On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

Krzyżanowski, Piotr

doi:10.1002/nla.717

Cited by 13 publications

(12 citation statements)

References 36 publications

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“…Remark In the case in which the solution of the stationary problem is used inside a parametrized stent topology optimization loop, it is of course of interest to solve the problem numerically as efficiently as possible. In this context, it might be of interest to use iterative solution methods, particularly those that can provably converge in few iterations, which is frequently the case for quasidefinite matrices . Note that the matrices which we produce are degenerate quasidefinite matrices in that the leading diagonal block is no longer positive definite but positive semidefinite.…”

Section: Piecewise Polynomial Discretizationmentioning

confidence: 99%

“…In this context, it might be of interest to use iterative solution methods, particularly those that can provably converge in few iterations, which is frequently the case for quasidefinite matrices. [24][25][26]29,30 Note that the matrices which we produce are degenerate quasidefinite matrices in that the leading diagonal block is no longer positive definite but positive semidefinite. However, the displacement vector u h that we compute is always unique and it is for the costate p h where we lack uniqueness.…”

Section: Figurementioning

confidence: 99%

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Direct solution method for the equilibrium problem for elastic stents

Grubišić

Tambača

2019

Numerical Linear Algebra App

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Summary We are interested in numerical methods for approximating vector‐valued functions on a metric graph. As a model problem, we formulate and analyze a numerical method for the solution of the stationary problem for the one‐dimensional elastic stent model. The approximation is built using the mixed finite element method. The discretization matrix is a symmetric saddle‐point matrix, and we discuss sparse direct methods for the fast and robust solution of the associated equilibrium system. The convergence of the numerical method is proven and the error estimate is obtained. Numerical examples confirm the theoretical estimates.

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Section: Piecewise Polynomial Discretizationmentioning

confidence: 99%

Section: Figurementioning

confidence: 99%

Direct solution method for the equilibrium problem for elastic stents

Grubišić

Tambača

2019

Numerical Linear Algebra App

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“…T factorization (1.9) has proved to be an effective way of constructing preconditioners for saddle point problems-see, e.g., [7,16,27,30,31,43,51]. The key component of such methods is an approximation of the matrix S, which is often dense.…”

Section: Null-space Preconditioners Taking Incomplete Versions Of Thmentioning

confidence: 99%

Null-Space Preconditioners for Saddle Point Systems

Pestana¹,

Rees²

2016

SIAM J. Matrix Anal. & Appl.

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Abstract. The null-space method is a technique that has been used for many years to reduce a saddle point system to a smaller, easier to solve, symmetric positive definite system. This method can be understood as a block factorization of the system. Here we explore the use of preconditioners based on incomplete versions of a particular null-space factorization and compare their performance with the equivalent Schur complement based preconditioners. We also describe how to apply the nonsymmetric preconditioners proposed using the conjugate gradient method (CG) with a nonstandard inner product. This requires an exact solve with the (1,1) block, and the resulting algorithm is applicable in other cases where Bramble-Pasciak CG is used. We verify the efficiency of the newly proposed preconditioners on a number of test cases from a range of applications.

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“…More examples of CG methods in nonstandard inner products have been extensively studied for in the last years; see, for example, and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

Shen

Jian

Bao

et al. 2013

Numer. Linear Algebra Appl.

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In this paper, we derive bounds for the complex eigenvalues of a nonsymmetric saddle point matrix with a symmetric positive semidefinite .2, 2/ block, that extend the corresponding previous bounds obtained by Bergamaschi. For the nonsymmetric saddle point problem, we propose a block diagonal preconditioner for the conjugate gradient method in a nonstandard inner product. Numerical experiments are also included to test the performance of the presented preconditioner.A large variety of applications and numerical solution methods of (1) or (2) have been comprehensively reviewed by Benzi, Golub, and Liesen [1]. Recently, the eigenvalue distribution of the nonsymmetric saddle point matrix A has been deeply studied. On one hand, eigenvalue bounds for A can be used to analyze the spectral properties of preconditioners such as symmetric indefinite preconditioners, inexact constraint preconditioners and primal-based penalty preconditioners for (1); see, for example, [3][4][5][6]. On the other hand, eigenvalue estimates for A can give theoretical basis for the CG method for (2) in a nonstandard inner product; see, for example, [3,[6][7][8]. This paper aims to provide further bounds for the complex eigenvalues of A that extend the corresponding ones in [4].An efficient implementation of the CG method for (2) has been established by Liesen and Parlett [7]. The authors pointed out that preconditioning should be studied in practical applications,(1) If Im( )=0, then minf% 1 , & 1 g 6 6 maxf% n , m g.

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On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

Cited by 13 publications

References 36 publications

Direct solution method for the equilibrium problem for elastic stents

Direct solution method for the equilibrium problem for elastic stents

Null-Space Preconditioners for Saddle Point Systems

On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

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