Minimum residual methods for augmented systems

Fischer, Bernd; Ramage, Alison; Silvester, David J.; Wathen, Andrew J.

doi:10.1007/bf02510258

Cited by 167 publications

(104 citation statements)

References 15 publications

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“…Later we will also require that A be positive definite and C be positive semidefinite. The spectral properties of the problem above have been studied in [12] for A = ηI n (η > 0) and C = O and in [28] for C = O. The results given here are more general and complete; for instance, our conditions for the reality of the spectrum do not appear to have been given before.…”

Section: Analysis Of the Eigenvaluesmentioning

confidence: 97%

“…When A = ηI , the matrix G given above reduces to the one given in [12]. Note that the condition λ min (A) > 4λ max (B A −1 B T ) cannot be relaxed, in general.…”

Section: Working With a Definite Inner Productmentioning

confidence: 99%

“…Preconditioners of the form P ± with A = 1 η 2 A (η > 0) and S ≈ S (a symmetric and positive definite approximation to S) have been studied in [12] for the case C = O. If L denotes the Cholesky factor of A and L s the Cholesky factor of S, then symmetrically applying the preconditioner P ± and dividing through by η results in preconditioned matrices of the form…”

Section: Block Diagonal Preconditioningmentioning

confidence: 99%

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On the eigenvalues of a class of saddle point matrices

2006

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We study spectral properties of a class of block 2 × 2 matrices that arise in the solution of saddle point problems. These matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system. We give conditions for having a (positive) real spectrum and for ensuring diagonalizability of the matrix. In particular, we show that these properties hold for the discrete Stokes operator, and we discuss the implications of our characterization for augmented Lagrangian formulations, for Krylov subspace solvers and for certain types of preconditioners.Mathematics Subject Classifications: Primary 65F10 · 65N22 · 65F50 · Secondary 76D07

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Section: Analysis Of the Eigenvaluesmentioning

confidence: 97%

“…When A = ηI , the matrix G given above reduces to the one given in [12]. Note that the condition λ min (A) > 4λ max (B A −1 B T ) cannot be relaxed, in general.…”

Section: Working With a Definite Inner Productmentioning

confidence: 99%

Section: Block Diagonal Preconditioningmentioning

confidence: 99%

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On the eigenvalues of a class of saddle point matrices

2006

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“…Remark 2.1. As discussed in [7] there is a close relationship between the convergence of the Krylov subspace method minres applied to the preconditioned system in Proposition 2.2, and that of the conjugate gradient method applied to the Schur complement system BM −1 B t , when it is preconditioned with the operator S * = K * M −1 I K * . Based on the bounds in Proposition 2.2, both the preconditioned minres and the preconditioned CG method will converge in at most O(h −1/2 ) iterations (minres will typically take twice as many iterations to reach the same tolerance, see, [7,Sect.…”

Section: A Conventional Block Preconditionermentioning

confidence: 78%

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Silvester

Mihajlović

2004

BIT Numerical Mathematics

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Abstract.We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any "black-box" multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems. (2000): 65F10, 65N12, 65N22, 65N55. AMS subject classification

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“…where A ∈ R m×m is symmetric and positive definite (SPD), and B ∈ R m×n , appears in many different applications such as the finite-element method for solving the NavierStokes equation (Elman & Golub, 1994;Elman & Silvester, 1996;Elman et al, 1997;Fischer et al, 1998). When A and B are large and sparse, iterative methods for solving system (1.1) are effective because of storage requirements and preservation of sparsity.…”

Section: Introductionmentioning

confidence: 99%

A note on an SOR-like method for augmented systems

Li¹,

Zheng²,

Evans³

et al. 2003

IMA Journal of Numerical Analysis

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Golub et al. (2001, BIT, 41, 71-85) gave a generalized successive over-relaxation method for the augmented systems. In this paper, the connection between the SOR-like method and the preconditioned conjugate gradient (PCG) method for the augmented systems is investigated. It is shown that the PCG method is at least as accurate (fast) as the SOR-like method. Numerical examples demonstrate that the PCG method is much faster than the SOR-like method.

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Minimum residual methods for augmented systems

Cited by 167 publications

References 15 publications

On the eigenvalues of a class of saddle point matrices

On the eigenvalues of a class of saddle point matrices

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

A note on an SOR-like method for augmented systems

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