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

Abstract

Summary 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. Copyright © 2016 John Wiley & Sons, Ltd.

Quasi-Newton approaches to interior point methods for quadratic problems

“…Based on the preliminary results, the default implementation of Algorithm 2, denoted qnHOPDM from now on, uses update U2 for solving (16) the step, strategy (ii) to improve quasi-Newton directions and Criteria 1 and 3 to decide when to use quasi-Newton at step 3. By default, HOPDM uses multiple centrality correctors, which were shown to improve convergence of the algorithm [7].…”

“…, , so it is possible to replace H k− by different matrices without updating the whole structure. This is suitable to be applied in a limited-memory scheme [16]. Third, the computation of H k v can be efficiently implemented in a scheme similar to the BFGS update described in [12], as we show in Algorithm 1.…”

BFGS‐like updates of constraint preconditioners for sequences of KKT linear systems in quadratic programming

“…Theorem 5. Let E upd , G, andÊ be the matrices in (27), (6), and (17), respectively. Then, any eigenvalue of E u d −1 G satisfies the following:…”

On preconditioner updates for sequences of saddle-point linear systems

“…These sequences appear, e.g., in interior-point methods for quadratic programming and in Lagrangian approaches for the solution of PDE problems, with applications to optimal control, elasticity, polycrystalline aggregates, etc. [1][2][3][4][5].…”

“…where F i and W i are suitable symmetric positive definite matrices and κ is a scalar [6,7], and to sparse approximate factorizations of M i . This updating technique is inspired by limited-memory quasi-Newton methods and the resulting preconditioners are called limited-memory preconditioners [2,4,10,11]. The reader is referred to [1,6,7] for details on the blocks of P c i , P d i and P t i and the spectral properties of the preconditioned matrices.…”