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Spectral two-level preconditioners for sequences of linear systems

Martin, Emeric. Spectral two-level preconditioners for sequences of linear systems. PhD, Institut National Polytechnique de Toulouse, 2005

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Official URL: http://ethesis.inp-toulouse.fr/archive/00000165/

Abstract

Many numerical simulations in scientific and engineering applications require the solution of a set of large linear systems involving the same coefficient matrix but different right-hand sides. Efficient methods for tackling this problem attempt to benefit from the previously solved right-hand sides for the solution of the next ones. This goal can be achieved either by recycling Krylov subspaces or by building preconditioner updates based on near invariant subspace information. In this thesis, we focus our attention on this last approach that attempts to improve a selected preconditioner. In the first part, we consider only one update of the preconditioner for all the systems. This update consists of a spectral low-rank correction that shifts by one the smallest eigenvalues in magnitude of the matrix of the original preconditioned system. We perform experiments in the context of the GMRES method preconditioned by an approximate inverse preconditioner. The spectral information is computed by an eigensolver in a preprocessing phase. In the second part, we consider an update of the preconditioner between each system. An incremental spectral correction of the preconditioner is proposed. We perform experiments using the GMRES-DR method, thanks to its efficiency as a linear solver and its ability to recover reliable approximations of the desired eigenpairs at run time. Suitable strategies are investigated for selecting reliable eigenpairs. The efficiency of the proposed approaches is in particular assessed for the solution of large and challenging problems in electromagnetic applications. For this purpose, they have been implemented in a parallel industrial code developed by EADS-CCR.

Item Type:PhD Thesis
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Institution: Université de Toulouse > Institut National Polytechnique de Toulouse - INPT
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Research Director:
Giraud, Luc
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Deposited On:21 Nov 2012 13:45

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