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Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

Abstract : This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right-hand sides. These preconditioners based on limited memory quasi-Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first-level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half-plane, respectively. Third, we focus on theoretical properties of the Ritz-LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large-scale symmetric saddle-point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications.
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Contributor : Françoise Grélaud Connect in order to contact the contributor
Submitted on : Wednesday, March 10, 2021 - 12:50:58 PM
Last modification on : Wednesday, November 3, 2021 - 7:17:19 AM

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Serge Gratton, Sylvain Mercier, Nicolas Tardieu, Xavier Vasseur. Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics. Numerical Linear Algebra with Applications, Wiley, 2016, 23 (5), pp.865--887. ⟨10.1002/nla.2058⟩. ⟨hal-03165024⟩



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