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

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 using the ranges of scaled derivatives of the spectral projector at a reference point in the parameter domain. For a prescribed order $n$, under local ${\mathrm{C}}^{n+1}$-regularity, we prove local convergence of orders $n+1$ and $2n+2$ for the eigenspace and the Ritz-value sum, respectively, and present a computationally efficient procedure for assembling the Taylor reduced basis space. Since this method is tightly connected to multivariate perturbation theory, we also analyze a projector approximation obtained from the finite Taylor polynomial of the eigenprojector and compare it with the Taylor-RBM.

arXiv commenttheoretical framework substantially revised. now 31 pages, 5 figures