Boundary Value Caching for Walk on Spheres

TL;DR

This paper introduces a caching strategy for walk on spheres Monte Carlo methods that leverages boundary solution estimates to efficiently evaluate solutions inside the domain, reducing noise and computational cost.

Contribution

The proposed method provides a bias-free, efficient caching scheme for elliptic PDE solutions that handles complex geometries without global solves or boundary repairs.

Findings

Reduces variance in Monte Carlo solutions

Handles imperfect geometries without boundary repair

Enables fast, output-sensitive interior evaluations

Abstract

Grid-free Monte Carlo methods such as walk on spheres can be used to solve elliptic partial differential equations without mesh generation or global solves. However, such methods independently estimate the solution at every point, and hence do not take advantage of the high spatial regularity of solutions to elliptic problems. We propose a fast caching strategy which first estimates solution values and derivatives at randomly sampled points along the boundary of the domain (or a local region of interest). These cached values then provide cheap, output-sensitive evaluation of the solution (or its gradient) at interior points, via a boundary integral formulation. Unlike classic boundary integral methods, our caching scheme introduces zero statistical bias and does not require a dense global solve. Moreover we can handle imperfect geometry (e.g., with self-intersections) and detailed boundary/source terms without repairing or resampling the boundary representation. Overall, our scheme is similar in spirit to virtual point light methods from photorealistic rendering: it suppresses the typical salt-and-pepper noise characteristic of independent Monte Carlo estimates, while still retaining the many advantages of Monte Carlo solvers: progressive evaluation, trivial parallelization, geometric robustness, etc. We validate our approach using test problems from visual and geometric computing.