Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning

TL;DR

This paper introduces bias-reducing discretization methods for stochastic processes in the Feynman-Kac formula and employs temporal difference learning to improve efficiency over traditional Monte Carlo approaches for solving elliptic PDEs.

Contribution

It presents novel discretization techniques to reduce bias and applies temporal difference learning to enhance sampling efficiency in solving elliptic equations via stochastic processes.

Findings

Bias reduction in stochastic discretization methods.

Improved efficiency with temporal difference learning.

Potential advantages over traditional Monte Carlo methods.

Abstract

The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, na\"ive numerical implementations suffer from statistical bias and sampling error. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.