Grid-free Monte Carlo for PDEs with spatially varying coefficients

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.

A bidirectional formulation for Walk on Spheres

“…To implement our method one needs to evaluate 𝒢 ( x → y ) and 𝒫 ( x → y ). The concrete values of these depend on the PDE one is solving and we refer to the appendix in Sawhney et al [SSJC22] for a comprehensive listing.…”

“…However, the FEM-based nature of these methods does not fit well with Monte Carlo rendering methods. We believe that in the future, these methods could be replaced by our proposed WoS algorithm extended to heterogeneous domains using the concurrent work of Sawhney et al [SSJC22].…”

“…Deterministic solvers vs. Monte Carlo. The trade-offs between traditional, finite element method (FEM)-based PDE solvers and Monte Carlo solvers are well explored, and we refer to Sawhney and Crane [SC20] and Sawhney et al [SSJC22] for a thorough discussion. In this paper, we focus on problems with sparse sources.…”

“…The recursion terminates when samples x ′ or y ′ land on the boundary. In general, the probability of this event is almost zero, and we follow prior WoS work [SC20; SSJC22] and instead terminate if the sample lands within a small distance ∊ of the boundary, at the cost of a small amount of bias. Fig.…”

“…Catalyzed by the recent introduction of the walk‐on‐spheres (WoS) algorithm [Mul56] to graphics [SC20; SSJC22], a similar development is now happening for numerical solvers of partial differential equations (PDEs). Analogous to path tracing in rendering, WoS uses random walks to compute point estimates of the solution of harmonic PDEs [Eva10].…”