A Spectral Analysis of Subspace Enchanced Preconditioners

TL;DR

This paper analyzes how subspace-enhanced preconditioners affect the spectrum of linear systems, revealing the impact of coarse space alignment and inverse projection accuracy on convergence.

Contribution

It provides a spectral analysis of deflation, coarse correction, and adapted deflation preconditioners, highlighting factors influencing their effectiveness.

Findings

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

Inverse projection matrix accuracy influences the preconditioned system spectrum.

Numerical experiments support the theoretical spectral 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 correction and adapted deflation preconditioners. Our analysis shows that the spectrum of the preconditioned system is highly impacted by the angle between the coarse 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 the accuracy of the inverse of projection matrix also impacts the spectrum of the preconditioned system. Numerical experiments emphasized the theoretical analysis.