Grid-free Monte Carlo for PDEs with spatially varying coefficients

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