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

This paper introduces a novel, adaptive algorithm that combines the AAA rational approximation of randomly scalarized boundary integral resolvents with a refined search strategy to accurately and efficiently compute acoustic and electromagnetic resonances as poles of rational approximants for both open and closed cavities.

Original authors: Oscar P. Bruno, Manuel A. Santana, Lloyd N. Trefethen

Published 2026-06-02

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CC BY 4.0

Original authors: Oscar P. Bruno, Manuel A. Santana, Lloyd N. Trefethen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to find the specific notes a musical instrument can play. If you have a closed drum, it hums at certain clear pitches. If you have a drum with a small hole in it, it still hums, but the sound "leaks" out, making the pitch slightly fuzzy and unstable. In physics and engineering, finding these specific pitches (called resonances) for complex shapes—like a kite, a rocket, or a circle with a tiny gap—is a difficult math problem.

This paper introduces a new, clever "tuner" to find these pitches quickly and accurately, whether the shape is a closed drum or an open one with a hole.

Here is how the method works, broken down into simple concepts:

1. The Problem: Finding the "Ghost" Notes

The scientists are looking for "resonances."

Closed Cavity: Think of a sealed room. It has specific natural frequencies (like a guitar string). These are easy to find.
Open Cavity: Think of a room with a door slightly ajar. The sound leaks out. The "notes" still exist, but they are faint and complex (mathematically, they are "complex numbers"). Finding them is like trying to hear a whisper in a noisy room.

Traditional methods to find these notes are like trying to map a forest by walking in perfect circles around every tree. It works, but it's slow and rigid. If you want to zoom in on a specific area, you have to restart your whole walk.

2. The Solution: The "Magic Guess" (AAA Algorithm)

The authors use a tool called the AAA algorithm. Imagine you are trying to guess the shape of a hidden object by poking it with a stick at random spots.

Instead of poking in a rigid grid, the AAA algorithm is a "smart guesser." It picks a few random spots, makes a guess about the shape, sees where it was wrong, and then picks the next spot specifically where the guess was most likely to be off.
It keeps doing this, refining its guess, until it has drawn a perfect map of the object using very few pokes. In the paper, this "object" is a mathematical function that describes the sound waves.

3. The Strategy: The Adaptive Search (The Zoom Lens)

Once the "smart guesser" has a rough map, the team uses an adaptive search strategy.

Imagine you are looking for a lost coin in a large field. Instead of checking every square inch, you divide the field into big rectangles.
You check the big rectangles first. If a rectangle looks "busy" (meaning it might contain a coin), you split it into four smaller rectangles and check those.
You keep zooming in only on the busy areas until you find the exact spot.
This allows the computer to find every resonance in a large area without wasting time on empty spaces.

4. The Final Polish: The "Secant Method"

The "smart guesser" (AAA) gets you very close to the answer, but maybe not perfect. To get it to "machine precision" (the highest accuracy a computer can hold), they use a secant method.

Think of this as a final, microscopic adjustment. Once the algorithm finds a resonance, it takes a tiny step back and forth around that spot to lock onto the exact frequency.
This step also acts as a quality control check. Sometimes the "smart guesser" might see a "ghost" (a fake resonance that isn't real). The final polish step confirms whether the note is real or a ghost and throws the fake ones away.

5. What They Found

The paper tests this new tuner on various shapes:

Simple Shapes: Circles and kites.
Hard Shapes: Rockets and circles with tiny gaps.
High Frequencies: Very high-pitched sounds (high frequencies) where the waves are tiny and hard to track.

The Results:

Speed: It found hundreds of resonances in seconds.
Accuracy: It found them with extreme precision, matching the theoretical "perfect" answers.
Flexibility: Unlike old methods that required walking in perfect circles, this method can search in rectangles, lines, or any shape, making it much easier to use for complex problems.
The "Gap" Effect: They studied what happens when a closed circle slowly opens up into a gap. They found that as the gap gets smaller, the "fuzzy" notes slowly turn back into the "clear" notes of the closed circle, and they could measure exactly how fast this happens.

Summary

The authors built a new digital tool that combines a smart, self-improving guesser with a zoom-in search strategy and a final precision lock. This allows them to find the hidden "notes" of complex shapes (like open or closed cavities) faster and more accurately than previous methods, even when the shapes are weird or the sounds are very high-pitched.

Technical Summary: Evaluation of Resonances via Adaptive AAA Rational Approximation

Problem Statement
The paper addresses the computational challenge of evaluating acoustic and electromagnetic resonances in open and closed cavities, as well as other scattering structures. Mathematically, these resonances correspond to solutions $(u, k)$ of the Helmholtz eigenvalue problem $\Delta u + k^2 u = 0$ subject to homogeneous boundary conditions (Dirichlet, Neumann, or Zaremba) on a domain $\Omega$ .

For interior problems, $k$ represents real eigenvalues.
For exterior and open-cavity problems, $k$ represents complex "scattering poles" or "complex eigenvalues" located in the lower half of the complex plane ( $\Im k \le 0$ ).

The problem is reformulated as a Nonlinear Eigenvalue Problem (NEP) for a matrix-valued function $F(k)$ , derived from the discretization of boundary integral operators. The goal is to find complex values $k$ such that $F(k)w = 0$ for some nonzero vector $w$ .

Methodology
The proposed algorithm combines three primary components: a randomly scalarized resolvent, the AAA rational approximation algorithm, and an adaptive search strategy with a secant-method refinement stage.

Scalarized Resolvent: Instead of solving the full matrix NEP directly, the method evaluates a scalar function $S(k) = u^* F(k)^{-1} v$ , where $u$ and $v$ are fixed random vectors in $\mathbb{C}^d$ . The poles of $S(k)$ correspond to the eigenvalues of the original problem.
AAA Rational Approximation: The function $S(k)$ is approximated by a rational function $r(k)$ using the AAA algorithm. This algorithm greedily selects interpolation points and weights to construct a barycentric rational approximant. The poles of this rational function $r(k)$ serve as initial approximations for the resonances.
Adaptive Search Strategy: To ensure all resonances within a search domain are captured, the algorithm employs a recursive dyadic subdivision of the domain (rectangles in the complex plane or intervals on the real line).
- Algorithm 1 (Basic): Evaluates $S(k)$ at a set of points, constructs the AAA approximant, and returns poles within the domain.
- Algorithm 2 (Adaptive): Recursively subdivides the domain. If the number of poles found in a sub-region changes upon further subdivision, the process repeats. This ensures that no resonances are missed due to insufficient sampling density.
Refinement and Verification:
- Secant Method: Once initial poles are identified, a secant-method termination stage is applied to $1/S(k)$ to achieve exponential convergence to near machine precision. This step also provides reliable error estimation and helps screen for spurious eigenvalues that the AAA algorithm might occasionally produce.
- Eigenfunctions: Corresponding eigenfunctions are recovered by computing the nullspace of $F(k)$ at the refined eigenvalues using Gaussian elimination with pivoting.

Key Contributions

Direct Pole Extraction: Unlike previous AAA-based approaches that use rational approximation to linearize the NEP, this method directly uses the poles of the rational approximant of the scalarized resolvent as the eigenvalue estimates.
Geometric Flexibility: The reliance on rational approximation rather than contour integration allows the use of arbitrary search domains, including rectangular regions and real intervals. This facilitates the design of the adaptive subdivision strategy, which is difficult to implement efficiently with traditional contour integration methods (e.g., trapezoidal rule on rectangles).
Robustness and Efficiency: The combination of adaptive search and secant refinement ensures high accuracy (near machine precision) for both low- and high-frequency states. The method requires only the evaluation of the scalarized resolvent at various wavenumbers, which is embarrassingly parallelizable.
Handling Open Arcs: The paper extends the theoretical framework for relating resolvent poles to eigenvalues to open-arc problems, utilizing specific jump relations for the normal derivative of the single-layer potential to handle edge singularities.

Numerical Results
The paper presents a variety of numerical experiments demonstrating the method's performance:

General NEPs: The algorithm successfully identified all eigenvalues in the "CD player" (60 eigenvalues) and "Butterfly" (256 eigenvalues) problems from the NLEVP collection.
Low-Frequency Scattering: Accurate computation of eigenvalues for kite-shaped and circular cavities with apertures. The method handled both real eigenvalues (interior) and complex poles near the real axis (open cavities).
Comparison with Contour Integration: Comparisons with the block SS method and Beyn's "Integral Algorithm 1" showed that while Beyn's method can converge faster without refinement, the proposed method achieves comparable accuracy with local refinement. Crucially, the proposed method avoids the need for multiple matrix solves per frequency point (required by block SS/Beyn when $L > 1$ ), offering potential advantages for large-scale 3D problems where LU factorization is unavailable.
Gap Size Asymptotics: The method was used to study the convergence of scattering poles as an open arc approaches a closed curve. Results showed that decay rates ( $\Im k$ ) converge at rates of $O(\theta^4)$ or $O(\theta^8)$ depending on the mode alignment with the gap, while spatial frequencies ( $\Re k$ ) converge at $O(\theta^2)$ or $O(\theta^4)$ .
High-Frequency Performance: The algorithm successfully computed 1,244 distinct interior eigenvalues for a unit circle in the interval $[1, 100]$ with near-machine accuracy. It also visualized high-frequency eigenfunctions for kite and rocket-shaped domains, showing strong correspondence between open-cavity eigenfunctions and scattering solutions under plane-wave illumination.

Significance and Claims
The authors claim that the proposed approach is simple, easy to implement, and rapidly convergent. Its primary significance lies in its ability to provide highly accurate approximations of scattering resonances and NEP solutions across a wide frequency range without the geometric constraints imposed by contour integration methods. The adaptive strategy ensures exhaustive search capabilities, while the secant-refinement stage guarantees high precision and reliability. The paper positions this method as a flexible alternative to existing contour-integration-based eigensolvers, particularly advantageous for problems involving complex geometries or where computational resources for repeated matrix factorizations are limited.