On the eigenvalues of a class of saddle point matrices

Benzi, Michele; Simoncini, Valeria

doi:10.1007/s00211-006-0679-9

Cited by 110 publications

(118 citation statements)

References 29 publications

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“…There are quite many works developing eigenvalue analyzes for these types of preconditioner; see [5,10] for block diagonal preconditioners, [7,32] for block triangular preconditioners, [8,15,36] for inexact Uzawa preconditioners, and [2,3,36] for block approximate factorization preconditioners -to mention just a few; we refer to [4] for many more references and historical remarks.…”

Section: 2)mentioning

confidence: 99%

A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

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We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these latter based on symmetric block Gauss-Seidel type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow to localize accurately both real and non real eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

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Section: 2)mentioning

confidence: 99%

A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

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“…A CP variant where D instead of H is approximated has been analysed in [28]; the resulting preconditioner is generally more expensive to apply, unless optimization problems with a particular structure are considered. The spectral properties of the preconditioned matrix P −1 K have been investigated in many papers [6,8,23,28,31,54,57]. The following result holds for the case D = 0 [54].…”

Section: Preconditioning the Kkt Systemmentioning

confidence: 88%

“…In [57] an inexact CP for the case D = 0 is analysed, which is obtained by replacing the zero (2,2)-block with a diagonal positive definite matrix D 1 and by applying an incomplete Cholesky factorization of the Schur complement M + J T D −1 1 J. In [6], an inexact CP is obtained without changing a priori the block D = 0 in the preconditioner, but using an incomplete Cholesky factorization directly on the Schur complement JM −1 J T . Spectral bounds for the preconditioned matrices are provided in both cases.…”

Section: Preconditioning the Kkt Systemmentioning

confidence: 99%

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

2008

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The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. In this paper we discuss the mutual impact of linear algebra and optimization, focusing on interior point methods and on the iterative solution of the KKT system. Three critical issues are addressed: preconditioning, termination control for the inner iterations, and inertia control.

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“…The minus sign that now appears in the second block row is not essential, but it will be used henceforth. Note that for β ≤ 0 or β > 0 and sufficiently small, the spectrum of the coefficient matrix of (3) is entirely contained in the right half-plane; see, e.g., [3,6]. For the case β = 0, it was shown in [5] that a block triangular preconditioner of the type…”

Section: The Augmented Lagrangian Preconditionermentioning

confidence: 99%

Block preconditioning for saddle point systems with indefinite (1, 1) block

Benzi

Liu

2007

International Journal of Computer Mathematics

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We investigate the solution of linear systems of saddle point type with an indefinite (1, 1) block by preconditioned iterative methods. Our main focus is on block matrices arising from eigenvalue problems in incompressible fluid dynamics. A block triangular preconditioner based on an augmented Lagrangian formulation is shown to result in fast convergence of the GMRES iteration for a wide range of problem and algorithm parameters. Some theoretical estimates for the eigenvalues of the preconditioned matrices are given. Inexact variants of the preconditioner are also considered.

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On the eigenvalues of a class of saddle point matrices

Cited by 110 publications

References 29 publications

A New Analysis of Block Preconditioners for Saddle Point Problems

A New Analysis of Block Preconditioners for Saddle Point Problems

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

Block preconditioning for saddle point systems with indefinite (1, 1) block

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