Fast Iterative Solution of Stabilised Stokes Systems Part II: Using General Block Preconditioners

Silvester, David J.; Wathen, Andrew J.

doi:10.1137/0731070

Cited by 361 publications

(332 citation statements)

References 14 publications

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“…In the case of the generalized Stokes problem, spectrally equivalent approximations A and S are known that lead to asymptotically optimal preconditioners, i.e., preconditioners that result in rates of convergence independent of the discretization parameter h; see, e.g., [9,10,21,29]. One can also consider indefinite block diagonal preconditioners of the form…”

Section: Block Diagonal Preconditioningmentioning

confidence: 99%

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: Block Diagonal Preconditioningmentioning

confidence: 99%

On the eigenvalues of a class of saddle point matrices

2006

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“…Hence, from the definition of H δ we have ( Taking into account (2.23), the bound (2.29) is sharper than that provided in [23]. Also, our results are a generalization of those given in [21].…”

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confidence: 55%

Regularization and preconditioning of KKT systems arising in nonnegative least‐squares problems

Bellavia

Gondzio

Morini

2008

Numerical Linear Algebra App

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Abstract. A regularized Newton-like method for solving nonnegative least-squares problems is proposed and analysed in this paper. A preconditioner for KKT systems arising in the method is introduced and spectral properties of the preconditioned matrix are analysed. A bound on the condition number of the preconditioned matrix is provided. The bound does not depend on the interior-point scaling matrix. Preliminary computational results confirm the effectiveness of the preconditioner and fast convergence of the iterative method established by the analysis performed in this paper.

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“…Following Lemma 2.1 in Rusten and Winther [33] (see also Lemma 2.1 in Silvester and Wathen [35]) we provide bounds for these eigenvalues.…”

Section: Assumptionsmentioning

confidence: 99%

Matrix-free interior point method

Gondzio

2010

Comput Optim Appl

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In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.

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Fast Iterative Solution of Stabilised Stokes Systems Part II: Using General Block Preconditioners

Cited by 361 publications

References 14 publications

On the eigenvalues of a class of saddle point matrices

On the eigenvalues of a class of saddle point matrices

Regularization and preconditioning of KKT systems arising in nonnegative least‐squares problems

Matrix-free interior point method

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