non-Hermitian Saddle Point Problems Research Articles

Class of Uzawa-NPHSS iteration method for solving nonsingular and singular saddle point problems

ABSTRACT Based on the non-alternating preconditioned HSS iteration scheme, a class of the Uzawa-NPHSS iteration method for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite is established. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular ones of the proposed method are carefully discussed under suitable conditions. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analysed in detail. Additionally, the parameter selection strategy for the Uzawa-NPHSS iteration method is provided. Numerical experiments are implemented to confirm the theoretical results, which show the feasibility and effectiveness of the proposed method.

On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems

In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems.

On Uzawa-SSI method for non-Hermitian saddle point problems

Based on the single-step iteration (SSI) method for the non-Hermitian positive definite linear systems, we propose a Uzawa-SSI method for solving the saddle point problems with non-Hermitian positive definite (1, 1) block in this paper. The convergence and semi-convergence properties of the Uzawa-SSI method, respectively, for nonsingular and singular cases are analyzed. Numerical examples with experiments are given to show the robustness and the efficiency of Uzawa-SSI method.

A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models

Based on a general splitting of the (1,1) leading block matrix, we first construct a general class of shift-splitting (GCSS) preconditioners for non-Hermitian saddle point problems. Convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the GCSS preconditioned saddle point matrices are analyzed. Then the GCSS preconditioner is specifically applied to the non-Hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time-harmonic eddy current models. With suitable choices of the splittings, the new GCSS preconditioners are easier to implement and have faster convergence rates than the existing shift-splitting preconditioner and its modified variant. Two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners.

Regularized DPSS preconditioners for non-Hermitian saddle point problems

Based on a new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) of the coefficient matrix, a RDPSS preconditioner is introduced for non-Hermitian saddle point problems. From implementation aspects, the RDPSS preconditioner has better computing efficiency than the regularized Hermitian and skew-Hermitian preconditioner studied recently (Bai and Benzi, 2017). It is proved that the corresponding RDPSS stationary iteration method is convergent unconditionally. In addition, clustering property of the eigenvalues of the RDPSS preconditioned matrix is carefully studied.

The Uzawa-PPS iteration methods for nonsingular and singular non-Hermitian saddle point problems

In this paper, based on the positive-definite and positive-semidefinite splitting (PPS) iteration scheme, we establish a class of Uzawa-PPS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite. Theoretical analyses show that the convergence and semi-convergence properties of the proposed methods can be guaranteed under suitable conditions. Furthermore, we consider acceleration of the Uzawa-PPS methods by Krylov subspace (like GMRES) methods and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners.

A New Uzawa-Type Iteration Method for Non-Hermitian Saddle-Point Problems

AbstractBased on a preconditioned shift-splitting of the (1,1)-block of non-Hermitian saddle-point matrix and the Uzawa iteration method, we establish a new Uzawa-type iteration method. The convergence properties of this iteration method are analyzed. In addition, based on this iteration method, a preconditioner is proposed. The spectral properties of the preconditioned saddle-point matrix are also analyzed. Numerical results are presented to verify the robustness and the efficiency of the new iteration method and the preconditioner.

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.

A class of preconditioned generalized local PSS iteration methods for non-Hermitian saddle point problems

In this paper, a class of new methods based on the positive-definite and skew-Hermitian splitting scheme, called preconditioned generalized local positive-definite and skew-Hermitian splitting (PGLPSS) methods, are considered to solve non-Hermitian saddle point problems. The convergence properties of the proposed methods are analyzed, which prove that the PGLPSS methods are convergent if the iteration parameters and parameter matrices satisfy appropriate conditions. Some numerical experiments are provided to verify the efficiency of the proposed method, showing the competitiveness and efficiency of this novel method over other testing methods, whether it served as a preconditioned iteration method or as a preconditioner to the Krylov subspace method.

A generalization of the HSS-based sequential two-stage method for solving non-Hermitian saddle point problems

For non-Hermitian saddle point problems with non-Hermitian positive definite (1,1)-block, Zhu et al. studied the HSS-based sequential two-stage method (see Zhu et al. Appl. Math. Comput. 242, 907---916 19). However, this approach may not work when the (1,1)-block of the saddle point problems is weakly Hermitian or skew-Hermitian dominant. By introducing a new preconditioning matrix, a generalization of the HSS-based sequential two-stage method is proposed for solving non-Hermitian saddle-point problems with non-Hermitian positive definite and Hermitian or skew-Hermitian dominant (1,1)-block. Theoretical analysis shows that the proposed iterative method is convergent. Numerical experiments are provided to confirm the theoretical results, which demonstrate that the generalized method is effective and feasible for solving saddle point problems with non-Hermitian positive definite and Hermitian or skew-Hermitian dominant (1,1)-block.

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].

A generalized relaxed positive-definite and skew-Hermitian splitting preconditioner for non-Hermitian saddle point problems

For non-Hermitian saddle point problems with the non-Hermitian positive definite (1,1)-block, Zhang et al. (2014) presented a relaxed positive-definite and skew-Hermitian splitting (RPSS) preconditioner to accelerate the convergence rates of the Krylov subspace iteration methods such as GMRES. In this paper, the convergence property of the GRPSS iteration method is proved and a generalized RPSS (GRPSS) preconditioner is proposed. The GRPSS preconditioner is much closer to the coefficient matrix than the RPSS preconditioner in certain norm, which straightforwardly results in an GRPSS iteration method. We employing the GRPSS preconditioner to accelerate some Krylov subspace methods (like GMRES). The spectral distribution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained. Finally, numerical experiments of a model Navier–Stokes equation are presented to illustrate the efficiency of the GRPSS preconditioner.

PU-STS method for non-Hermitian saddle-point problems

Based on the skew-Hermitian triangular splitting (STS) of the (1,1) part of saddle-point coefficient matrix, a modified Uzawa method is proposed for solving non-Hermitian saddle-point problems with non-Hermitian positive definite and skew-Hermitian dominant (1,1) part. Convergence properties of this method are analyzed and the corresponding convergence result is derived under suitable conditions. Numerical experiments are provided to confirm the theoretical results, which demonstrate that this method is effective and feasible for saddle-point problems with non-Hermitian positive definite and skew-Hermitian dominant (1,1) part.

On HSS-based sequential two-stage method for non-Hermitian saddle point problems

For large sparse saddle point problems with symmetric positive definite (1,1)-block, Li et al. studied an efficient iterative method (see Li et al. (2011)) [25]. By making use of the same preconditioning technique and a new matrix splitting based on the Hermitian and skew-Hermitian splitting (HSS) of the (1,1)-block of the preconditioned non-Hermitian saddle point systems, an efficient sequential two-stage method is proposed for solving the non-Hermitian saddle point problems. Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter α approaches infinity. Numerical experiments arising from Naiver–Stokes problem are provided to show that the new iterative method is feasible, effective and robust.

A preconditioned GLHSS iteration method for non-Hermitian singular saddle point problems

In this paper, we propose a new preconditioned generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration method for solving the non-Hermitian saddle point problems. The semi-convergence of this method is discussed. Theoretical analysis shows that the semi-convergence of this new method can be guaranteed by suitable choices of the parameters and parameter matrices. Numerical examples are used to illustrate the theoretical results and examine the numerical effectiveness of the GLHSS iteration method served either as a preconditioner for GMRES or as a solver.

A generalization of the local Hermitian and skew-Hermitian splitting iteration methods for the non-Hermitian saddle point problems

For large sparse saddle point problems, Jiang and Cao studied a class of local Hermitian and skew-Hermitian splitting (LHSS) iteration methods (see M.-Q. Jiang, Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math. 231 (2009) 973–982). In this paper, we generalized these methods and propose a class of generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration schemes for solving the non-Hermitian saddle point problems. We derive conditions for guaranteeing the convergence of these iterative methods. With different choices of the parameter matrices, the generalized iterative methods lead to a series of existing and new iterative methods. Numerical experiments for a model Stokes problem are provided, further show that the new iteration methods are feasible and effective.