MSc Defence: Improving Boundary Value Estimates at Domain Corners in a Reinforcement Learning Based Elliptic PDE Solver

CANDIDATE: Sam Vermeulen

ABSTRACT: The Feynman-Kac formula links the solution of linear elliptic partial differential equations (PDEs) to the expectation of stochastic processes. However, discretization of Brownian motion near domain corners introduces significant error in boundary value estimation, as the true exit position becomes obscured. This thesis presents a method that improves boundary value estimation at domain corners by utilizing Brownian bridge statistics to compute the probability that a discretized Brownian walker hits a boundary before another. Our approach decomposes the hitting time distributions into “early”, “late”, and “miss” scoring components to estimate crossing probability. We demonstrate reduced error in boundary value estimation and incorporate this method into an existing TD learning algorithm for solving elliptic PDEs.