On augmentation block triangular preconditioners for regularized saddle point problems

Shen, Shu-Qian; Bao, Wendi; Jian, Ling

doi:10.1016/j.camwa.2015.02.007

Cited by 8 publications

(4 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed the augmented Lagrangian preconditioner has been applied widely in optimization: one of the first such applications was to primal-dual interior point methods for linear and quadratic programming problems [288], see also [285,289] for other applications in optimization. The augmented Lagrangian preconditioner idea has been extended to nonsymmetric saddle point systems in [143,290], and has also found applicability to regularized systems where the (2, 2)-block of the saddle point system is nonzero [289,291]. It is also possible to construct block triangular analogues of (10) for saddle point systems [9,143,288,290,291].…”

Section: Augmented Lagrangian Preconditioningmentioning

confidence: 99%

“…The augmented Lagrangian preconditioner idea has been extended to nonsymmetric saddle point systems in [143,290], and has also found applicability to regularized systems where the (2, 2)-block of the saddle point system is nonzero [289,291]. It is also possible to construct block triangular analogues of (10) for saddle point systems [9,143,288,290,291].…”

Section: Augmented Lagrangian Preconditioningmentioning

confidence: 99%

See 1 more Smart Citation

Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

View full text Add to dashboard Cite

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.

show abstract

Section: Augmented Lagrangian Preconditioningmentioning

confidence: 99%

Section: Augmented Lagrangian Preconditioningmentioning

confidence: 99%

Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

View full text Add to dashboard Cite

show abstract

“…It is easy to obtain that this saddle point problem is equivalent to the following symmetric positive definite systems

\begin{matrix} aligned \\ right (D + \frac{1}{μ} K^{T} K) p & left = \frac{1}{μ} K^{T} f - g, & right \\ right u & left = \frac{1}{μ} MathClass-open(f - Kp MathClass-close) . \end{matrix}

However, the reduced system may be rather ill‐conditioned, so it can not be solved easily because D is not a scalar matrix in general; see some related arguments in . Some preconditioned iterative methods have been proposed to solve ; see for example, .…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…As constructed in ,Section 3.1, we choose μ = 0.001, D = diag( d 11 , ⋯ , d nn ) with the spectral condition number 10 6 .

K MathClass-rel= (k_{ij}) MathClass-rel\in {double-struckR}^{n MathClass-bin\times n}

is an ill‐conditioned Toeplitz matrix with

k_{ij} MathClass-rel= \frac{1}{2 \sqrt{2 π}} e^{MathClass-bin- MathClass-rel| i MathClass-bin- j {MathClass-rel|}^{2} MathClass-bin∕ 8} MathClass-punc.

The vectors f and g are taken as f = (1,2, … , n ) T and g = 0, respectively.…”

Section: Numerical Experimentsmentioning

confidence: 99%

On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

Shen

Jian

Bao

et al. 2013

Numer. Linear Algebra Appl.

Self Cite

View full text Add to dashboard Cite

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.

show abstract