Match-Based Adaptive Contour Eigensolver

• Match-based adaptive contour eigensolver is a numerical method that dynamically adjusts integration contours to accurately isolate eigenvalues in complex, large-scale, or parametric eigenproblems.
• It leverages spectral matching, moment-based subspace acceleration, and adaptive partitioning to optimize computational efficiency and ensure numerical stability across diverse matrix types.
• Empirical results indicate significant speedups and robust eigenvalue recovery, making the method practical for applications in mechanics, quantum chemistry, and electromagnetic analysis.

A match-based adaptive contour eigensolver is a class of numerical methods designed to isolate and accurately compute eigenvalues of (potentially large-scale or parametric/nonlinear) eigenproblems by adaptively partitioning the spectral domain using data-driven, structure-aware, or iterative schemes. The hallmark of these methods is their dynamic, iterative adaptation of the integration contour(s) used in contour-integral eigensolvers—guided by spectral information, moment-matching, kernel density estimation, or parameter matching heuristics—to ensure efficiency, numerical stability, and broad applicability across Hermitian, non-Hermitian, linear, and nonlinear eigenvalue problems.

1. Mathematical Foundations of Contour-Based Eigensolvers

The core of contour-based eigensolvers is the use of spectral projectors defined through complex contour integrals. For a generalized eigenvalue problem (GEP)

$$A\mathbf{x} = \lambda B \mathbf{x}, \quad A,B \in \mathbb{C}^{n\times n},$$

the spectral projector associated with eigenvalues inside a target region $\Gamma\subset\mathbb{C}$ is

$$P = \frac{1}{2\pi i} \oint_\Gamma (z B - A)^{-1} B \, dz.$$

This operator projects onto the invariant subspace of the included eigenpairs. When the problem is parameter-dependent, e.g. $A(\lambda,p)\mathbf{v}=0$, the integral generalizes to $$P(p) = (1/2\pi i) \oint_\Gamma A(\sigma,p)^{-1} d\sigma$$, maintaining the property that it selects the eigenspace corresponding to eigenvalues inside $\Gamma$ (Pradovera et al., 2023, Chen et al., 2 Nov 2025, Huber et al., 2020).

To render the calculation tractable, $\Gamma$ is discretized into quadrature nodes $\{z_j, \omega_j\}_{j=1}^q$, reducing the continuous projector to a sum over a finite set of shifted linear systems. Eigenvalues are then extracted by subspace projection (e.g., Rayleigh–Ritz on the range of $P$), or by the moment method of Sakurai–Sugiura (forming block-Hankel moment matrices to capture multiplicities and resolve clusters) (Huber et al., 2020, Liu et al., 15 Mar 2025).

2. Adaptive Contour Construction and Spectral Matching Techniques

A critical challenge is the a priori selection of $\Gamma$ such that all desired eigenvalues are enclosed, while minimizing unnecessary computational cost. Match-based adaptive approaches address this by matching the contour or partitioning the spectral domain according to eigenvalue density, moment information, or parametric data.

Modern frameworks such as DeepContour (Chen et al., 2 Nov 2025) employ a deep learning-based spectral predictor (Fourier Neural Operator) combined with kernel density estimation (KDE) on the predicted eigenvalues:

• The FNO rapidly infers the likely spectral support $\hat\Lambda$ from the system parameters.
• KDE is applied to $\hat\Lambda$, constructing a smoothed spectral density.
• Intervals with too many eigenvalues (as indicated by KDE) are recursively subdivided at points of maximal sparsity; small intervals are wrapped in tight contours (typically circles or boxes), ensuring well-localized and balanced quadrature and subspace sizes.

An alternative is region partitioning: recursively splitting the domain based on moment or singular value information derived from the contour-integral method (Liu et al., 15 Mar 2025). Here, division continues until individual subregions are guaranteed (by rank, residual, or SVD criteria) to contain only a manageable number of eigenvalues.

For parameter-dependent problems, match-based solvers (Pradovera et al., 2023) track eigenvalue trajectories as parameters change, aligning eigenvalues across samples by solving combinatorial assignment problems on cost matrices constructed from consecutive spectra. Special care is given to bifurcations and eigenvalue migrations, with interpolation techniques for eigenvalue curves and polynomial-based handling of defectivity.

3. Subspace Acceleration and Moment Matching

To reduce computational cost, moment-matching schemes generate richer subspaces without increasing the number of expensive right-hand-sides in linear solves. The Sakurai–Sugiura moment method uses powers of $z$ within the contour integral: $$M_k = \sum_{j=1}^q \omega_j z_j^k (z_j B - A)^{-1} B Y,\quad k=0,\ldots,s-1,$$ where $Y$ is a block of random vectors. Collecting several moments ($s>1$) expands the subspace, improving convergence and potentially reducing the number of overall shifted linear solves required (Huber et al., 2020).

Match-based adaptive algorithms dynamically adjust the number of moments, quadrature order, and whether to operate in multi-moment or single-moment mode according to the observed convergence:

• If residual decrease stagnates, the method switches mode or adjusts the quadrature accordingly.
• Locking strategies ensure that converged eigenpairs are excluded from subsequent iterations, further increasing efficiency (Huber et al., 2020).

4. Rational Filter Optimization and Partitioning for Load Balancing

Rational filters, notably in the FEAST and CIRR frameworks, serve as practical approximations of spectral projectors. Optimal rational filter design aims to minimize the worst-case convergence ratio (e.g., Zolotarev filtering, SLiSe/ESLiSe filter optimization) for interior eigenvalues over the desired spectral interval: $$w_G(r) = \frac{\max_{|x| \ge 1/G} |r(x)|}{\min_{|x| \le G} |r(x)|}$$ where $r$ denotes the rational filter and $G$ the spectral gap parameter (Kollnig et al., 2020, Chen et al., 2023).

Adaptive match-based eigensolvers balance computational load across segments by partitioning the spectrum such that each slice admits a similar number of eigenvalues (estimated via stochastic trace or Lanczos methods), and then applying (possibly parallelized) FEAST-like eigensolves on each partition using the same optimized filter. This ensures both efficiency and load balancing even with highly nonuniform eigenvalue distributions.

Partitioning criteria can be further improved by automated region splitting based on robust rank-revealing methods, as in recursive Beyn's method, which eliminates the need for hand-tuned parameters in partitioning (Liu et al., 15 Mar 2025). This yields reliable extraction even near singularities and accumulation points.

5. Robustness, Numerical Accuracy, and Error Analysis

The accuracy and robustness of match-based contour eigensolvers hinge on several interrelated factors:

• Contour proximity to the spectrum: The quadrature error in the spectral projector decays exponentially as the integration contour is separated from the nearest eigenvalue. By adaptively shrinking region sizes or using KDE-based intervals, a safety buffer $\delta$ is maintained, ensuring numerical stability (Chen et al., 2 Nov 2025, Kollnig et al., 2020).
• Subspace dimension and moment order: Adaptive schemes match subspace size and basis width with the expected multiplicity and density of eigenvalues, adjusting on-the-fly as spectral data become available (Huber et al., 2020).
• Error bounds: For projectors, $\|P-P_N\| \leq C e^{-c N_q}$, $N_q$ being the quadrature order and $c$ a constant, provided the contour remains separated from the spectrum (Chen et al., 2 Nov 2025).
• Numerical experiments confirm high accuracy (e.g., residuals $\leq 10^{-12}$; missed-eigenvalue rates zero with adaptive KDE contours) and significant speedups versus fixed, non-adaptive approaches (e.g., DeepContour achieves up to $5.63\times$ speedup on high-dimensional GEPs) (Chen et al., 2 Nov 2025).

6. Practical Applications and Benchmark Results

Match-based adaptive contour eigensolvers have demonstrated broad applicability:

• Linear GEPs from mechanics, quantum chemistry, and electromagnetic cavity analysis, where spectral densities are highly heterogeneous (Chen et al., 2 Nov 2025).
• Parametric and nonlinear eigenproblems (e.g., waveguide modes, RF-gun cavities) where eigenvalue curves migrate, merge, or bifurcate as system parameters vary—requiring robust matching and partitioning (Pradovera et al., 2023, Liu et al., 15 Mar 2025).
• High-performance parallel settings, where partitioned filters and match-based slicing enable strong and weak scaling due to load balancing and localized communication (Kollnig et al., 2020).

Empirical studies reveal:

• Speedups of $2\times{-}5.63\times$ over standard subspace or scouting baseline methods for large matrices ($N \sim 5\times 10^4$; $M=500$ target eigenvalues) (Chen et al., 2 Nov 2025).
• Drastic reduction in total right-hand-sides and wall time in multi-moment acceleration frameworks (Huber et al., 2020).
• Robust recovery of all true eigenvalues (residuals $\leq 10^{-12}$) even in the presence of multiplicities, near-pole behavior, or dense clusters (Liu et al., 15 Mar 2025).
• Near-optimal convergence rates for rational filter-based segmentation, empirically matching theoretical predictions from Zolotarev analysis (Chen et al., 2023, Kollnig et al., 2020).

7. Extensions and Future Directions

Emerging developments target several natural generalizations:

• 2D kernel density estimation for complex spectra in genuinely non-Hermitian or nonlinear eigenvalue problems (Chen et al., 2 Nov 2025).
• Active learning frameworks for spectral predictors to improve generalization across domains.
• Fully non-intrusive eigensolvers for high-dimensional parametric NEPs, exploiting match-based adaptation for both spectral and parameter domains (Pradovera et al., 2023).
• Region partitioning approaches for NEPs that circumvent all problem-dependent thresholding while maintaining accuracy in singular and accumulation regions (Liu et al., 15 Mar 2025).
• Advanced rational filter design leveraging nested (composite) quadrature or optimization for further reduction in computational overhead and improved load balancing (Chen et al., 2023, Kollnig et al., 2020).

A plausible implication is the continued evolution toward hybrid schemes integrating machine learning-based spectral surrogates with optimal rational filtering and adaptive partitioning, yielding fast, robust, and scalable eigensolvers for broad classes of operator pencil problems.