Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions

Bailey Miller*	Rohan Sawhney*	Keenan Crane†	Ioannis Gkioulekas†

ACM Trans. Graph. (2023)

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. This paper introduces the walk on stars (WoSt) method, 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 by locating the closest visible point on the geometric silhouette. Overall, WoSt retains many attractive features of other grid-free Monte Carlo methods, such as progressive and view-dependent evaluation, trivial parallelization, and sublinear scaling to increasing geometric detail.

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Bailey Miller*, Rohan Sawhney*, Keenan Crane†, Ioannis Gkioulekas† (2023). Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions. ACM Trans. Graph..

@article{Sawhney:2023:WoSt,
author = {Bailey Miller* and Rohan Sawhney* and Keenan Crane† and Ioannis Gkioulekas†},
title = {Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions},
journal = {ACM Trans. Graph.},
year = {2023},
publisher = {ACM},
address = {New York, NY, USA},
}