Structured preconditioners for nonsingular matrices of block two-by-two structures

Bai, Zhong‐Zhi

doi:10.1090/s0025-5718-05-01801-6

Cited by 236 publications

(129 citation statements)

References 44 publications

Supporting

Mentioning

129

Contrasting

Order By: Relevance

“…For all cases with uniform grids, we can see that both DS and RDF preconditioned GMRES are essentially independent of h. Table 5 compare favorably with those obtained with DS (cf. Table 5 in [7]), especially for small m. (4) An important property of the RDF preconditioner is that both the optimal a and the performance of the preconditioner remain virtually unchanged throughout the solution of the Navier-Stokes equation by Picard iteration. For the Q2-Q1 discretization of the lid driven cavity problem on a 128 Â 128 grid, this phenomenon is illustrated in Table 10, which displays the optimal value of a and the number of linear iterations required at each of the first five Picard steps needed to solve the Navier-Stokes equations with m = 0.1, 0.01 and m = 0.001.…”

Section: The Leaky Lid Driven Cavity Problem Discretized By Q2-q1 Finmentioning

confidence: 99%

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

Benzi

Niu

et al. 2011

Journal of Computational Physics

102

View full text Add to dashboard Cite

a b s t r a c tIn this paper we introduce a Relaxed Dimensional Factorization (RDF) preconditioner for saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related Dimensional Splitting (DS) preconditioner recently introduced by Benzi and Guo [7]. Numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the RDF preconditioner with respect to both mesh size and viscosity.

show abstract

Section: The Leaky Lid Driven Cavity Problem Discretized By Q2-q1 Finmentioning

confidence: 99%

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

Benzi

Niu

et al. 2011

Journal of Computational Physics

102

View full text Add to dashboard Cite

show abstract

“…. , ω 2 q ) ∈ R q×q is a nonnegative diagonal matrix, G = (g ij ) ∈ R q×q , and ⊗ denotes the Kronecker product; we refer the reader to [50] for details. In our computations we take θ = 1 and g ij = 1 (i+j) 2 .…”

Section: Examples and Numerical Experimentsmentioning

confidence: 99%

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

Huang

Carpentieri

2016

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

In the present paper, we introduce a new extension of the conjugate residual (CR) for solving non-Hermitian linear systems with the aim of developing an alternative basic solver to the established biconjugate gradient (BiCG), biconjugate residual (BiCR) and biconjugate A-orthogonal residual (BiCOR) methods. The proposed Krylov subspace method, referred to as the BiCGCR2 method, is based on short-term vector recurrences and is mathematically equivalent to both BiCR and BiCOR. We demonstrate by extensive numerical experiments that the proposed iterative solver has often better convergence performance than BiCG, BiCR and BiCOR. Hence, it may be exploited for the development of new variants of non-optimal Krylov subspace methods.

show abstract

“…For example, Uzawa-like methods ( [8,10,12]), SOR-like methods ( [7,14]), RPCG methods ( [6,9]), HSS-like methods ( [2][3][4][5]) and so on. We refer to [2] for algebraic properties for saddle point problem (1.7). In this paper, we will focus on the systems that arise in the context of PDE-constrained optimization.…”

Section: Introductionmentioning

confidence: 99%

Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems

Zhong¹

2013

JCM

View full text Add to dashboard Cite

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained optimization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are derived. Numerical implementations show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.Mathematics subject classification: 65F10, 65F50, 65F08, 65F22, 65F35, 65N22.

show abstract

Structured preconditioners for nonsingular matrices of block two-by-two structures

Cited by 236 publications

References 44 publications

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

A Relaxed Dimensional Factorization preconditioner for the incompressible Navier–Stokes equations

BiCGCR2: A new extension of conjugate residual method for solving non-Hermitian linear systems

Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems

Contact Info

Product

Resources

About