Evaluation of Resonances via AAA Rational Approximation of Randomly Scalarized Boundary Integral Resolvents

This paper presents a novel algorithm, based on use of rational approximants of randomly scalarized boundary integral resolvents, for the evaluation of acoustic and electromagnetic resonances in open and closed cavities; for simplicity we restrict treatment to cavities in two-dimensional space. The desired open cavity resonances (also known as ``eigenvalues'' for interior problems, and ``scattering poles'' for exterior and open problems) are obtained as the poles of associated rational approximants; both the approximants and their poles are obtained by means of the recently introduced AAA rational-approximation algorithm. In fact, the proposed resonance-search method applies to any nonlinear eigenvalue problem (NEP) associated with a given function $F: U \to \mathbb{C}^{n\times n}$, wherein a complex value $k$ is sought for which $F_kw = 0$ for some nonzero $w\in \mathbb{C}^n$. For the cavity problems considered in this paper, $F_k$ is taken as a spectrally discretized version of a Green function-based boundary integral operator at spatial frequency $k$. In all cases, the scalarized resolvent is given by an expression of the form $u^* F_k^{-1} v$, where $u,v \in \mathbb{C}^n$ are fixed random vectors. A variety of numerical results are presented for both scattering resonances and other NEPs, demonstrating the accuracy of the method even for high frequency states.