Walk on Heat Stars provides a boundary-integral Monte Carlo solver for parabolic PDEs with Neumann conditions via exact heat-ball sampling that yields unbiased estimators.
Walking on Spheres and Talking to Neighbors: Variance Reduction for Laplace's Equation
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abstract
Walk on Spheres algorithms leverage properties of Brownian Motion to create Monte Carlo estimates of solutions to a class of elliptic partial differential equations. We propose a new caching strategy which leverages the continuity of paths of Brownian Motion. In the case of Laplace's equation with Dirichlet boundary conditions, our algorithm has improved asymptotic runtime compared to previous approaches. Until recently, estimates were constructed pointwise and did not use the relationship between solutions at nearby points within a domain. Instead, our results are achieved by passing information from a cache of fixed size. We also provide bounds on the performance of our algorithm and demonstrate its performance on example problems of increasing complexity.
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A relaxed Picard iteration plus heteroscedastic boundary denoising lets Monte Carlo PDE solvers solve heat equations with nonlinear radiation boundary conditions more accurately than linearization.
RQMC applied to walk-on-spheres for harmonic functions yields median variance decay slightly better than O(n^{-1.1}) and reduction factors 1.8-10.7 across four methods and five examples.
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Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions
Walk on Heat Stars provides a boundary-integral Monte Carlo solver for parabolic PDEs with Neumann conditions via exact heat-ball sampling that yields unbiased estimators.
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Monte Carlo PDE Solvers for Nonlinear Radiative Boundary Conditions
A relaxed Picard iteration plus heteroscedastic boundary denoising lets Monte Carlo PDE solvers solve heat equations with nonlinear radiation boundary conditions more accurately than linearization.
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Randomized quasi-Monte Carlo for walk on spheres
RQMC applied to walk-on-spheres for harmonic functions yields median variance decay slightly better than O(n^{-1.1}) and reduction factors 1.8-10.7 across four methods and five examples.