Calculating the minimal/maximal eigenvalue of symmetric parametrized matrices using projection

TL;DR

This paper introduces a projection-based method for efficiently estimating the minimal eigenvalue of parametrized symmetric matrices, incorporating multiple eigenvectors and derivatives to improve accuracy, especially useful in uncertainty quantification.

Contribution

The paper presents a novel projection technique that uses multiple eigenvectors and their derivatives to accurately approximate minimal eigenvalues of large parametrized matrices.

Findings

Adding multiple eigenvectors improves approximation accuracy.

Including derivatives of eigenvectors enhances the subspace quality.

Numerical experiments confirm the method's efficiency and effectiveness.

Abstract

In applications of linear algebra including nuclear physics and structural dynamics, there is a need to deal with uncertainty in the matrices. We focus on matrices that depend on a set of parameters $\omega$ and we are interested in the minimal eigenvalue of a matrix pencil $(A,B)$ with $A,B$ symmetric and $B$ positive definite. If $\omega$ can be interpreted as the realisation of random variables, one may be interested in statistical moments of the minimal eigenvalue. In order to obtain statistical moments, we need a fast evaluation of the eigenvalue as a function of $\omega$. Since this is costly for large matrices,we are looking for a small parametrized eigenvalue problem whose minimal eigenvalue makes a small error with the minimal eigenvalue of the large eigenvalue problem. The advantage, in comparison with a global polynomial approximation (on which, e.g., the polynomial chaos approximation relies), is that we do not suffer from the possible non-smoothness of the minimal eigenvalue. The small scale eigenvalue problem is obtained by projection of the large scale problem. Our main contribution is that for constructing the subspace we use multiple eigenvectors as well as derivatives of eigenvectors.We provide theoretical results and document numerical experiments regarding the beneficial effect of adding multiple eigenvectors and derivatives.