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Inverse problems, from 3D reconstruction to PDE-constrained optimization, are fundamental across science and engineering, relying on accurate gradients that link physical measurements to parameters like geometry, materials, and boundary conditions. These gradients drive optimization for tasks such as calibrating digital twins and designing devices with specific physical properties. However, differentiating traditional mesh-based simulations becomes computationally prohibitive as geometric complexity increases, largely due to the expenses of meshing and re-meshing.
Monte Carlo methods offer a compelling alternative by reformulating these problems—including light transport and many elliptic PDEs—as high-dimensional integrations that inherently handle intricate geometries without requiring explicit discretization. Building on this principle, this dissertation introduces a suite of novel Differentiable Monte Carlo estimators that compute gradients efficiently through both path-space light-transport simulations and stochastic PDE solvers.
Our core contributions include: 1. A unified differential path-integral framework for rendering that supports both interfacial and volumetric light transport. This framework delivers robust gradient estimation for complex geometries and light transport effects. We further enhance this by introducing a novel method for shape differentiation of translucent objects and integrating antithetic sampling for variance reduction in pixel reconstruction filters. 2. Novel differentiable Monte Carlo solvers for second-order elliptic PDEs using walk-on-spheres (WoS) and walk-on-stars (WoSt) processes. These solvers enable differentiation with respect to arbitrary parameters, including domain shapes and mixed boundary conditions, thereby significantly enhancing derivative accuracy.
These advancements provide efficient, highly parallelizable, and scalable alternatives for tackling inverse rendering and PDE-constrained optimization problems. The proposed differentiable Monte Carlo framework delivers low-variance gradient computations that scale effectively with geometric complexity, paving the way for sophisticated gradient-based design and analysis in computer graphics, computational physics, vision, and engineering.