Spectral discretization errors in filtered subspace iteration
Provides theoretical error bounds for a known numerical method, but the analysis is incremental and domain-specific to spectral approximation.
This paper analyzes how discretization errors in filtered subspace iteration affect eigenvalue approximations for selfadjoint operators, providing theoretical bounds on the Hausdorff distance between computed and exact eigenvalue clusters. Numerical experiments confirm the sharpness of the estimates.
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.