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This 3 hour course provides a detailed overview of grid-free Monte Carlo methods for solving partial differential equations (PDEs) based on the walk on spheres (WoS) algorithm, with a special emphasis on problems with high geometric complexity. PDEs are a basic building block of models and algorithms used throughout science, engineering and visual computing. Yet despite decades of research, conventional PDE solvers struggle to capture the immense geometric complexity of the natural world. A perennial challenge is spatial discretization: traditionally, one must partition the domain into a high-quality volumetric mesh—a process that can be brittle, memory intensive, and difficult to parallelize. WoS makes a radical departure from this approach, by reformulating the problem in terms of recursive integral equations that can be estimated using the Monte Carlo method, eliminating the need for spatial discretization. Since these equations strongly resemble those found in light transport theory, one can leverage deep knowledge from Monte Carlo rendering to develop new PDE solvers that share many of its advantages: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations. The course is divided into two parts. Part I will cover the basics of using WoS to solve fundamental PDEs like the Poisson equation. Topics include formulating the solution as an integral equation, generating samples via recursive random walks, and employing accelerated distance and ray intersection queries to efficiently handle complex geometries. Participants will also gain experience setting up demo applications involving data interpolation, heat transfer, and geometric optimization using the open-source “Zombie” library, which implements various grid-free Monte Carlo PDE solvers. Part II will feature a mini-panel of academic and industry contributors covering advanced topics including variance reduction, differentiable and multi-physics simulation, and applications in industrial design and robust geometry processing.
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Grid-free Monte Carlo for PDEs with spatially varying coefficients
Partial differential equations (PDEs) with spatially varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem---leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend thewalk on spheres (WoS)algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired bynull-scatteringyields unbiased Monte Carlo estimators for a large class of 2nd order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.
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- PAR ID:
- 10603116
- Publisher / Repository:
- Association for Computing Machinery (ACM)
- Date Published:
- Journal Name:
- ACM Transactions on Graphics
- Volume:
- 41
- Issue:
- 4
- ISSN:
- 0730-0301
- Format(s):
- Medium: X Size: p. 1-17
- Size(s):
- p. 1-17
- Sponsoring Org:
- National Science Foundation
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