Match-based solution of general parametric eigenvalue problems

TL;DR

A novel algorithm for solving general parametric (nonlinear) eigenvalue problems and proposes an adaptive strategy that allows one to effectively apply the method even without any a priori information on the behavior of the sought-after eigenvalues.

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.

Figures (16)

• Figure 1: Sample view of a collection of eigenvalue curves. The arrows denote the "direction of movement" along each curve as $p$ increases. The shown curves have been obtained by applying our proposed approximation algorithm to the numerical example described in \ref{['sec:5:3']}.
• Figure 1: Example of match and approximation of the eigenvalue curves. Exact eigenvalues at $p=p_1$ and $p=p_2$ are shown as circles and squares, respectively. The line segments are the approximate eigenvalue curves $\tilde{\lambda}^i$. On the right, the cost matrix (with entries truncated at the first decimal digit). The highlighted entries correspond to the optimal match, which is reported below the matrix.
• Figure 1: Example of local behavior around an order-3 bifurcation (left plot). Arrows show the direction of eigenvalue changes as $p$ increases. Squares and dots denote eigenvalues at $p=p_1$ and $p=p_2$, respectively. Performing a match and linking the eigenvalues 1-to-1 (as described in \ref{['sec:2:1']}) leads to the middle figure. The all-at-once implicit approximate representation of the eigenvalues (described in \ref{['sec:3:0']}) gives the right figure.
• Figure 1: The testing error at $\hat{p}_j=\frac{p_j+p_{j+1}}{2}$ is zero. This prevents refinements near the mislabeled eigenvalue crossing.
• Figure 1: Real (left) and imaginary (right) parts of the solutions of \ref{['eq:5:1:evp']}. The curves are plotted only over the region of interest, i.e., $\left|\lambda\right|\leq4$.

Theorems & Definitions (11)

• Theorem 2.1: Keldysh
• Remark 2.2
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 4.1
• Remark 4.2
• Proposition 4.3
• Proof 1: Sketch of proof
• Remark 4.4