A modified projected walk on spheres method for elliptic equations on high-dimensional embedded manifolds: algorithm and error estimates

Abstract

In this paper, we propose a modified projected Walk on Spheres method (MPWoS) 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. The proposed approach can be viewed as a high-dimensional extension and modification of the projected Walk on Spheres method introduced for surface PDEs in [Sugimoto et al., SIGGRAPH Asia 2024 Conference Papers, pp. 1-10], with three main distinctions: a compensation term that corrects the discrepancy between the ambient Laplacian applied to the closest-point extension and the intrinsic Laplace-Beltrami operator on the manifold, an adaptive radius strategy determined by local geometric and boundary information, and a rigorous error analysis for the proposed algorithm. Under assumptions on the geometric projection and the 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.