skew-Hermitian Splitting Research Articles

A new block preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix, discuss the eigenvalue distribution and derive an upper bound for the degree of its minimal polynomial. Finally, some numerical examples are provided to show the effectiveness and robustness of our proposed preconditioner.

Preconditioning of GMRES by skew-Hermitian iterations

A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on Hermitian and skew-Hermitian splitting of the initial matrix. A generalization for saddle point systems having semidefinite or singular (1, 1) blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners can be efficiently used for the iterative solution of systems of linear algebraic equations by the GMRES method.

Two new variants of the HSS preconditioner for regularized saddle point problems

Two new preconditioners, which can be viewed as improved variants of the Hermitian and skew-Hermitian splitting (HSS) preconditioner, are presented for regularized saddle point problems. The unconditionally convergent properties of the corresponding iteration methods are deduced and the selection of optimal iteration parameters resulting in fast convergence for the two iteration methods are discussed. Moreover, eigenvalue bounds of the preconditioned matrices and upper bounds of the degree of their minimal polynomials are obtained. Compared with the HSS preconditioner, they are much better approximations to the coefficient matrix of the saddle point problem and they have better convergence properties and spectrum distributions. Numerical experiments from the Stokes problem are presented, which illustrates the advantages of the two preconditioners over the HSS preconditioner and some other preconditioners to accelerate the convergence rate of GMRES.

Improved convergence theorems for new Hermitian and skew-Hermitian splitting methods

In this note, based on the previous work by Pour and Goughery (New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems Numer. Algor. 69 (2015) 207–225), we further discuss this new Hermitian and skew-Hermitian splitting (described as NHSS) methods for non-Hermitian positive definite linear systems. Some new convergence conditions of the NHSS method are obtained, which are superior to the results in the above paper.

Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse nonHermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

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.

On Preconditioned MHSS Real-Valued Iteration Methods for a Class of Complex Symmetric Indefinite Linear Systems

AbstractA generalized preconditioned modified Hermitian and skew-Hermitian splitting (GPMHSS) real-valued iteration method is proposed for a class of complex symmetric indefinite linear systems. Convergence theory is established and the spectral properties of an associated preconditioned matrix are analyzed. We also give several variants of the GPMHSS preconditioner and consider the spectral properties of the preconditioned matrices. Numerical examples illustrate the effectiveness of our proposed method.

A class of generalized relaxed PSS preconditioners for generalized saddle point problems

In this paper we introduce a class of generalized relaxed positive semi-definite and skew-Hermitian splitting (GRPSS) preconditioners for generalized saddle point problems. Properties of the preconditioned matrix are analyzed and compared with those of the closely related positive semi-definite and skew-Hermitian (PSS) preconditioner introduced by Shen (2014). Numerical results of a model Navier–Stokes equation demonstrate the effectiveness of the proposed preconditioners.

Accelerated PMHSS iteration methods for complex symmetric linear systems

In this paper, we introduce and analyze an accelerated preconditioning modification of the Hermitian and skew-Hermitian splitting (APMHSS) iteration method for solving a broad class of complex symmetric linear systems. This accelerated PMHSS algorithm involves two iteration parameters ź,β and two preconditioned matrices whose special choices can recover the known PMHSS (preconditioned modification of the Hermitian and skew-Hermitian splitting) iteration method which includes the MHSS method, as well as yield new ones. The convergence theory of this class of APMHSS iteration methods is established under suitable conditions. Each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. Theoretical analyses show that the upper bound ź1(ź,β) of the asymptotic convergence rate of the APMHSS method is smaller than that of the PMHSS iteration method. This implies that the APMHSS method may converge faster than the PMHSS method. Numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method.

An iterative method for solving the continuous sylvester equation by emphasizing on the skew-hermitian parts of the coefficient matrices

ABSTRACTWe present an iterative method based on the Hermitian and skew-Hermitian splitting (HSS) for solving the continuous Sylvester equation. By using the HSS of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a conjugate gradient on the normal equations (CGNR)-like method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent splitting of the coefficient matrices. Via this method, a Sylvester equation with coefficient matrices and (which are the skew-Hermitian part of A and B, respectively) is solved iteratively by a CGNR-like method. Convergence conditions of this method are studied and numerical examples show the efficiency of this method. In addition, we show that the quasi-Hermitian splitting can induce accurate, robust and effective preconditioned Krylov subspace methods.

A variant of the deteriorated PSS preconditioner for nonsymmetric saddle point problems

For nonsymmetric saddle point problems, Pan et al. (Appl Math Comput 172:762–771, 2006) proposed a deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner. In this paper, a variant of the DPSS preconditioner is proposed to accelerate the convergence of the associated Krylov subspace methods. The new preconditioner is much closer to the coefficient matrix than the DPSS preconditioner. The spectral properties of the new preconditioned matrix are analyzed. Theorem which provides the dimension of the Krylov space for the preconditioned matrix is obtained. Numerical experiments of a model Navier–Stokes problem are presented to illustrate the effectiveness of the new preconditioner.

Spectral properties of a class of matrix splitting preconditioners for saddle point problems

Based on the accelerated Hermitian and skew-Hermitian splitting iteration scheme (Bai and Golub, 2007), we propose a new two-parameter matrix splitting preconditioner in this paper. Spectral properties of the preconditioned matrix are analyzed in detail. Furthermore, based on this preconditioner, an improved version of matrix splitting preconditioner is presented and analyzed. Finally, performance of the preconditioners is compared by using GMRES(m) as an iterative solver on linear systems arising from the discretization of Stokes and Navier–Stokes equations.

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 modified HSS method for singular complex symmetric linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting, we give a generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method to solve singular complex symmetric linear systems, this method has two parameters. We give the semi-convergent conditions, and some numerical experiments are given to illustrate the efficiency of this method.

A simplified HSS preconditioner for generalized saddle point problems

For generalized saddle point problems, we propose a simplified Hermitian and skew-Hermitian splitting (SHSS) preconditioner which is much closer to the generalized saddle point matrix than the HSS preconditioner. It is proved that all eigenvalues of the SHSS preconditioned matrix are real and nonunit eigenvalues are located in a positive interval. We also study the eigenvector distribution and the degree of the minimal polynomial of the preconditioned matrix. Numerical examples of a model Stokes problem show the effectiveness of the SHSS preconditioner.

Preconditioned AHSS-PU alternating splitting iterative methods for saddle point problems

In order to solve large sparse saddle point problems (SPP) quickly and efficiently, Wang and Zhang recently studied the preconditioned accelerated Hermitian and skew-Hermitian splitting (PAHSS) methods. Through accelerating the PAHSS iteration algorithms by using parameterized Uzawa (PU) method, a preconditioned AHSS-PU alternating splitting iterative method (PAHSS-PU method) for solving saddle point problems is proposed in this paper. The convergence results of this new method are given under some suitable conditions. Moreover, we can obtain that if the parameters are suitable selected, then the PAHSS-PU algorithm will outperform the PAHSS algorithm and some Uzawa-type methods in the same precision condition. Numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the PAHSS-PU method.

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

For non-Hermitian saddle point linear systems, Pan, Ng and Bai presented a positive semi-definite and skew-Hermitian splitting (PSS) preconditioner (Pan et al. Appl. Math. Comput. 172, 762---771 2006), to accelerate the convergence rate of the Krylov subspace iteration methods like the GMRES method. In this paper, a relaxed positive semi-definite and skew-Hermitian (RPSS) splitting preconditioner based on the PSS preconditioner for the non-Hermitian generalized saddle point problems is considered. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Finally, numerical experiments of a model Navier-Stokes equation are presented to illustrate the efficiency of the RPSS preconditioner compared to the PSS preconditioner, the block diagonal preconditioner (BD), and the block triangular preconditioner (BT) in terms of the number of iteration and computational time.

The generalized HSS method with a flexible shift-parameter for non-Hermitian positive definite linear systems

Based on the Hermitian and skew-Hermitian splitting (HSS), we come up with a generalized HSS iteration method with a flexible shift-parameter for solving the non-Hermitian positive definite system of linear equations. This iteration method utilizes the optimization technique to obtain the optimal value of the flexible shift-parameter at iteration process. Both theory and experiment have shown that the new strategy is efficient.

A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations

This paper is concerned with a generalization of the Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations. The analysis shows that the GHSS iteration method will converge under certain assumptions. Numerical results show that this new method is more efficient and robust than the existing ones.

A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the Navier-Stokes equation

In this paper, we extend the relaxed positive-definite and skew-Hermitian splitting preconditioner (RPSS) for generalized saddle-point problems in [J.-L. Zhang, C.-Q. Gu and K. Zhang, Appl. Math. C...