Skip to Main content Skip to Navigation
Journal articles

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.
Document type :
Journal articles
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-03165024
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

Links full text

Identifiers

Citation

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⟩

Share

Metrics

Record views

85