Solving nonlinear eigenvalue problems via contour integration and region partitioning

TL;DR

This paper addresses solving nonlinear eigenvalue problems $T(\lambda)v=0$ within a region $\Omega$ by fusing contour-integral techniques with region partitioning. The authors introduce a Beyn-based partitioning criterion that replaces problem-dependent indicators, combining Beyn's contour-moment approach with a region-splitting strategy to robustly identify many interior eigenvalues, even near singularities or accumulation points, and to recover eigenvectors alongside eigenvalues. Two algorithmic variants are developed: (i) RIM by numerical rank and (ii) Recursive Beyn's method, with an optional acceleration via Infinite GMRES to speed up linear solves. Numerical experiments on gun, photonics_1, and photonics_2 problems demonstrate accurate eigenpair recovery (including challenging multiplicities and poles), superior counting robustness over standard partitioning criteria, and substantial speedups when using infGMRES, indicating strong practical potential for quasi-normal mode analysis and related NEP applications.

Abstract

In this work, we combine Beyn's method and the recently developed recursive integral method (RIM) to propose a contour integral-based, region partitioning eigensolver for nonlinear eigenvalue problems. A new partitioning criterion is employed to eliminate the need for a problem-dependent parameter, making our algorithm much more robust compared to the original RIM. Moreover, our algorithm can be directly applied to regions containing singularities or accumulation points, which are typically challenging for existing nonlinear eigensolvers to handle. Comprehensive numerical experiments are provided to demonstrate that the proposed algorithm is particularly well suited for dealing with regions including many eigenvalues.

Figures (11)

• Figure 1: Gauss--Legendre quadrature rule on the boundary of a rectangular region.
• Figure 2: Partitioning pattern.
• Figure 3: Using recursive Beyn's method to solve the gun problem. The region of interest is $[1.25\times10^4,1.125\times10^5]\times[-5\times10^4,5\times10^4]$ in the complex plane. The parameters are set as: maximum eigenvalues in each subregion $k_{\mathrm{sub}}=5$, number of quadrature nodes on each segment $N=32$, and maximum search depth $d_{\mathrm{max}}=6$. All $22$ eigenvalues as well as their corresponding eigenvectors are found. The branch cut $(-\infty,108.8774^2]$ is indicated by a line with a cross mark on its endpoint.
• Figure 4: Geometric distribution of the photonics_1 and photonics_2 problems.
• Figure 5: Using recursive Beyn's method to solve the photonics_1 problem. The region of interest is $[0,0.188365]\times[0,0.384419]$ in the complex plane. The parameters are set as: maximum eigenvalues in each subregion $k_{\mathrm{sub}}=5$, number of quadrature nodes on each segment $N=32$, and maximum search depth $d_{\mathrm{max}}=6$. A total of $22$ different nonzero eigenvalues as well as their corresponding eigenvectors are found. The eigenvalue with high multiplicity $\lambda^{(0)}$ and the pole $\lambda^{(\infty)}$ are also marked in the figure.

Theorems & Definitions (4)

• Remark 1: How to divide the region
• Remark 2
• Remark 3
• Remark 4