Monte Carlo physical simulation

Partial differential equations (PDEs) play a central role in physics-based modeling across vision, graphics, and robotics, but conventional grid-based solvers often struggle with scalability and complex geometry. This tutorial introduces grid-free Monte Carlo methods for solving PDEs, focusing on algorithms such as walk on spheres and walk on stars that eliminate the need for spatial discretization. It presents the theoretical foundations of these methods alongside practical techniques for efficient sampling, variance reduction, and differentiable simulation. The tutorial also highlights applications in vision and robotics, including inverse problems and physics-based learning, and provides hands-on guidance for implementing Monte Carlo PDE solvers in real-world systems.