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

Summary · qwen2.5:32b

The article introduces a new caching strategy for Walk on Spheres algorithms that leverages Brownian Motion continuity to improve variance reduction in solving Laplace's equation with Dirichlet boundary conditions, achieving better asymptotic runtime than previous methods. This approach differs from earlier pointwise estimation by utilizing relationships between nearby points through a fixed-size cache. The algorithm’s performance and bounds are demonstrated across problems of increasing complexity.

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Excerpt

arXiv:2404.17692v3 Announce Type: cross 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.