Walk on Spheres Algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence

TL;DR

This paper introduces a novel Walk On Spheres algorithm for solving Helmholtz and Yukawa equations by leveraging the Duffin correspondence, transforming these problems into Laplace equations for efficient Monte Carlo simulation.

Contribution

It develops a new Monte Carlo method based on the Duffin correspondence to solve Helmholtz and Yukawa equations, extending classical WOS techniques.

Findings

The Duffin WOS algorithm effectively solves Helmholtz and Yukawa equations.

Comparison shows the new method's advantages over existing algorithms.

The approach simplifies complex boundary value problems into Laplace equations.

Abstract

We show that a constant-potential time-independent Schr\"odinger equation with Dirichlet boundary data can be reformulated as a Laplace equation with Dirichlet boundary data. With this reformulation, which we call the Duffin correspondence, we provide a classical Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the boundary value problem. We compare the obtained Duffin WOS algorithm with existing modified WOS algorithms.