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

In this paper, we further investigate the local Hermitian and skew-Hermitian splitting (LHSS) iteration method and the modified LHSS (MLHSS) iteration method for solving generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. When A is non-symmetric positive definite, the convergence conditions are obtained, which generalize some results of Jiang and Cao [M.-Q. Jiang and Y. Cao, On local Hermitian and Skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 2009(231): 973-982] for the generalized saddle point problems to generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. Numerical experiments show the effectiveness of the iterative methods.

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.

In this paper, we consider the regularized deteriorated positive and skew-Hermitian splitting (RDPSS) iteration method to solve singular non-Hermitian saddle point problems arising from the steady Navier–Stokes equation, which was first presented by Cao (Appl. Math. Lett., 2018). The unconditional semi-convergence property of the RDPSS iteration method and some spectral properties of the corresponding preconditioned matrix are analysed detailedly. Meanwhile, the strategies to estimate the parameters of the RDPSS preconditioner are discussed. In the end, some numerical experiments are performed to verify the feasibility and effectiveness of the proposed method and outperform several other commonly used ones.

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

Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems

In this paper, a modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving the complex linear matrix equation $AXB = C$ has been presented. As the theoretical analysis shows, the MHSS iteration method will converge under certain conditions. Each iteration in this method requires the solution of four linear matrix equations with real symmetric positive definite coefficient matrices, although the original coefficient matrices are complex and non-Hermitian. In addition, the optimal parameter of the new iteration method is proposed. Numerical results show that MHSS iteration method is efficient and robust.

The Modified Hermitian and skew-Hermitian splitting (MHSS) iteration method was presented and studied by Bai, Benzi and Chen (Computing, 87(2010), 93-111) for solving a class of complex symmetric linear systems. In this paper, using the properties of Toeplitz matrix, we propose a class of structured MHSS iteration methods for solving the complex Toeplitz linear system. Theoretical analysis shows that the structured MHSS iteration method is unconditionally convergent to the exact solution. When the MHSS iteration method is used directly to complex symmetric Toeplitz linear systems, the computational costs can be considerately reduced by use of Toeplitz structure. Finally, numerical experiments show that the structured MHSS iteration method and the structured MHSS preconditioner are efficient for solving the complex Toeplitz linear system.

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

On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation [formula omitted

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

Recently, by combining the preconditioned accelerated Hermitian and skew-Hermitian splitting (PAHSS) and the parameterized Uzawa (PU) methods, Zheng and Ma (Appl Math Comput 273, 217–225, 2016b) presented the PAHSS–PU method for saddle point problems. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PAHSS–PU iterative scheme, the modified PAHSS–PU (MPAHSS–PU) iteration method is proposed in this paper, which has a faster convergence rate than the PAHSS–PU one. Furthermore, changing the position of the parameters in the MPAHSS–PU method, we develop another new method referred to as the modified PPHSS-SOR (MPPHSS-SOR) iteration method for solving saddle point problems. We provide the convergence properties of the MPAHSS–PU and the MPPHSS-SOR iteration methods, which show that the new methods are convergent if the related parameters satisfy suitable conditions. Meanwhile, practical ways to choose iteration parameters for the proposed methods are developed. Finally, numerical experiments demonstrate that the MPAHSS–PU and the MPPHSS-SOR methods outperform some existing ones both on the number of iterations and the computational times.

Anderson acceleration is a kind of effective method for improving the convergence of the general fixed point iteration. In the linear case, Anderson acceleration can be used to improve the convergence rate of matrix splitting based iterative methods. In this paper, by using Anderson acceleration on general splitting iterative methods for linear systems, three classes of methods are given. The first one is obtained by directly applying Anderson acceleration on splitting iterative methods. For the second class of methods, Anderson acceleration is used periodically in the splitting iteration process. The third one is constructed by combining the Anderson acceleration and split iteration method in each iteration process. The key of this class of method is to determine a combination coefficient for Anderson acceleration and split iteration method. One optimal combination coefficient is given. Some theoretical results about the convergence of the considered three methods are established. Numerical experiments show that the proposed methods are effective.

In this paper, we firstly give a scaled diagonal‐times‐Toeplitz splitting (SDTS) iteration method for solving the discretization system from one‐dimensional spatial fractional diffusion equations of variable coefficients. The SDTS iteration methods can be used to solve the one‐dimensional spatial fractional diffusion equations of both anisotropic and isotropic, while the respectively scaled HSS (RSHSS) iteration method can be used only to solve anisotropic spatial fractional diffusion equations. Therefore, the SDTS iteration method is of wider range of applications than the RSHSS iteration method. Furthermore, the SDTS iteration method will naturally lead to a scaled diagonal‐times‐circulant splitting (SDCS) preconditioner. Theoretical analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments are provided, which demonstrate the feasibility and effectiveness of the SDCS preconditioned GMRES method for solving the proposed examples.

A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems