
%
|
||
|
|
|
||
|
|
\title{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}
|