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

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.