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Conference Papers Year : 2013

Solving large problems in linear algebra


Many numerical simulations end up on a problem of linear algebra involving an operator which is expressed after discretization by a very large sparse matrix. Typically, the problems include linear system solving, computation of eigenvalues and corresponding eigenvectors, and application of a function of the matrix on a given vector. To solve such problems, the methods based on Krylov subspaces have the advantage of not requiring a transformation of the matrix since they only use the matrix as an operator, i.e. through the multiplication of the matrix by a vector. The classical procedure to run these methods is the Arnoldi process which iteratively builds an orthonormal basis of the Krylov subspace. Unfortunately, this procedure has a limited potential for parallelism. To get rid of the bottleneck of the Gram-Schmidt procedure which is the heart of the Arnoldi process, non-orthonormal bases of Krylov subspaces are considered. The difficulty is then to avoid construction of too ill-conditioned bases. In this talk, we propose two types of three-term recurrences to generate such bases. In the last part and as an illustration, we present GPREMS, a parallel GMRES method preconditioned by a Multiplicative block-Schwarz iteration.
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hal-00903742 , version 1 (21-11-2013)


  • HAL Id : hal-00903742 , version 1


Bernard Philippe. Solving large problems in linear algebra. Fourth Annual meeting of the "Lebanese Society for the Mathematical Sciences" LSMS-2013, 2013, Beirut, Lebanon. ⟨hal-00903742⟩
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