Inexact subspace iteration to accelerate the solution of linear systems with multiple right-hand sides - Balsa, Carlos (2014)
We analyze the convergence and propose some strategy to monitor an inexact subspace iteration type of algorithm called BlockCGSI. This algorithm is purely iterative and combines the block Conjugate Gradient (blockCG) algorithm with the Subspace Iteration. We proceed to an inner-outer convergence analyze and exploit the possibility of reducing the total amount of computational work by controlling the accuracy during the solution of linear systems at each inverse iteration. The proposed method can be adequate for large scale problems where we need to solve consecutively several linear systems with the same coefficient matrix (or with very close spectral properties) but with changing right-hand sides. The BlockCGSI algorithm can be used to compute some spectral information, which is then used to remove the effect of the smallest eigenvalues in two different ways: either by building a Spectral Low Rank Update (SLRU) preconditioner that basically adds the value 1 to these eigenvalues, or by performing a deflation of the initial residual in order to remove part of the solution corresponding to the smallest eigenvalues. Both techniques can reduce substantially the total number of iterations and computational work in each subsequent runs of the Conjugate Gradient algorithm.