skew-Hermitian Splitting Iteration Method Research Articles

Semiconvergence of the extended PSS method for singular generalized saddle point problems

Recently, an extension of the positive definite and skew-Hermitian splitting (EPSS) iteration method for nonsingular generalized saddle point problems has been stated by Masoudi and Salkuyeh. In this article, we study the semiconvergence of the EPSS method for singular generalized saddle problems. Then a special case of the EPSS (SEPSS) preconditioner is applied to accelerate the convergence of the GMRES method for solving singular generalized saddle point problems. Some numerical results are given to demonstrate the robustness of the preconditioner.

A simplified PSS preconditioner for non-Hermitian generalized saddle point problems

The preconditioner for positive-definite and skew-Hermitian splitting (PSS) iteration method has been used to solve saddle point problems. Firstly, we propose a simplified PSS preconditioner for non-Hermitian generalized saddle point problems, which is much closer to original matrix than other PSS preconditioners. Moreover, the spectral properties of preconditioned matrix are also discussed. Finally, we give an example to illustrate the efficiency of the new preconditioner which is used to accelerate the convergence rate of the Krylov subspace, such as GMRES method, it has an obvious advantage than other preconditioners.

Lopsided modified Euler-extrapolated Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems

In this paper, a lopsided modified Euler-extrapolated Hermitian and skew-Hermitian splitting (LME-HS) iteration method is introduced for solving the complex symmetric linear systems. Under a loose restriction on parameter $\theta$, we demonstrate that LME-HS iteration method is convergent. Moreover, we present the optimal parameter ${\theta}^{*}$ of the LME-HS method and discuss the spectral properties of corresponding preconditioned matrix. Finally, the numerical experiments are used to verify the effectiveness of the proposed method.

Convergence of a generalized PMHSS method for a class of singular block two-by-two linear systems

In this work, we introduce a generalization of the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, named as generalized PMHSS (GPMHSS) iteration method, to solve a class of singular block two-by-two system of linear equations. Theoretical analyses show that the GPMHSS iteration method converges unconditionally to the minimum norm least squares solution for any initial guess no matter the system is consistent or inconsistent. Besides, with the preconditioner derived from the GPMHSS iteration method, the preconditioned generalized minimal residual (GMRES) method also determines the minimum norm least squares solution of the consistent singular block two-by-two linear systems at breakdown. Numerical experiments are presented to show the effectiveness and the robustness of the GPMHSS iteration method and the corresponding preconditioner.

On preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting method for complex symmetric linear systems

In this paper, we introduce a preconditioned Euler-extrapolated single-step Hermitian and skew-Hermitian splitting (PE-SHSS) iteration method for solving a class of complex symmetric system of linear equations. The convergence properties of the PE-SHSS iteration method are investigated under suitable restrictions. In addition, the spectral properties of the corresponding preconditioned matrix are discussed. Finally, three numerical examples are also used to verify the effectiveness of the PE-SHSS iteration method.

Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems

For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods.

Preconditioned modified Hermitian and skew-Hermitian splitting iteration methods for fractional nonlinear Schrödinger equations

A variant of preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method is proposed for a class of Toeplitz-like complex linear equations arising from the space fractional coupled nonlinear Schrödinger equations. Theoretical analysis show that the new method is convergent unconditionally. Moreover, the choice of the theoretical optimal parameter is further studied in detail, which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the Toeplitz matrix and the extreme absolute values of the diagonal matrix. In addition, the splitting can be used as a preconditioner for Krylov subspaces method. Numerical experiments show that the proposed method is efficient, and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem.

ERHSS iteration method for PDE optimal control problem

An extended regularized Hermitian and skew-Hermitian splitting iteration method, for solving the double saddle point linear system arising from PDE optimal control problem, is proposed. The unconditional convergence of the method is proved. Furthermore, some spectral properties of the preconditioned matrix are investigated. Numerical experiments show that the iteration method is feasible and efficient.

How to compute the minimum norm least squares solution of singular linear system by using the preconditioned HSS method?

For the singular, non-Hermitian, and positive semi-definite system of linear equations Ax = b, we introduce a kind of preconditioners for the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method. Then, the iteration sequence generated by the corresponding PHSS iteration method converges to the minimum norm least squares solution A‡b for any initial guess no matter the singular system of linear equations is consistent or inconsistent. In addition, a kind of PHSS preconditioners are derived from the PHSS iteration method. The PHSS preconditioned generalized minimum residual (PGMRES) methods also determine the minimum norm least squares solution of the consistent singular linear system at breakdown. Numerical experiments are used to verify the effectiveness and the robustness of the PHSS iteration method and the PHSS preconditioner.

Modified special HSS method for discrete ill-posed problems and image restoration

ABSTRACT For solving ill-posed inverse problems, we establish a modified version of special Hermitian and skew-Hermitian splitting (SHSS) iteration method, called the modified SHSS (MSHSS) method in this paper. Based on an augmented system formulation, we apply the modified splitting to solve ill-posed inverse problems, and theoretically study its convergence and determine its optimal iteration parameters. Furthermore, the SOR acceleration scheme for MSHSS iterative method is discussed. Finally, numerical experiments arising from a Fredholm integral equation of the first kind and image restoration are used to further examine the effectiveness of the new method.

An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems

In this paper, we exploit the numerical solvers for a fractional differential equations (FDE) optimization problem with constraints given by fractional elliptic state equations. The discretized linear system can be rewritten as a two-by-two linear system. By making use of the strategy of modified Hermitian and skew-Hermitian splitting (MHSS) iteration method proposed by Bai, we propose an MHSS-like iteration method. Comparing with the MHSS iteration method, the MHSS-like method only requires one to solve the two-by-two linear systems by a sparse Cholesky factorization and a fast Fourier transform. Hence, the MHSS-like method has less workload than the MHSS method. The convergence properties and the quasi-optimal parameters are analyzed in detail. Numerical examples are used to testify the efficiency of the new method.

Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems

By applying the minimum residual technique to the Hermitian and skew-Hermitian splitting (HSS) iteration scheme, we introduce a non-stationary iteration method named minimum residual HSS (MRHSS) iteration method to solve non-Hermitian positive definite linear systems. The convergence property of the MRHSS iteration method together with the property of the iteration parameters are carefully studied. Numerical results verify the effectiveness and robustness of the MRHSS iteration method.

Regularized HSS iteration methods for stabilized saddle-point problems

Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

How to compute the optimal parameters of the parameterized PMHSS preconditioner?

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method and its alternating-directional variant, a parameterized PMHSS (P2MHSS) preconditioner is proposed for a class of block two-by-two systems of linear equations. To optimize the preconditioning effect of the P2MHSS preconditioner, practical formulas for computing the optimal parameters of the P2MHSS preconditioner are derived by employing the scaled norm minimization method. Finally, we test the feasibility and the effectiveness of the P2MHSS preconditioner.

On the semi-convergence of regularized HSS iteration methods for singular saddle point problems

Recently, Bai and Benzi proposed a class of regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for solving the nonsingular saddle point problem. In this paper, we apply this method to solve the singular saddle point problem. In the process of the semi-convergence analysis, we get that the RHSS method and the HSS method are unconditionally semi-convergent, which has improved the previous results. Then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed. Finally, some numerical experiments on linear systems arising from the discretization of the Stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners.

Modified PHSS iterative methods for solving nonsingular and singular saddle point problems

For large sparse saddle point problems, we establish a new version of the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method, called the modified PHSS (MPHSS) method in this paper. Then, we theoretically study its convergence and semi-convergence properties and determine its optimal iteration parameter and corresponding optimal convergence factor. Furthermore, the spectral properties of the MPHSS preconditioned matrix are discussed in detail. Numerical experiments show that the MPHSS iteration method is effective and robust when it is used either as a solver or as a matrix splitting preconditioner for the generalized minimal residual (GMRES) method.

On preconditioned normal and skew-Hermitian splitting iteration method for continuous Sylvester equations AX + XB = C*

In this paper, we present a preconditioned normal and skew-Hermitian splitting (PNSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. Theoretical analysis shows that the PNSS methods will converge unconditionally to the exact solution of the continuous Sylvester equations. An inexact variant of the PNSS iteration method(IPNSS) and the analysis of its convergence property in detail have been established. Numerical experiments further show that this new method is more efficient and robust than the existing ones.

Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants.

On the semi-convergence of preconditioned GLHSS iteration method for non-Hermitian singular saddle point problem

Recently, Fan and Zheng studied the preconditioned generalized local Hermitian and skew-Hermitian splitting (GLHSS) iteration method for non-Hermitian singular saddle point problem, and given its semi-convergence conditions; see Fan and Zheng (2014). In this note, we prove the semi-convergence of preconditioned GLHSS method by another method. The obtained result shows that the conditions for guaranteeing its semi-convergence are easy to check and more weaker.

Modified Newton-NSS method for solving systems of nonlinear equations

By making use of the normal and skew-Hermitian splitting (NSS) method as the inner solver for the modified Newton method, we establish a class of modified Newton-NSS method for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. Under proper conditions, the local convergence theorem is proved. Furthermore, the successive-overrelaxation (SOR) technique has been proved quite successfully in accelerating the convergence rate of the NSS or the Hermitian and skew-Hermitian splitting (HSS) iteration method, so we employ the SOR method in the NSS iteration, and we get a new method, which is called modified Newton SNSS method. Numerical results are given to examine its feasibility and effectiveness.