  Grid-free Monte Carlo methods based on the walk on spheres (WoS) algorithm
solve fundamental partial differential equations (PDEs) like the Poisson
equation without discretizing the problem domain or approximating functions in
a finite basis. Such methods hence avoid aliasing in the solution, and evade
the many challenges of mesh generation. Yet for problems with complex geometry,
practical grid-free methods have been largely limited to basic Dirichlet
boundary conditions. We introduce the walk on stars (WoSt) algorithm, which
solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary
conditions. The key insight is that one can efficiently simulate reflecting
Brownian motion (which models Neumann conditions) by replacing the balls used
by WoS with star-shaped domains. We identify such domains via the closest point
on the visibility silhouette, by simply augmenting a standard bounding volume
hierarchy with normal information. Overall, WoSt is an easy modification of
WoS, and retains the many attractive features of grid-free Monte Carlo methods
such as progressive and view-dependent evaluation, trivial parallelization, and
sublinear scaling to increasing geometric detail.
