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

2024-10-04

ACM SIGGRAPH Conference and Exhibition on Computer Graphics and Interactive Techniques in Asia

AI Summary

This work addresses the mesh-free, pointwise solution of surface partial differential equations (PDEs) under Dirichlet boundary conditions. We propose Projected Walk on Spheres (PWoS), a grid-free Monte Carlo method. PWoS extends the classical spherical walk to arbitrary embedded surfaces by introducing a geometry-adaptive projection mechanism grounded in the closest-point extension theory—enabling support for mixed-codimensional surfaces and unstructured inputs such as triangle meshes and point clouds. The method integrates local feature size estimation, normal distance field construction, and mean filtering for acceleration, enabling efficient computation of diffusion curves, geodesic distances, and wave propagation. Theoretical convergence is established, and empirical robustness is demonstrated across diverse geometric domains. Compared with conventional discretization-based solvers, PWoS achieves a superior trade-off between memory efficiency and numerical accuracy.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Content-based information diffusion

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.

Problem

• Solving surface PDEs with Dirichlet boundaries
• Extending walk on spheres to surfaces
• Handling surfaces represented as meshes or point clouds

Innovation

• Projected Walk on Spheres Monte Carlo solver
• Adapts recursive WoS to surface neighborhood
• Projects sample points back to surface