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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. 

Physics-based differentiable rendering is becoming increasingly crucial for tasks in inverse rendering and machine learning pipelines. To address discontinuities caused by geometric boundaries and occlusion, two classes of methods have been proposed: 1) the edge-sampling methods that directly sample light paths at the scene discontinuity boundaries, which require nontrivial data structures and precomputation to select the edges, and 2) the reparameterization methods that avoid discontinuity sampling but are currently limited to hemispherical integrals and unidirectional path tracing. We introduce a new mathematical formulation that enjoys the benefits of both classes of methods. Unlike previous reparameterization work that focused on hemispherical integral, we derive the reparameterization in the path space. As a result, to estimate derivatives using our formulation, we can apply advanced Monte Carlo rendering methods, such as bidirectional path tracing, while avoiding explicit sampling of discontinuity boundaries. We show differentiable rendering and inverse rendering results to demonstrate the effectiveness of our method. 

Pixel reconstruction filters play an important role in physics-based rendering and have been thoroughly studied. In physics-based differentiable rendering, however, the proper treatment of pixel filters remains largely under-explored. We present a new technique to efficiently differentiate pixel reconstruction filters based on the path-space formulation. Specifically, we formulate the pixel boundary integral that models discontinuities in pixel filters and introduce new antithetic sampling methods that support differentiable path sampling methods, such as adjoint particle tracing and bidirectional path tracing. We demonstrate both the need and efficacy of antithetic sampling when estimating this integral, and we evaluate its effectiveness across several differentiable- and inverse-rendering settings. 

Boundary integrals are unique to physics-based differentiable rendering and crucial for differentiating with respect to object geometry. Under the differential path integral framework---which has enabled the development of sophisticated differentiable rendering algorithms---the boundary components are themselves path integrals. Previously, although the mathematical formulation of boundary path integrals have been established, efficient estimation of these integrals remains challenging. In this paper, we introduce a new technique to efficiently estimate boundary path integrals. A key component of our technique is a primary-sample-space guiding step for importance sampling of boundary segments. Additionally, we show multiple importance sampling can be used to combine multiple guided samplings. Lastly, we introduce an optional edge sorting step to further improve the runtime performance. We evaluate the effectiveness of our method using several differentiable-rendering and inverse-rendering examples and provide comparisons with existing methods for reconstruction as well as gradient quality.