AAA least squares solution of Helmholtz problems

TL;DR

This paper introduces an adaptive, meshless method for solving exterior Helmholtz scattering problems by automating singularity placement through rational approximation, improving accuracy and robustness for complex geometries.

Contribution

It develops the AAALS-Helmholtz algorithm, connecting Helmholtz and Laplace problems, and provides a formal basis for the double poles technique, advancing boundary integral methods.

Findings

Robust source placement for complex geometries

Automated singularity detection via AAA algorithm

Theoretical link between Helmholtz and Laplace solutions

Abstract

This paper presents an adaptive numerical framework for solving exterior "sound-soft" scattering problems governed by the Helmholtz equation. By interpreting the Method of Fundamental Solutions through the lens of rational approximation, we introduce an automated strategy for singularity placement based on the analytic continuation of boundary data. The proposed AAALS-Helmholtz algorithm leverages a "continuum" variant of the AAA algorithm to identify the singularities limiting analytic extension, and to ensure an optimal source distribution even for complex, non star-shaped geometries. Furthermore, we establish a formal connection between the Helmholtz and Laplace problems, providing a theoretical justification for the "double poles" technique. The approach offers a robust, meshless alternative to heuristic source placement.