Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients

TL;DR

This paper introduces a grid-free Monte Carlo method for solving complex PDEs with spatially varying coefficients, eliminating the need for discretization and enabling efficient, unbiased solutions in intricate geometries.

Contribution

It extends the walk on spheres algorithm to variable-coefficient PDEs using volumetric rendering techniques, providing an unbiased, mesh-free Monte Carlo solver.

Findings

Exact solutions in expectation for complex geometries.

No meshing or global system solving required.

Efficient pointwise evaluation of PDE solutions.

Abstract

Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem -- leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend the walk on spheres (WoS) algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired by null-scattering yields unbiased Monte Carlo estimators for a large class of 2nd-order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.