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

Abstract: 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 $F(k)w=0$ 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 $u^∗F^{-1}(k)v$, 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.

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