2 days ago
- Elliptic PDEs are crucial in graphics and engineering but challenging on complex geometries; traditional discretizations (FEM/FDM) require heavy meshing, while grid-free Monte Carlo methods (WoS/WoSt) suffer from long walks and high variance, especially with Neumann boundaries.
- The proposed hybrid method combines Monte Carlo's geometric flexibility with deterministic global solves, decomposing the domain into regular subdomains and using MC to estimate local first-passage solution operators (Poisson kernels) with controlled walk lengths.
- Local operators are assembled into a sparse global system solved deterministically, replacing discrete random walks and trading stochastic variance for fixed discretization bias, yielding stable, reusable solution operators.
- Shortening walks via domain decomposition (dividing into tiles) and coupling tiles via an absorbing Markov chain enables efficient deterministic solve of interface values, followed by parallel interior reconstruction using precomputed operators.
- The method adapts to geometric complexity (only tiles intersecting geometry need estimation), allows adjustable trade-off between MC effort and global solve size, and supports efficient re-solving for changing geometry.
- The approach reconciles grid-free Monte Carlo with grid-based solvers, leveraging the fast convergence of deterministic methods while preserving geometric flexibility, with potential for further generalization to other PDEs and boundary conditions.