Singular Saddle Point Problems Research Articles

A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

AbstractBased on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

The generalized modified shift-splitting preconditioners for nonsymmetric saddle point problems

For a nonsymmetric saddle point problem, the modified shift-splitting (MSS) preconditioner has been proposed by Zhou et al. By replacing the parameter α in (2,2)-block in the MSS preconditioner by another parameter β, a generalized MSS (GMSS) preconditioner is established in this paper, which results in a fixed point iteration called the GMSS iteration method. We provide the convergent and semi-convergent analysis of the GMSS iteration method, which show that this method is convergence and semi-convergence if the related parameters satisfy suitable restrictions. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. Finally, numerical examples show that the GMSS method is more feasibility and robustness than the MSS, Uzawa-HSS and PU-STS methods as a solver, and the GMSS preconditioner outperforms the GSOR, Uzawa-HSS, MSS and LMSS preconditioners for the GMRES method for solving both the nonsingular and the singular saddle point problems with nonsymmetric positive definite and symmetric dominant (1,1) parts.

The generalized Uzawa-SHSS method for non-Hermitian saddle-point problems

Recently, Li and Wu (2015) proposed the single-step Hermitian and skew-Hermitian splitting (SHSS) method for solving the non-Hermitian positive definite linear systems. Based on the single-step Hermitian and skew-Hermitian splitting of the (1,1) part of the saddle-point coefficient matrix, a new Uzawa-type method is proposed for solving a class of saddle-point problems with non-Hermitian positive definite (1,1) parts. Convergence (Semi-convergence) properties of this new method for nonsingular (singular) are derived under suitable conditions. Numerical examples are implemented to confirm the theoretical results and verify that this new method is more feasibility and robustness than the new HSS-like (NHSS-like), the Uzawa-HSS and the parameterized Uzawa-skew-Hermitian triangular splitting (PU-STS) methods for solving both the nonsingular and the singular saddle-point problems with non-Hermitian positive definite and Hermitian dominant (1,1) parts.

Eigenvalue bounds of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices

For singular nonsymmetric saddle-point problems, a shift-splitting preconditioner was studied in (Appl. Math. Comput. 269:947-955, 2015). To further show the efficiency of the shift-splitting preconditioner, we provide eigenvalue bounds for the nonzero eigenvalues of the shift-splitting preconditioned singular nonsymmetric saddle-point matrices. For real parts of the eigenvalues, the bound is provided by valid inequalities. For eigenvalues having nonzero imaginary parts, the bound is a combination of two inequalities proving their clustering in a confined region of the complex plane. Finally, two numerical examples are presented to verify the theoretical results.

On semi-convergence and inexact iteration of the GSS iteration method for nonsymmetric singular saddle point problems

In this paper, the generalized shift-splitting (GSS) iteration method is used to solve nonsymmetric singular saddle point systems with the positive real (1, 1)-block sub-matrix. By a careful analysis of the spectral properties of the corresponding iteration matrix, we investigate the semi-convergence property of the GSS iteration method, give the sharper bounds of the eigenvalues for the GSS iteration method and propose the inexact GSS iteration method, also. Numerical experiments are given to show the correctness of the theoretical analysis and the feasibility of the GSS preconditioner with appropriate parameters.

On semi-convergence of ULT iterative method for the singular saddle point problems

Zheng and Ma recently proposed an efficient upper and lower triangular (ULT) splitting iterative method for solving the large sparse nonsingular saddle point problems; see Zheng and Ma (2016). In this paper, we further prove the semi-convergence of this method when it is applied to solve the large sparse singular saddle point problems under suitable conditions. The characteristic of eigenvalues of the iteration matrix of the ULT method is analyzed. Also, the pseudo-optimal iteration parameters and the corresponding pseudo-optimal semi-convergence factor for some special cases of the ULT method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the ULT iterative method for solving singular saddle point problems.

Convergence behavior of generalized parameterized Uzawa method for singular saddle-point problems

In this paper, we will seek the least squares solution for singular saddle-point problems. The parameterized Uzawa (PU) method is further studied and a generalized PU (GPU) proper splitting is proposed. The convergence behavior of the corresponding GPU iteration is studied. It is proved that the GPU iteration method can converge to the best least squares solutions of the singular saddle-point problems. In addition, we prove that the GPU preconditioned GMRES for singular saddle-point problems will also determine the least squares solution at breakdown. The eigenvalue distributions of the GPU preconditioned matrix are derived. Numerical experiments are presented, which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the GPU iteration has better convergence behavior than the PU iteration, both as a solver and a preconditioner of GMRES.

Improved PPHSS iterative methods for solving nonsingular and singular saddle point problems

Based on the parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method (PPHSS), proposed by Li et al. (2014) for saddle point problems, an improvement on the PPHSS method (IPPHSS) is presented in this paper. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PPHSS iterative scheme, both the number of iterations and the consume time are decreased. We provide the convergence and semi-convergence analysis of the IPPHSS method, which show that this method is convergence and semi-convergence if the related parameters satisfy suitable restrictions. Furthermore, we discuss the spectral properties of the corresponding preconditioned matrix of the IPPHSS method. Finally, numerical examples show that the IPPHSS method is better than the PPHSS, PHSS and AHSS methods both as a solver and as a preconditioner for the GMRES method.

ACCELERATION OF ONE-PARAMETER RELAXATION METHODS FOR SINGULAR SADDLE POINT PROBLEMS

In this paper, we first introduce two one-parameter relaxation (OPR) iterative methods for solving singular saddle point problems whose semi-convergence rate can be accelerated by using scaled preconditioners. Next we present formulas for finding their optimal parameters which yield the best semi-convergence rate. Lastly, numerical experiments are provided to examine the efficiency of the OPR methods with scaled preconditioners by comparing their performance with the parameterized Uzawa method with optimal parameters.

A generalized modified SOR-like method for the singular saddle point problems

Recently, Guo et al. proposed a modified SOR-like (MSOR-like) iteration method for solving the nonsingular saddle point problem. In this paper, we further prove the semi-convergence of this method when it is applied to solve the singular saddle point problems under suitable conditions on the involved iteration parameters. Moreover, the optimal iteration parameters and the corresponding optimal semi-convergence factor for the MSOR-like method are determined. In addition, numerical experiments are used to show the feasibility and effectiveness of the MSOR-like method for solving singular saddle point problems, arising from the incompressible flow problems.

A generalized preconditioned parameterized inexact Uzawa method for singular saddle point problems

In this short paper, we introduce and analyze a generalized preconditioned parameterized inexact Uzawa method for solving singular saddle point problems, this method is a generalization of the PPIU method (see, Gao and Kong, 2010), then we give the semi-convergent conditions of this method, which are more practical for the applications.

On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems

Recently, a class of generalized shift-splitting iteration methods were proposed for solving nonsingular nonsymmetric saddle point problems (Cao et al., 2015). In this paper, the generalized shift-splitting iteration method is extended to solve singular nonsymmetric saddle point problems. Semi-convergence of the generalized shift-splitting iteration method is carefully analyzed, which shows that the iterative sequence generated by the generalized shift-splitting iteration method converges to a solution of the singular saddle point problem unconditionally. Numerical results verify the robustness and efficiency of the generalized shift-splitting iteration method.

A triple-parameter modified SSOR method for solving singular saddle point problems

In this paper, we study a triple-parameter modified SSOR (TMSSOR) method for solving singular saddle point problems. We prove the semi-convergence of the TMSSOR method under suitable restrictions on the iteration parameters, and obtain the local optimal parameters which minimize the pseudo-spectral radii of the associated iteration matrices. Finally, numerical experiments demonstrate the effectiveness of the TMSSOR method for solving singular saddle point problems.

Generalized ASOR and Modified ASOR Methods for Saddle Point Problems

Recently, the accelerated successive overrelaxation- (SOR-) like (ASOR) method was proposed for saddle point problems. In this paper, we establish a generalized accelerated SOR-like (GASOR) method and a modified accelerated SOR-like (MASOR) method, which are extension of the ASOR method, for solving both nonsingular and singular saddle point problems. The sufficient conditions of the convergence (semiconvergence) for solving nonsingular (singular) saddle point problems are derived. Finally, numerical examples are carried out, which show that the GASOR and MASOR methods have faster convergence rates than the SOR-like, generalized SOR (GSOR), modified SOR-like (MSOR-like), modified symmetric SOR (MSSOR), generalized symmetric SOR (GSSOR), generalized modified symmetric SOR (GMSSOR), and ASOR methods with optimal or experimentally found optimal parameters when the iteration parameters are suitably chosen.

A Note on Parameterized Preconditioned Method for Singular Saddle Point Problems

Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singular saddle point problems. We prove the semi-convergence of the PPHSS method under some conditions. Numerical experiments are given to illustrate the efficiency of the method with appropriate parameters.

Performance comparison of relaxation methods with singular and nonsingular preconditioners for singular saddle point problems

Performance comparison of relaxation methods with singular and nonsingular preconditioners for singular saddle point problems

Augmented block splitting preconditioner for singular saddle point problems

In this paper, the augmented block splitting preconditioner is considered to solve singular saddle point problems. This preconditioner will result in a proper splitting for the saddle point matrix and the corresponding iteration method is proved to converge to the least squares solution of the singular saddle point problem. Besides, GMRES with this preconditioner will also converge to the least squares solution. Numerical experiments are presented to illustrate its effectiveness for solving singular saddle point problems.

A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems

In this paper, based on the positive definite and skew-Hermitian splitting iteration scheme and the Uzawa iteration method, we propose a class of Uzawa-PSS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with (1,1) block of the saddle point matrix being non-Hermitian positive definite. We derive conditions of the new iteration method for guaranteeing the convergence for nonsingular saddle point problems and its semi-convergence for singular saddle point problems, respectively. Numerical experiments of a model Navier–Stokes problem are presented to illustrate the effectiveness of the Uzawa-PSS iteration method. Numerical results show that the new method is much efficient than the Uzawa-HSS iteration method for solving both the nonsingular saddle point problems Yang and Wu, (2014) [28] and the singular saddle point problems Yang, Li and Wu, (2015) [30].

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

In this paper, based on the SOR and SSOR splittings of the (1,1) part of saddle-point coefficient matrix, some variants of the accelerated parameterized inexact Uzawa (VAPIU) method are proposed for solving nonsingular and singular saddle-point problems. By choosing different parameter matrices, we derive some existing and new iterative methods. The corresponding convergence and semi-convergence of the VAPIU methods for solving nonsingular and singular saddle-point problems are studied in depth, respectively. The preconditioning strategies based on the VAPIU splittings of the coefficient matrices are presented. Numerical experiments are provided, which confirms that these new methods need less CPU times per iteration step comparing with some other methods for solving both nonsingular and singular saddle-point problems.

SEMI-CONVERGENCE OF THE PARAMETERIZED INEXACT UZAWA METHOD FOR SINGULAR SADDLE POINT PROBLEMS

In this paper, we provide semi-convergence results of the parameterized inexact Uzawa method with singular preconditioners for solving singular saddle point problems. We also provide numerical experiments to examine the effectiveness of the parameterized inexact Uzawa method with singular preconditioners.