Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace's Equation

TL;DR

This paper introduces a caching strategy for Walk on Spheres algorithms that leverages Brownian Motion continuity, improving the efficiency of Monte Carlo estimates for Laplace's equation with Dirichlet boundary conditions.

Contribution

A novel caching approach that utilizes path continuity to enhance the asymptotic runtime of Walk on Spheres algorithms for Laplace's equation.

Findings

Improved asymptotic runtime over previous methods

Effective caching reduces computational complexity

Demonstrated performance gains on complex problems

Abstract

Walk on Spheres algorithms leverage properties of Brownian Motion to create Monte Carlo estimates of solutions to a class of elliptic partial differential equations. We propose a new caching strategy which leverages the continuity of paths of Brownian Motion. In the case of Laplace's equation with Dirichlet boundary conditions, our algorithm has improved asymptotic runtime compared to previous approaches. Until recently, estimates were constructed pointwise and did not use the relationship between solutions at nearby points within a domain. Instead, our results are achieved by passing information from a cache of fixed size. We also provide bounds on the performance of our algorithm and demonstrate its performance on example problems of increasing complexity.