Match-based solution of general parametric eigenvalue problems

TL;DR

The paper introduces a new algorithm for solving general parametric eigenvalue problems by combining high-accuracy non-parametric solutions and global approximation techniques, effectively handling eigenvalue migration and bifurcations.

Contribution

It presents a novel match-based algorithm that integrates contour-integration methods with adaptive strategies for efficient parametric eigenvalue approximation.

Findings

Achieves high-accuracy eigenvalue approximations.

Effectively handles eigenvalue migration and bifurcations.

Demonstrates strong numerical performance across tests.

Abstract

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.