An iterative method for Helmholtz boundary value problems arising in wave propagation

TL;DR

This paper introduces a novel probabilistic domain decomposition approach to solve the Helmholtz boundary value problem, recasting it into Poisson equations and analyzing convergence conditions using stochastic methods.

Contribution

It presents a new probabilistic framework for Helmholtz problems, offering a different perspective and conditions for convergence based on stochastic and discretization techniques.

Findings

Identifies a predictable range of wave numbers where the method converges

Recasts Helmholtz as a sequence of Poisson problems using probabilistic arguments

Provides a necessary and sufficient condition for convergence

Abstract

The complex Helmholtz equation $(\Delta + k^2)u=f$ (where $k\in{\mathbb R},u(\cdot),f(\cdot)\in{\mathbb C}$) is a mainstay of computational wave simulation. Despite its apparent simplicity, efficient numerical methods are challenging to design and, in some applications, regarded as an open problem. Two sources of difficulty are the large number of degrees of freedom and the indefiniteness of the matrices arising after discretisation. Seeking to meet them within the novel framework of probabilistic domain decomposition, we set out to rewrite the Helmholtz equation into a form amenable to the Feynman-Kac formula for elliptic boundary value problems. We consider two typical scenarios, the scattering of a plane wave and the propagation inside a cavity, and recast them as a sequence of Poisson equations. By means of stochastic arguments, we find a sufficient and simulatable condition for the convergence of the iterations. Upon discretisation a necessary condition for convergence can be derived by adding up the iterates using the harmonic series for the matrix inverse -- we illustrate the procedure in the case of finite differences. From a practical point of view, our results are ultimately of limited scope. Nonetheless, this unexpected -- even paradoxical -- new direction of attack on the Helmholtz equation proposed by this work offers a fresh perspective on this classical and difficult problem. Our results show that there indeed exists a predictable range $k<k_{max}$ in which this new ansatz works with $k_{max}$ being far below the challenging situation.