Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix

TL;DR

This paper introduces a greedy projection method for uniformly approximating the smallest eigenvalues and singular values of large parameter-dependent matrices, with proven convergence and practical efficiency.

Contribution

It develops a novel greedy iterative approach for uniform eigenvalue approximation that maximizes surrogate errors over the entire parameter space, unlike classical methods.

Findings

The method converges uniformly to the true eigenvalue function.

It significantly reduces matrix size while maintaining high accuracy.

Numerical examples demonstrate effectiveness on discretized parametric PDEs.

Abstract

We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of a greedy strategy; at each iteration the parameter where a surrogate error is maximal is computed and the eigenvectors associated with the smallest eigenvalues at the maximizing parameter value are added to the subspace. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We formally prove that the projected eigenvalue function converges to the actual eigenvalue function uniformly. In the second part, we focus on the uniform approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian. The proposed frameworks on numerical examples, including those arising from discretizations of parametric PDEs, reduce the size of the large matrix-valued function drastically, while retaining a high accuracy over all permissible parameter values.