Inexact subspace iteration for the consecutive solution of linear systems with changing right-hand sides

Abstract

We propose a two-phase acceleration technique for the solution of Symmetric and Positive Definite linear systems with multiple right-hand sides. In the first phase we compute some partial spectral information related to the ill conditioned part of the given coefficient matrix and, in the second phase, we use this information to improve the convergence of the Conjugate Gradient algorithm. This approach is adequate for large scale problems, like the simulation of time dependent differential equations, where it is necessary to solve consecutively several linear systems with the same coefficient matrix (or with matrices that present very close spectral properties) but with changing right-hand sides. To compute the spectral information, in the first phase, we combine the block Conjugate Gradient algorithm with the Inexact Subspace Iteration to build a purely iterative algorithm, that we call BlockCGSI. We proceed to an inner-outer convergence analysis and we show that it is possible to determine when to stop the inner iteration in order to achieve the targeted invariance in the outer iteration. The spectral information is used in a second phase to remove the effect of the smallest eigenvalues in two different ways: either by building a Spectral Low Rank Update preconditioner, or by performing a deflation of the initial residual in order to remove part of the solution corresponding to the smallest eigenvalues.