  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.
