鞍点问题PHSS方法预条件的新取法（英文）

Bai, Golub and Pan presented a preconditioned Hermitian and skew-Hermitian splitting(PHSS) method [Numerische Mathematik, 2004, 32: 1-32] for non-Hermitian positive semidefinite linear systems. We improve the method to solve saddle point systems whose(1,1) block is a symmetric positive definite M-matrix with a new choice of the preconditioner and compare it with other preconditioners. The results show that the new preconditioner outperforms the previous ones.     

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.     

Semi-convergence analysis of preconditioned deteriorated PSS iteration method for singular saddle point problems

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.     

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

Based on the variant of the deteriorated positive-definite and skew-Hermitian splitting (VDPSS) preconditioner developed by Zhang and Gu (BIT Numer. Math. 56:587–604, 2016), a generalized VDPSS (GVDPSS) preconditioner is established in this paper by replacing the parameter α in (2,2)-block of the VDPSS preconditioner by another parameter β. This preconditioner can also be viewed as a generalized form of the VDPSS preconditioner and the new relaxed HSS (NRHSS) preconditioner which has been exhibited by Salkuyeh and Masoudi (Numer. Algorithms, 2016). The convergence properties of the GVDPSS iteration method are derived. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. We also study the upper bounds on the degree of the minimum polynomial of the preconditioned matrix. Numerical experiments are implemented to illustrate the effectiveness of the GVDPSS preconditioner and verify that the GVDPSS preconditioned generalized minimal residual method is superior to the DPSS, relaxed DPSS, SIMPLE-like, NRHSS, and VDPSS preconditioned ones for solving saddle point problems in terms of the iterations and computational times.     

广义鞍点问题基于PSS的约束预条件子

对于(1,1)块为非Hermitian阵的广义鞍点问题,本文给出了一种基于正定和反对称分裂(Positive definite andskew-Hermitian splitting, PSS)的约束预条件子.该预条件子的(1,1)块由求解非Hermitian正定线性方程组时的PSS迭代法所构造得到.文中分析了PSS约束预条件子的一些性质并证明了预处理迭代法的收敛性.最后用数值算例验证了该预条件子的有效性.     

On the Generalized Deteriorated Positive Semi-Definite and Skew-Hermitian Splitting Preconditioner

For nonsymmetric saddle point problems, Huang et al. in [Numer. Algor. 75 (2017), pp. 1161-1191] established a generalized variant of the deteriorated positive semi-definite and skew-Hermitian splitting (GVDPSS) preconditioner to expedite the convergence speed of the Krylov subspace iteration methods like the GMRES method. In this paper, some new convergence properties as well as some new numerical results are presented to validate the theoretical results.     

An inexact modified relaxed splitting preconditioner for the generalized saddle point problems from the incompressible Navier-Stokes equations

Based on the modified relaxed splitting (MRS) preconditioner proposed by Fan and Zhu (Appl. Math. Lett. 55, 18–26 2016), an inexact modified relaxed splitting (IMRS) preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier-Stokes equations. The eigenvalues and eigenvectors of the preconditioned matrix are analyzed, and the convergence property of the corresponding iteration method is also discussed. Numerical experiments are presented to show the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.     

Regularized DPSS Preconditioners for Singular Saddle Point Problems

Recently, Cao proposed a regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for the non-Hermitian nonsingular saddle point problem. In this paper, we consider applying RDPSS preconditioner to solve the singular saddle point problem. Moreover, we propose a two-parameter accelerated variant of the RDPSS (ARDPSS) preconditioner to further improve its efficiency. Theoretical analysis proves that the RDPSS and ARDPSS methods are semi-convergent unconditionally. Some spectral properties of the corresponding preconditioned matrices are analyzed. Numerical experiments indicate that better performance can be achieved when applying the ARDPSS preconditioner to accelerate the GMRES method for solving the singular saddle point problem.     

The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems

In this paper, we consider the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems with nonzero (2, 2) blocks. The spectral property of the preconditioned matrix is studied in detail. Under certain conditions, all eigenvalues of the preconditioned matrix with the original system being non-Hermitian will form two tight clusters, one is near (0, 0) and the other is near (2, 0) as the iteration parameter approaches to zero from above, so do all eigenvalues of the preconditioned matrix with the original system being Hermitian. Numerical experiments are given to demonstrate the results.     

Improved PHSS iterative methods for solving saddle point problems

An improvement on a generalized preconditioned Hermitian and skew-Hermitian splitting method (GPHSS), originally presented by Pan and Wang (J. Numer. Methods Comput. Appl. 32, 174–182, 2011), for saddle point problems, is proposed in this paper and referred to as IGPHSS for simplicity. After adding a matrix to the coefficient matrix on two sides of first equation of the GPHSS iterative scheme, both the number of required iterations for convergence and the computational time are significantly decreased. The convergence analysis is provided here. As saddle point problems are indefinite systems, the Conjugate Gradient method is unsuitable for them. The IGPHSS is compared with Gauss-Seidel, which requires partial pivoting due to some zero diagonal entries, Uzawa and GPHSS methods. The numerical experiments show that the IGPHSS method is better than the original GPHSS and the other two relevant methods.     

A generalized preconditioned HSS method for singular saddle point problems

Li et al. recently studied the generalized HSS (GHSS) method for solving singular linear systems (see Li et al., J. Comput. Appl. Math. 236, 2338–2353 (2012)). In this paper, we generalize the method and present a generalized preconditioned Hermitian and skew-Hermitian splitting method (GPHSS) to solve singular saddle point problems. We prove the semi-convergence of GPHSS under some conditions, and weaken some semi-convergent conditions of GHSS, moreover, we analyze the spectral properties of the corresponding preconditioned matrix. Numerical experiments are given to illustrate the efficiency of GPHSS method with appropriate parameters both as a solver and as a preconditioner.     

Preconditioned AHSS iteration method for singular saddle point problems

In this paper, for solving the singular saddle point problems, we present a new preconditioned accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration method. The semi-convergence of this method and the eigenvalue distribution of the preconditioned iteration matrix are studied. In addition, we prove that all eigenvalues of the iteration matrix are clustered for any positive iteration parameters α and β. Numerical experiments illustrate the theoretical results and examine the numerical effectiveness of the AHSS iteration method served either as a preconditioner or as a solver.     

关于广义鞍点问题的HSS迭代方法的收缩因子(英文)

白中治等提出了解非埃尔米特正定线性方程组的埃尔米特和反埃尔米特分裂(HSS)迭代方法(Bai Z Z,Golub G H,Ng M K.Hermitian and skew-Hermitian splitting methodsfor non-Hermitian positive definite linear systems.SIAM J.Matrix Anal.Appl.,2003,24:603-626).本文精确地估计了用HSS迭代方法求解广义鞍点问题时在加权2-范数和2-范数下的收缩因子.在实际的计算中,正是这些收缩因子而不是迭代矩阵的谱半径,本质上控制着HSS迭代方法的实际收敛速度.根据文中的分析,求解广义鞍点问题的HSS迭代方法的收缩因子在加权2-范数下等于1,在2-范数下它会大于等于1,而在某种适当选取的范数之下,它则会小于1.最后,用数值算例说明了理论结果的正确性.     

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

     

Skew-Hermitian Triangular Splitting Iteration Methods for Non-Hermitian Positive Definite Linear Systems of Strong Skew-Hermitian Parts

By further generalizing the skew-symmetric triangular splitting iteration method studied by Krukier, Chikina and Belokon (Applied Numerical Mathematics, 41 (2002), pp. 89–105), in this paper, we present a new iteration scheme, called the modified skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this new method in depth. Moreover, when it is applied to precondition the Krylov subspace methods like GMRES, the preconditioning property of the modified skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that, as both solver and preconditioner, the modified skew-Hermitian triangular splitting iteration method is very effective for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems

For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSS) preconditioner as well as the MGSS iteration method is derived in this paper, which generalize the modified shift-splitting (MSS) preconditioner and the MSS iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 317:535–546, 2017), respectively. The convergent and semi-convergent analyses of the MGSS iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. Meanwhile, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSS preconditioner and the MGSS iteration method and also illustrate that the MGSS iteration method outperforms the generalized shift-splitting (GSS) and the generalized modified shift-splitting (GMSS) iteration methods, and the MGSS preconditioner is superior to the shift-splitting (SS), GSS, modified SS (M-SS), GMSS and MSS preconditioners for the generalized minimal residual (GMRES) method for solving the nonsymmetric saddle point problems.     

ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS

We present a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi-definite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.     

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 α,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 σ₁(α,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.     

一类非线性方程组的Newton-PSS迭代法

正定反Hermite分裂(PSS)方法是求解大型稀疏非Hermite正定线性代数方程组的一类无条件收敛的迭代算法.将其作为不精确Newton方法的内迭代求解器,我们构造了一类用于求解大型稀疏且具有非Hermite正定Jacobi矩阵的非线性方程组的不精确Newton-PSS方法,并对方法的局部收敛性和半局部收敛性进行了详细的分析.数值结果验证了该方法的可行性与有效性.