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.

翻译：本文提出一种新颖算法，用于评估开放和封闭腔体中的声学与电磁共振，该算法基于随机标量化边界积分预解式的有理逼近；为简化起见，我们将处理范围限定于二维空间中的腔体。所需开放腔体共振（在内部问题中亦称“本征值”，在外部及开放问题中亦称“散射极点”）通过相关有理逼近的极点获得；逼近式及其极点均借助近期引入的AAA有理逼近算法求得。实际上，所提出的共振搜索方法适用于与给定函数$F: U \to \mathbb{C}^{n\times n}$相关的任何非线性本征值问题，其中需寻找使$F_kw = 0$对某个非零$w\in \mathbb{C}^n$成立的复数值$k$。针对本文考虑的腔体问题，$F_k$被视为空间频率$k$下基于格林函数的边界积分算子的谱离散化版本。在所有情况下，标量化预解式由形如$u^* F_k^{-1} v$的表达式给出，其中$u,v \in \mathbb{C}^n$为固定随机向量。本文展示了针对散射共振及其他非线性本征值问题的多种数值结果，证明了该方法即使对高频态仍具有较高精度。