Model Order Reduction for Parametric Hermitian Eigenvalue Problems: Local Acceleration with Taylor-Reduced Basis Method

TL;DR

This paper introduces the Taylor-reduced basis method (Taylor-RBM) for efficiently approximating eigenspaces of large parametric Hermitian matrices, leveraging derivatives of spectral projectors for local model reduction.

Contribution

The paper provides a new local model order reduction technique, Taylor-RBM, with error analysis and an efficient basis assembly procedure for parametric eigenvalue problems.

Findings

Taylor-RBM effectively approximates eigenspaces with derivative information.

Error analysis justifies the accuracy of the Taylor-RBM approach.

Comparison shows spectral approximation via Taylor-RBM aligns with multivariate perturbation theory.

Abstract

This paper is concerned with the Taylor-reduced basis method (Taylor-RBM) for the efficient approximation of eigenspaces of large scale parametric Hermitian matrices. The Taylor-RBM is a local model order reduction method, which constructs an approximation space by capturing derivatives information of the spectral projector at a reference point in the parameter domain. We perform a concise error analysis to justify the Taylor-RBM for eigenvalue problems, and we present a computationally efficient procedure to assemble the Taylor-reduced basis space. Since this method is tightly connected to the classical multivariate analytic perturbation theory, we also provide a detailed analysis of the spectral approximation using the truncated power series of the eigenprojector, and compare this with the approximation obtained from the Taylor-RBM.