AAA least squares solution of Helmholtz problems

TL;DR

The paper tackles exterior sound-soft scattering for the Helmholtz problem by integrating the meshless Method of Fundamental Solutions with continuum AAA rational approximation to adaptively place singularities based on analytic continuation. The AAALS-Helmholtz framework uses a unit-circle rational approximation, conformal mapping, and a shielding curve to generate a robust, singularity-adapted distribution of sources, achieving high accuracy for complex geometries and high wavenumbers. It provides both theoretical justification (linking to Schwarz function geometry and Laplace limits) and extensive numerical validation, including comparisons to high-accuracy baselines. The work offers a practical, scalable meshless solver and lays groundwork for future enhancements such as FMM acceleration and broader boundary conditions (Neumann, transmission).

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.

Figures (10)

• Figure 1: Transformation of the unit circle $C(t)$ in the $\zeta$-plane onto an analytic Jordan curve $\Gamma(t)$ in the $z$-plane (thick black lines) via a generic mapping function $M:\zeta \rightarrow z$. This implicitly defines a conformal map between the regions $A \supset C$ and $M(A) \supset \Gamma$ (shaded yellow). On the left: a circular annular region (gray grid) centered on $C$, the zeros of $M'$ (red circles), and two curves $\gamma$ and $\delta$ (thick blue) connecting the zeros nearest to $C$; these bound a domain of conformality $A$ larger than the annulus, though not necessarily maximal. On the right: their respective images through $M$ and the branch cuts of the Schwarz function for $\Gamma$, delineated by black dots.
• Figure 2: Approximation of $v(z)=(\Re(z))^2$ on the parametric starfish curve $\Gamma(t) = (1+0.3\cos(5\pi\,t))e^{i\pi\,t}$ using the Matlab routine aaazp. Smaller black dots represent samples (381) generated on the fly, larger black dots are support points (95), and red dots are poles (94) of the rational approximation. The tolerance was loosened to $10^{-6}$ to clearly visualize the curve samples; accordingly, the computing time is reduced to half a second.
• Figure 3: Parametric mapping from the unit circle $C$ to the starfish curve $\Gamma$ of Figure (\ref{['fig:starfish-aaazp']}). On the left, red circles represent the zeros of $M'$. On the right, the starfish is shown with the branch cuts of the rational approximation of its Schwarz function (delineated by streams of black dots) and the images of the zeros of $M'$ through $M$ (red circles). With this particular parametrization, the zeros identify the branch points of $S(z)$. Only 45 sample points are necessary to reach a maximum error of $1.8\cdot10^{-15}$, explaining the seemingly coarse resolution of the curve.
• Figure 4: "Sound-soft" scattering problems for incident plane waves with angle $\pi/3$ and different wave numbers. On the left, the blue line $\gamma$ bounds the zeros of $M'$ (red circles) and the poles (blue dots) of the rational approximation of $v = \Re(u_{inc}(\Gamma))$ interior to $C$; black dots are support points of the rational approximation and red dots are their radial projections at an optimal distance between $\gamma$ and $C$. On the right, the solutions; red dots are scattering sources and black dots delineate the branch structure of the Schwarz function.
• Figure 5: The same curve $\Gamma$ of figure \ref{['fig:rand_wave']} for two point sources of equal wave number.