Parallel Multi-Step Contour Integral Methods for Nonlinear Eigenvalue Problems

TL;DR

This work addresses the problem of finding all eigenvalues $\lambda$ of a nonlinear matrix-valued function $T(\lambda)$ within a bounded region $\Omega\subset\mathbb{C}$ without requiring prior eigenvalue counts. It introduces two parallel multi-step contour-integral methods, pmCIMa and pmCIMb, based on domain decomposition: pmCIMa combines the spectral indicator method (SIM) with a linear eigensolver, and pmCIMb combines SIM with Beyn's method in subregions followed by a verification step. Both methods exploit parallelism over subregions and rely on quadrature-based contour integrals with the exponential convergence of the trapezoidal rule to recover eigenpairs robustly and accurately. Numerical experiments on a quadratic eigenvalue problem and on scattering poles demonstrate accurate eigenpairs and substantial parallel speedups (around $11\times$ on 12 workers), validating the approach's scalability for large-scale nonlinear eigenvalue problems in wave propagation and resonances.

Abstract

We consider nonlinear eigenvalue problems to compute all eigenvalues in a bounded region on the complex plane. Based on domain decomposition and contour integrals, two robust and scalable parallel multi-step methods are proposed. The first method 1) uses the spectral indicator method to find eigenvalues and 2) calls a linear eigensolver to compute the associated eigenvectors. The second method 1) divides the region into subregions and uses the spectral indicator method to decide candidate regions that contain eigenvalues, 2) computes eigenvalues in each candidate subregion using Beyn's method; and 3) verifies each eigenvalue by substituting it back to the system and computes the smallest eigenvalue. Each step of the two methods is carried out in parallel. Both methods are robust, accurate, and does not require prior knowledge of the number and distribution of the eigenvalues in the region. Examples are presented to show the performance of the two methods.

Figures (1)

• Figure 1: Eigenvalues computed by pmCIMa and pmCIMb. Left: Quadratic eigenvalue problem ($\Omega = [-0.5, 0.5] \times [-0.5, 0.5]$). Right: Scattering poles ($\Omega = [0.2, 3.2] \times [-3.2, -0.2]$).

Theorems & Definitions (4)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4