A note on the positive stable block triangular preconditioner for generalized saddle point problems

Cao, Yang; Jiang, Mei-Qun; Zheng, Ying-Long

doi:10.1016/j.amc.2012.05.003

Cited by 10 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It should be noted that the inexact preconditioners are very important in practical computation since their iteration costs are much cheaper than exact preconditioners and the necessary iteration steps do not increase significantly. For the generalized saddle point problem (1), where the submatrix A is Hermitian positive definite, some inexact block triangular preconditioners are proposed in Simoncini, 18 Cao, 19 and Jiang et al 20 and improved eigenvalue bounds of the preconditioned matrices are studied in Cao et al 21 For a general block two-by-two non-Hermitian matrix, some inexact block triangular preconditioners are studied in Bai and Ng, 22 Peng and Li, 23 and Li et al 24 For standard saddle point problem where the (1,1) block matrix A is Hermitian positive definite and the (2,2) block matrix C is a zero matrix, an inexact constraint preconditioner is proposed in Bergamaschi et al 25 and spectral analysis of the inexact constraint preconditioned matrix is studied in Sesana and Simoncini. 26 In this article, we propose a class of inexact modified positive-definite and skew-Hermitian splitting (IMPSS) preconditioners for the generalized saddle point problem (1).…”

Section: Introductionmentioning

confidence: 99%

Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

Shen

Shi

2018

Advances in Mechanical Engineering

View full text Add to dashboard Cite

In this article, to better implement the modified positive-definite and skew-Hermitian splitting preconditioners studied recently (Numer. Algor., 72 (2016) 243-258) for generalized saddle point problems, a class of inexact modified positivedefinite and skew-Hermitian splitting preconditioners is proposed with improved computing efficiency. Some spectral properties, including the eigenvalue distribution, the eigenvector distribution, and an upper bound of the degree of the minimal polynomial of the inexact modified positive-definite and skew-Hermitian splitting preconditioned matrices are studied. In addition, a theoretical optimal inexact modified positive-definite and skew-Hermitian splitting preconditioner is obtained. Numerical experiments arising from a model steady incompressible Navier-Stokes problem are used to validate the theoretical results and illustrate the effectiveness of this new class of proposed preconditioners.

show abstract

Section: Introductionmentioning

confidence: 99%

Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

Shen

Shi

2018

Advances in Mechanical Engineering

View full text Add to dashboard Cite

show abstract

“…. 20) implies that the vectors a (k) (k = 1, 2, 3) are zero vectors. Recalling that in (3.20) the first matrix arises from the case θ k = 1 (k = 1, · · · , n), the second matrix from the case θ k = 1 (k = 1, · · · , i), and the last matrix from the case…”

Section: Analysis Of the Rhss Iterationmentioning

confidence: 99%

“…In the past few years, much work also has been devoted to developing efficient preconditioners for saddle point problems. A variety of preconditioners have been proposed and studied in many papers, such as block diagonal and block triangular preconditioners [7,19,20,24], constraint preconditioners [8,25], HSS-based preconditioners [3,6,12,13,30], dimensional split preconditioners [15,16], H−matrix preconditioners [17] and so on. In [14], Benzi, Golub and Liesen discussed a selection of numerical methods and useful preconditioners for saddle point problems.…”

mentioning

confidence: 99%

A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization

Cao¹,

Yao

Jiang

et al. 2013

JCM

Self Cite

View full text Add to dashboard Cite

In this paper, a relaxed Hermitian and skew-Hermitian splitting (RHSS) preconditioner is proposed for saddle point problems from the element-free Galerkin (EFG) discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS preconditioner to accelerate the convergence of some Krylov subspace methods (like GMRES) is also studied. Theoretical analyses show that the eigenvalues of the RHSS preconditioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.Mathematics subject classification: 65F10.

show abstract

A note on parameterized block triangular preconditioners for generalized saddle point problems

2013

Applied Mathematics and Computation

View full text Add to dashboard Cite

A note on the positive stable block triangular preconditioner for generalized saddle point problems

Cited by 10 publications

References 20 publications

Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

Inexact modified positive-definite and skew-Hermitian splitting preconditioners for generalized saddle point problems

A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization

A note on parameterized block triangular preconditioners for generalized saddle point problems

Contact Info

Product

Resources

About