Evaluation of resonances: adaptivity and AAA rational approximation of randomly scalarized boundary integral resolvents

This paper presents a novel algorithm, based on use of rational approximants of a randomly scalarized boundary integral resolvent in conjunction with an adaptive search strategy and an exponentially convergent secant-method termination stage, for the evaluation of acoustic and electromagnetic resonances in open and closed cavities. The desired cavity resonances 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 associated with a given function $F: U \to \mathbb{C}^{d\times d}$, wherein, denoting $F(k) = F_k$, a complex value $k$ is sought for which $F_kw = 0$ for some nonzero $w\in \mathbb{C}^d$. For the scattering problems considered in this paper, $F_k$ is taken to equal 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 $u,v \in \mathbb{C}^d$ are fixed random vectors. The proposed adaptive search strategy relies on use of a rectangular subdivision of the resonance search domain which is locally refined to ensure that all resonances in the domain are captured. The approach works equally well in the case in which the search domain is an interval of the real line, in which case the rectangles used degenerate into subintervals of the search domain. A variety of numerical results are presented, including comparisons with well-known methods based on complex contour integration, and a discussion of the asymptotics that result as open cavities approach closed cavities -- in all, demonstrating the accuracy provided by the method, for low- and high-frequency states alike.