Inexact subspace iteration to accelerate the solution of linear systems with multiple right-hand sides
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resumo
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.
