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

Abstract

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.

Figures (8)

• Figure 1: Demonstration of Algorithm \ref{['alg:two']} on two problems from the NLEVP set nlevp. In both panels red points represent exact eigenvalues and black circles represent eigenvalues produced by the proposed numerical method. Black lines represent divisions related to the adaptive version of the algorithm. Left panel: The CD player problem, for which all $60$ eigenvalues in the interval $[-50,5]$ were found to at least $7$ digits. Right: The butterfly problem, for which all $256$ eigenvalues were found to at least $10$ digits. Accuracy near machine precision was subsequently obtained by increasing the discretization in each subregion or by using methods refinement secant- or AAA-based methods described in Sections \ref{['sec:NEP']} and \ref{['sec:lowfreq']}
• Figure 2: Low-frequency interior eigenvalue problems mentioned in Section \ref{['sec:lowfreq']}: the first ten interior eigenfunctions for the kite-shaped domain. The associated frequencies $k$ are listed in the left column of Table \ref{['tab:lowfreq_evals']}.
• Figure 3: Low-frequency open arc eigenproblems discussed in Section \ref{['sec:lowfreq']}: the first ten scattering poles of the open circle with imaginary part less than $-0.2i$ (ordered left-to-right and top-to-bottom with increasing values of the real part of the frequency). The associated frequencies $k$ are displayed in the right column of Table \ref{['tab:lowfreq_evals']}.
• Figure 4: Comparison of Algorithm \ref{['alg:one']} to the block SS and Beyn 1 methods on the problem of evaluation of scattering poles outside the kite. Left panel: Eigenvalues computed by all three methods. Right panel: Errors computed by comparison with a secant method evaluation of the eigenvalue near $2.299- 1.597i$. The curves labeled "with secant" were obtained by following the initial eigenvalue determination by four iterations of the secant method. For the curve labeled "with local AAA", four points were sampled on a circle of radius $1e-5$ around the initial AAA approximation of the pole together with a degree $1$ rational approximant. To avoid underflow the maximum between the error and machine precision is plotted in all cases.
• Figure 5: Convergence of $\Re(k)$ and $\Im(k)$ to their limiting values as the gap size $\theta$ shrinks to zero for the nearly degenerate open circle eigenfunctions displayed as the second and third images on the top row of Figure \ref{['fig:circles']}. For both modes the imaginary parts (decay rates) converge at twice the rates of the real parts (spatial frequencies), and both the rates for the real and imaginary parts for the third mode are twice those for the second mode. Note that for the third mode the approximate nodal line is aligned with the gap and thus results in a weaker coupling of interior and exterior fields and associated faster convergence.