Projected Walk on Spheres: A Monte Carlo Closest Point Method for Surface PDEs

TL;DR

The paper introduces PWoS, a Monte Carlo method for solving surface PDEs that projects points onto the surface during the walk, enabling efficient computations on complex surfaces without discretization.

Contribution

It extends the walk on spheres method to surface PDEs using a projection technique, with strategies for local feature size estimation and boundary distance computation.

Findings

Demonstrates convergence of PWoS

Applies PWoS to graphics tasks like diffusion and wave propagation

Works on various surface types, including mixed codimension

Abstract

We present projected walk on spheres (PWoS), a novel pointwise and discretization-free Monte Carlo solver for surface PDEs with Dirichlet boundaries, as a generalization of the walk on spheres method (WoS) [Muller 1956; Sawhney and Crane 2020]. We adapt the recursive relationship of WoS designed for PDEs in volumetric domains to a volumetric neighborhood around the surface, and at the end of each recursion step, we project the sample point on the sphere back to the surface. We motivate this simple modification to WoS with the theory of the closest point extension used in the closest point method. To define the valid volumetric neighborhood domain for PWoS, we develop strategies to estimate the local feature size of the surface and to compute the distance to the Dirichlet boundaries on the surface extended in their normal directions. We also design a mean value filtering method for PWoS to improve the method's efficiency when the surface is represented as a polygonal mesh or a point cloud. Finally, we study the convergence of PWoS and demonstrate its application to graphics tasks, including diffusion curves, geodesic distance computation, and wave propagation animation. We show that our method works with various types of surfaces, including a surface of mixed codimension.