Spectral discretization errors in filtered subspace iteration

Abstract

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters are obtained in terms of the discretization parameters within an abstract framework. A realization of the proposed approach for a model second-order elliptic operator using a standard finite element discretization of the resolvent is described. Some numerical experiments are conducted to gauge the sharpness of the theoretical estimates. arXiv:1709.06694v4 [math.NA]

Analysis of FEAST Spectral Approximations Using the DPG Discretization

“…In this section, we summarize the abstract framework of [9] for analyzing spectral discretization errors of the FEAST algorithm when applied to general selfadjoint operators. Accordingly, in this section, A is not restricted to the reaction-diffusion operator mentioned in Section 1.…”

“…When used as an algorithm for matrix eigenvalues, discretization errors are irrelevant, which explains the dearth of studies on discretization errors within such algorithms. However, in this paper, like in [9,14], we are interested in the eigenvalues of a partial differential operator on an infinite-dimensional space. In these cases, practical computations can proceed only after discretizing the resolvent of the partial differential operator by some numerical strategy, such as the finite element method.…”

“…According to [9], bounds on spectral errors can be obtained from bounds on the approximation of the resolvent z Þ Ñ Rpzq. This function maps complex numbers to bounded operators.…”

“…This function maps complex numbers to bounded operators. In [9], certain finite-rank computable approximations to Rpzq, denoted by R h pzq, were considered and certain abstract sufficient conditions were laid out for bounding the resulting spectral errors. (Here h represents some discretization parameter like the mesh size.)…”

“…The outline of the paper is as follows. In Section 2 we present the abstract theory from [9] pertaining to FEAST iterations using discretized resolvents of unbounded operators. In Section 3 we derive new estimates for discretizations of a resolvent by the DPG method.…”