Walk on Spheres Algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence
This work provides a simpler Monte Carlo method for solving Helmholtz and Yukawa boundary value problems, which is relevant for computational physics and engineering applications.
The authors reformulate the Helmholtz/Yukawa equation as a Laplace equation via the Duffin correspondence, enabling a classical Walk on Spheres algorithm for Monte Carlo simulation. They demonstrate the algorithm's effectiveness by comparing it with existing modified WOS methods.
We show that a constant-potential time-independent Schrödinger 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.