A parallel method for solving Laplace equations with Dirichlet data using local boundary integral equations and random walks

TL;DR

This paper introduces a parallel hybrid method combining boundary integral equations and Monte Carlo walks to efficiently solve Laplace equations with Dirichlet data in 3D domains, enabling scalable computation of boundary data.

Contribution

A novel hybrid approach that integrates local boundary integral equations with probabilistic Monte Carlo methods for parallel solution of Laplace equations.

Findings

Achieves parallel computation of Neumann data without data communication.

Uses Feynman-Kac formula with WOS algorithm for local solution evaluation.

Provides a scalable method for boundary value problems in 3D domains.

Abstract

In this paper, we will present a new approach for solving Laplace equations in general 3-D domains. The approach is based on a local computation method for the DtN mapping of the Laplace equation by combining a deterministic (local) boundary integral equation method and the probabilistic Feynman-Kac formula of PDE solutions. This hybridization produces a parallel algorithm where the bulk of the computation has no need for data communications. Given the Dirichlet data of the solution on a domain boundary, a local boundary integral equation (BIE) will be established over the boundary of a local region formed by a hemisphere superimposed on the domain boundary. By using a homogeneous Dirichlet Green's function for the whole sphere, the resulting BIE will involve only Dirichlet data (solution value) over the hemisphere surface, but over the patch of the domain boundary intersected by the hemisphere, both Dirichlet and Neumann data will be used. Then, firstly, the solution value on the hemisphere surface is computed by the Feynman-Kac formula, which will be implemented by a Monte Carlo walk on spheres (WOS) algorithm. Secondly, a boundary collocation method is applied to solve the integral equation on the aforementioned local patch of the domain boundary to yield the required Neumann data there. As a result, a local method of finding the DtN mapping is obtained, which can be used to find all the Neumann data on the whole domain boundary in a parallel manner. Finally, the potential solution in the whole space can be computed by an integral representation using both the Dirichlet and Neumann data over the domain boundary.