skew-Hermitian Splitting Iteration Research Articles

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 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.

Parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method for a class of linear matrix equations

Recently, Wang et al. (2013) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method to approximate the solution of linear matrix equation. To further improve the efficiency of the HSS method, we in this paper establish a parameterized preconditioned HSS (PPHSS) iteration method by introducing two Hermitian positive definite preconditioners and two more different iteration parameters for the HSS method. The convergence properties of the PPHSS method used for solving linear matrix equation AXB=C are strictly analyzed and the quasi-optimal values of the iteration parameters for the PPHSS method are also derived. Numerical results demonstrate the feasibility and the efficiency of the PPHSS iteration method.

A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

For Toeplitz systems of weakly nonlinear equations, combining the separability and strong dominance between the linear and the nonlinear terms with the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-cSSS and nonlinear cSSS-like iteration methods, which are based on a special case of the HSS, where the symmetric part H=12(A+AT) is a centrosymmetric matrix and the skew-symmetric part H=12(A−AT) is a skew-centrosymmetric matrix. The advantages of these methods are that they can transfer the linear sub-systems involved in inner iteration to two linear systems of half an order, besides, fast methods are available for computing the two half-steps involved in the inner iteration. Numerical results are provided, to further show that both Picard-cSSS and nonlinear cSSS-like iteration methods are feasible and effective.

An inexact relaxed DPSS preconditioner for saddle point problem

Based on the relaxed deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner, in this paper, we proposed a class of relaxed deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioner for solving the saddle point problem. The proposed RDPSS preconditioner is a technical modification of the deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner [36]. The PSS preconditioner is a straightforward application of the positive-definite and skew-Hermitian splitting (PSS) iteration method for solving non-Hermitian positive definite linear systems initially established by Bai et al. [37]. Numerical results have shown that the proposed RDPSS preconditioner is advantageous over the existing DPSS preconditioner.

On modified HSS iteration methods for continuous Sylvester equations

We introduce and analyze a modification of the Hermitian and skew-Hermitian splitting iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semidefinite matrices. It is found that the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent. Each iteration in this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices. These two systems can be solved inexactly. Numerical results show that the MHSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems

For a class of complex symmetric indefinite linear systems, we present two variants of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method. One is a generalization of the PMHSS iteration method and the other is a generalization of the lopsided PMHSS iteration method. We study the convergence properties of these two variants by analyzing the upper bounds of spectral radiuses of the corresponding iteration matrices and compare these upper bounds. Numerical examples are illustrated to show the effectiveness of the proposed iteration methods.

A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems

By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a non-alternating PHSS (NPHSS) iteration method for solving large sparse non-Hermitian positive definite linear systems. The convergence analysis demonstrates that the iterative series produced by the NPHSS method converge to the unique solution of the linear system when the parameters satisfy some moderate conditions. We also give a possible optimal upper bound for the iterative spectral radius. Moreover, to reduce the computational cost, we establish an inexact variant of the NPHSS (INPHSS) iteration method whose convergence property is studied. Both theoretical and numerical results validate that the NPHSS method outperforms the PHSS method when the Hermitian part of the coefficient matrix is dominant.

Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method

It is known that the Hermitian and skew-Hermitian splitting (HSS) iteration method is an efficient solver for non-Hermitian positive-definite linear system of equations. Benzi [A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 360–374] proposed a generalized HSS (GHSS) iteration method. In this paper, we present a two-parameter version of the GHSS (TGHSS) method and investigate its convergence properties. To show the effectiveness of the proposed method the TGHSS iteration method is applied to image restoration and convection–diffusion problems and the results are compared with those of the HSS and GHSS methods.

A single-step HSS method for non-Hermitian positive definite linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of the linear system. Furthermore, we derive an upper bound for the spectral radius of the SHSS iteration matrix, and the quasi-optimal parameter is obtained to minimize the above upper bound. Numerical experiments are reported to the efficiency of the SHSS method; numerical comparisons show that the proposed SHSS method is superior to the HSS method under certain conditions.

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

Based on the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, an efficient Uzawa–HSS iteration method has been proposed to solve the nonsingular saddle-point problems. In this paper, we discuss the feasibility of the Uzawa–HSS method used for solving singular saddle-point problems. The semi-convergence properties of the Uzawa–HSS iteration method are carefully analyzed, which show that the iterative sequence generated by the Uzawa–HSS method converges to a solution of the singular saddle-point problem if the iteration parameters satisfy suitable restrictions. Numerical results verify the robustness and efficiency of the Uzawa–HSS method.

The comparisons of two special Hermitian and skew-Hermitian splitting methods for image restoration

In this paper, on the basis of an augmented linear system we propose a new special Hermitian and skew-Hermitian splitting iteration method for solving the linear systems which come from image restoration. The convergence analysis and some comparisons are also shown.

On semi-convergence of modified HSS method for a class of complex singular linear systems

In this paper, we discuss the semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex singular linear systems. Some semi-convergence theories of the MHSS iteration method are established and are weaker than those appeared in previously published works.

The Uzawa–HSS method for saddle-point problems

Based on the Hermitian and skew-Hermitian splitting iteration scheme, we propose a Uzawa-type iteration method for solving a class of saddle-point problems whose coefficient matrix has non-Hermitian positive definite (1, 1)-block. The convergence properties of this novel method are analyzed, which show that the Uzawa-type iteration method is convergent if the iteration parameters satisfy suitable restrictions.

The Generalized Local Hermitian and Skew-Hermitian Splitting Iteration Methods for the Non-Hermitian Generalized Saddle Point Problems

For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 (2012) 8816–8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the convergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners. Mathematics subject classification: 65F10 65F50.

On Augmented Lagrangian Methods for Saddle-Point Linear Systems with Singular or Semidefinite (1, 1) Blocks

An effective algorithm for solving large saddle-point linear systems, presented by Krukier et al., is applied to the constrained optimization problems. This method is a modification of skew-Hermitian triangular splitting iteration methods. We consider the saddle-point linear systems with singular or semidefinite (1, 1) blocks. Moreover, this method is applied to precondition the GMRES. Numerical results have confirmed the effectiveness of the method and showed that the new method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse saddle-point linear systems. Mathematics subject classification: 65F10, 65F50.

Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations

In this paper, to solve a broad class of complex symmetric linear systems, we recast the complex system in a real formulation and apply the generalized successive overrelaxation (GSOR) iterative method to the equivalent real system. We then investigate its convergence properties and determine its optimal iteration parameter as well as its corresponding optimal convergence factor. In addition, the resulting GSOR preconditioner is used to precondition Krylov subspace methods such as the generalized minimal residual method for solving the real equivalent formulation of the system. Finally, we give some numerical experiments to validate the theoretical results and compare the performance of the GSOR method with the modified Hermitian and skew-Hermitian splitting iteration.

A modified GPSS method for non-Hermitian positive definite linear systems

This paper is concerned with a modified version of the generalization of positive-definite and skew-Hermitian splitting iteration (MGPSS) to solve the non-Hermitian positive definite linear systems. The theoretical analysis shows that the MGPSS iteration method will converge unconditionally. Numerical examples are reported to confirm the efficiency of the proposed methods.

On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations

This paper is concerned with some generalizations of the Hermitian and skew-Hermitian splitting (HSS) iteration for solving continuous Sylvester equations. The main contents we will introduce are the normal and skew-Hermitian splitting (NSS) iteration methods for the continuous Sylvester equations. It is shown that the new schemes can outperform the standard HSS method in some situations. Theoretical analysis shows that the NSS methods converge unconditionally to the exact solution of the continuous Sylvester equations. Moreover, we derive the upper bound of the contraction factor of the NSS iterations. Numerical experiments further show the effectiveness of our new methods.