Walk on Decomposed Subdomains: Faster Monte Carlo PDE Solvers

Monte Carlo methods like Walk on Spheres solve elliptic PDEs without meshing, but they suffer from high variance and slow convergence on complex geometries. This work introduces a decomposition strategy that splits the domain into smaller subdomains, dramatically shortening random walks and reducing variance. By coupling these subdomain walks with deterministic grid-based solvers, the method eliminates variance at the cost of controllable discretization bias, offering a practical bridge between grid-free and grid-based approaches.