Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems

Dollar, H. Sue; Gould, Nicholas I. M.; Schilders, W.H.A.; Wathen, Andrew J.

doi:10.1137/05063427x

Cited by 49 publications

(44 citation statements)

References 34 publications

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“…It is quite evident from (27), that the induced block factors L and D are different from the ones in [2,10,11,35], since they are deduced from A by using a different transformation operator T . Furthermore, in [2,10,11], suggestions for the blocks of L and D are provided such that LDL T approximates the block A by keeping the constraint matrix B intact, which is known as implicit factorization for preconditioners.…”

Section: Induced Block Factorizationmentioning

confidence: 99%

“…Furthermore, in [2,10,11], suggestions for the blocks of L and D are provided such that LDL T approximates the block A by keeping the constraint matrix B intact, which is known as implicit factorization for preconditioners. In contrary, the induced block factorization in (27) and the Schilders' factorization give the exact factorization of A.…”

Section: Induced Block Factorizationmentioning

confidence: 99%

“…For example, the Schilders' factorization [35] is a block 3 × 3 structured factorization with blocks of order m and n − m, applied to T TÅ T for a different T . Later in [10,11,13], it has been used as a basis for implicit factorization for constructing different families of constraint preconditioners for the saddle point matrices. We also produce such a 3 × 3 structured block factorization from…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparse block factorization of saddle point matrices

Lungten

Schilders

Maubach

2016

Linear Algebra and its Applications

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The factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices. A variant of Gaussian elimination equivalent to the Cholesky's factorization is suggested and implemented for factorizing the saddle point matrices block-wise with small blocks of order 1 and 2. The Gaussian elimination applied to these small blocks on block level also induces a block 3 × 3 structured factorization of which the blocks have special properties. We compare the new block factorization with the Schilders' factorization in terms of the sparsity of their factors and computational efficiency. The factorization can be used as a direct method, and also anticipate for preconditioning techniques.

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Section: Induced Block Factorizationmentioning

confidence: 99%

Section: Induced Block Factorizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Sparse block factorization of saddle point matrices

Lungten

Schilders

Maubach

2016

Linear Algebra and its Applications

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“…Our regularized augmented system equation (4.5) can be solved by the Projected Preconditioned Conjugate-Gradient (PPCG) method developed in [10,11]. PPCG provides the vectorq k , while the vectorp k is computed byp k = (C k ) −1 (S k )A Tq k .…”

Section: Iterative Linear Algebramentioning

confidence: 99%

“…We are not aware of any comparable result in the literature. Moreover, the proposed preconditioner allows us to use the short recurrence PPCG method [11] to solve the preconditioned linear system. Thus, we performed also the spectral analysis of the reduced preconditioned normal equation whose eigenvalues determine the convergence of the PPCG method.…”

mentioning

confidence: 99%

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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Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.

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Implicit-Factorization Preconditioning and Iterative Solvers for Regularized Saddle-Point Systems

Cited by 49 publications

References 34 publications

Sparse block factorization of saddle point matrices

Sparse block factorization of saddle point matrices

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

Preconditioners for Krylov subspace methods: An overview

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