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Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions
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Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions
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Grid-free Monte Carlo methods based on the walk on spheres (WoS) algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain or approximating functions in a finite basis. Such methods hence avoid aliasing in the solution, and evade the many challenges of mesh generation. Yet for problems with complex geometry, practical grid-free methods have been largely limited to basic Dirichlet boundary conditions. We introduce the walk on stars (WoSt) algorithm, which solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary conditions. The key insight is that one can efficiently simulate reflecting Brownian motion (which models Neumann conditions) by replacing the balls used by WoS with star-shaped domains. We identify such domains via the closest point on the visibility silhouette, by simply augmenting a standard bounding volume hierarchy with normal information. Overall, WoSt is an easy modification of WoS, and retains the many attractive features of grid-free Monte Carlo methods such as progressive and view-dependent evaluation, trivial parallelization, and sublinear scaling to increasing geometric detail.
Forward citations
Cited by 7 Pith papers
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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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Walk on spheres and Array-RQMC
Array-RQMC-WOS cuts Monte Carlo variance by 57-2290 times with empirical rates n^{-1.4} to n^{-1.8} and introduces a column-wise mean dimension to explain the gain.
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Monte Carlo Steklov Operators for Large-Scale Geometry Processing in the Wild
Monte Carlo estimation of volumetric Steklov operators enables robust spectral geometry processing at the scale of hundreds of thousands of in-the-wild meshes and supports contrastive 3D representation learning.
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Walk on spheres and Array-RQMC
Array-RQMC-WoS reduces Monte Carlo MSE/variance 71- to 3087-fold at n=2^17 on five Dirichlet problems, with empirical rates n^{-1.4} to n^{-1.8}, far above plain RQMC-WoS.
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Randomized quasi-Monte Carlo for walk on spheres
Randomized quasi-Monte Carlo applied to walk-on-spheres yields variance reduction factors between 1.8 and 10.7 and median convergence slightly better than O(n^{-1.1}) across five examples.
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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.
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