Abstract:In this paper, we propose a projected Walk on Spheres method (PWoS) for screened Poisson equations on embedded manifolds. The method employs local extensions together with the Green representation in local Euclidean balls, coupled with a closest-point projection that maps the boundary samples back to the manifold. This formulation yields a meshfree and highly parallelizable stochastic recursion in the ambient Euclidean space, rather than a direct discretization of the Laplace-Beltrami operator on the manifold. To recover the intrinsic geometric structure of the problem, we introduce a compensation term for the discrepancy between the ambient Laplacian of the closest-point extension and the Laplace-Beltrami operator on the manifold, together with an adaptive radius strategy determined by local geometric and boundary information. Under the geometric projection and prescribed compensation accuracy, we establish mean-square error estimates for the proposed Monte Carlo method in both the boundary and closed-manifold settings. Extensive numerical examples on parametrized, implicit, high-dimensional (up to 1000 dimensions), and point-cloud manifolds are presented to illustrate the convergence and efficiency of the proposed method across different geometries.
From: Changtao Sheng Prof (Asso) [view email]
[v1]
Sat, 20 Jun 2026 04:50:43 UTC (4,322 KB)
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