A Study on Perturbation Analysis of Spectral Preconditioners

TL;DR

This paper analyzes how the spectrum of preconditioned systems in Krylov methods is affected by the choice of coarse grid spaces and the accuracy of inverse projections, impacting convergence rates.

Contribution

It provides a theoretical investigation into the spectral effects of deflation and coarse grid correction preconditioners, highlighting the influence of subspace angles and inverse accuracy.

Findings

Spectrum is highly affected by the angle between coarse grid space and eigenvector subspace.

Inverse matrix approximation accuracy impacts the spectrum of the preconditioned system.

Numerical experiments support the theoretical analysis.

Abstract

It is well-known that the convergence of Krylov subspace methods to solve linear system depends on the spectrum of the coefficient matrix, moreover, it is widely accepted that for both symmetric and unsymmetric systems Krylov subspace methods will converge fast if the spectrum of the coefficient matrix is clustered. In this paper we investigate the spectrum of the system preconditioned by the deflation, coarse grid correction and adapted deflation preconditioners. Our analysis shows that the spectrum of the preconditioned system is highly impacted by the angle between the coarse grid space for the construction of the three preconditioners and the subspace spanned by the eigenvectors associated with the small eigenvalues of the coefficient matrix. Furthermore, we prove that with a certain restriction the accuracy of the inverse of projection matrix also impacts the spectrum of the preconditioned system. Numerical experiments emphasized the theoretical analysis.