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

Abstract

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.

On a Semi-analytical Method for Solution of the 2d Laplace Equation in Arbitrary Domains

“…The WoS method has recently gained popularity in the Computer Graphic community see, for example, ref. [9]. Despite its theoretical accuracy, WoS faces convergence issues which are problematical for the scientific applications that we are concerned with.…”

Patch Decomposition for Efficient Mesh Contours Extraction

“…Mesh contours have proved to be crucial for various applications. It is a key component in 3D non‐photorealistic rendering for line art stylization (e.g., [KMM*02, GTDS10]) and scientific visualization [LVPI18], but also for shadow‐volume algorithms [Cro77], spherical visibility computation [NBMJ14], next‐event estimation for many lights rendering [CEK18], differential rendering through edge sampling [LADL18, YLB*22] and grid‐free Monte‐Carlo methods with Neumann boundary conditions [SMGC23]. In all these cases, the common problem is to efficiently locate contour edges every time the viewpoint or mesh vertex positions change [IFH*03, BH19].…”