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.

翻译：本文提出一种新颖算法，用于计算开放与封闭腔体中的声学和电磁共振。该算法基于随机标量化边界积分预解式的有理逼近，结合自适应搜索策略及指数收敛的割线法终止阶段。目标腔体共振通过相关有理逼近的极点获得；逼近式及其极点均采用近期提出的AAA有理逼近算法计算。实际上，所提出的共振搜索方法适用于任意与给定函数$F: U \to \mathbb{C}^{d\times d}$相关的非线性特征值问题，其中记$F(k) = F_k$，需寻找使$F_kw = 0$对某个非零$w\in \mathbb{C}^d$成立的复数值$k$。针对本文研究的散射问题，$F_k$取为空间频率$k$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$u,v \in \mathbb{C}^d$为固定随机向量。所提出的自适应搜索策略依赖于对共振搜索域进行矩形剖分，并通过局部细化确保捕获域内所有共振。当搜索域为实轴区间时，所用矩形退化为搜索域的子区间，该方法同样有效。文中展示了多种数值结果，包括与基于复围道积分的经典方法的比较，以及开放腔体趋近封闭腔体时的渐近性讨论——所有结果均证明该方法对低频与高频态均具有高精度。