A projected walk on spheres method for elliptic equations on high-dimensional embedded manifolds
This work provides a novel stochastic solver for PDEs on manifolds, addressing the challenge of high-dimensional geometry for computational scientists.
The paper proposes a projected Walk on Spheres method for solving screened Poisson equations on high-dimensional embedded manifolds, achieving meshfree and parallelizable computation with demonstrated convergence up to 1000 dimensions.
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.