关于鞍点问题的广义预处理HSS-SOR交替分裂迭代方法

本文研究了鞍点问题的迭代法. 在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂（PHSS）迭代法的基础上,通过结合GSOR迭代格式,利用两个参数加速,提出了一种广义预处理HSS-SOR交替分裂迭代法,并研究了该方法的收敛性.数值结果表明本文所给方法是有效的.     

求解广义鞍点问题的混合并行迭代法(英文)

为了在高性能计算机上求解广义鞍点问题,对于合适的系数矩阵,本文提出混合并行迭代法及其加速形式.并详细讨论了新方法的收敛性.     

鞍点问题迭代解法收敛因子估计

分析Ｂａｎｋ（１９９０年）的论文中的求解鞍点问题的迭代解法,得到一个改进的收敛因子估计。     

关于鞍点问题的预处理HSS-SOR交替分裂迭代方法

为了高效地求解大型稀疏鞍点问题,在白中治,Golub和潘建瑜提出的预处理对称/反对称分裂(PHss)迭代法的基础上,通过结合SOR-like迭代格式对原有迭代算法进行加速,提出了一种预处理HSS-SOR交替分裂迭代方法,并研究了该算法的收敛性.数值例子表明:通过参数值的选择,新算法比SOR-like和PHSS算法都具有更快的收敛速度和更少的迭代次数,选择了合适的参数值后,可以提高算法的收敛效率.     

非Hermitian正定线性方程组的外推的HSS迭代方法

为了高效地求解大型稀疏非Hermitian正定线性方程组，在白中治、Golub和Ng提出的Hermitian和反Hermitian分裂（HSS）迭代法的基础上，通过引入新的参数并结合迭代法的松弛技术，对HSS迭代方法进行加速，提出了一种新的外推的HSS迭代方法（EHSS），并研究了该方法的收敛性.数值例子表明：通过参数值的选择，新方法比HSS方法具有更快的收敛速度和更少的迭代次数，选择了合适的参数值后，可以提高HSS方法的收敛效率.     

速度追踪问题中鞍点系统的新分裂迭代

 有限元离散一类速度追踪问题后得到具有鞍点结构的线性系统,针对该鞍点系统,本文提出了一种新的分裂迭代技术.证明了新的分裂迭代方法的无条件收敛性,详细分析了新的分裂预条件子对应的预处理矩阵的谱性质.数值结果验证了对于大范围的网格参数和正则参数,新的分裂预条件子在求解有限元离散速度追踪问题得到的鞍点系统时的可行性和有效性.     

一类求解鞍点问题的修正SOR迭代方法

针对鞍点问题,该文详细讨论和分析了修正SOR弛迭代方法的收敛性.理论分析表明,当选择合适的参数时,修正SOR迭代方法迭代方法是收敛的.进一步,我们得到了修正SOR迭代方法收敛时参数需要满足的条件.最后,数值算例表明了该方法的正确性以及有效性.     

鞍点问题的广义位移分裂预条件子

对于大型稀疏非Hermitian正定线性方程组,Bai等人提出了一种位移分裂预条件子(J.Comput.Math.,24(2006)539-552).本文将这种思想用到鞍点问题上并提出了一种广义位移分裂(Generalized Shift Splitting,GSS)预条件子,同时证明了该预条件子所对应分裂迭代法的无条件收敛性.最后用数值算例验证了新预条件子的有效性.     

鞍点问题的修正对称超松弛迭代算法

为了提高求解鞍点问题的迭代算法的速度,通过设置合适的加速变量,对修正超松弛迭代算法（简记作MSOR-like算法）和广义对称超松弛迭代算法（简记作GSSOR-like算法）进行了修正,给出了修正对称超松弛迭代算法,即MSSOR-like （modified symmetric successiveover-relaxation）算法,并研究了该算法收敛的充分必要条件.最后,通过数值例子表明,选择合适的参数后,新算法的迭代速度和迭代次数均优于MSOR-like （modified successive overrelaxation）和GSSOR-like （generalized symmetric successive over-relaxation）算法,因此,它是一种较好的解决鞍点问题的算法.     

Optimization of the generalized method of Hermitian and skew-Hermitian splitting iterations for solving symmetric saddle-point problems

An algorithm for solving a nonsingular symmetric system of linear equations with a saddle point is examined. This algorithm has two constant iteration parameters and is an extension of the algorithm of Hermitian and skew-Hermitian splitting iterations (the HSS algorithm). Analytical formulas are derived for the optimal values of the iteration parameters. The formulation of the optimization problem is a classical one for the saddle-point problems. The results obtained are sharp.     

Preconditioning of saddle point problems by the method of Hermitian and skew-Hermitian splitting iterations

For a nonsingular symmetric system of linear equations with a saddle point, a Hermitian and skew-Hermitian splitting (HSS) preconditioner is considered. For the preconditioned system, symmetrizability conditions are established under which estimates are derived for the spectrum and the convergence rate of Chebyshev-type algorithms and 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.     

Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems

By further generalizing the modified skew-Hermitian triangular splitting iteration methods studied in [L. Wang, Z.-Z. Bai, Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44 (2004) 363-386], in this paper, we present a new iteration scheme, called the product-type 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 method. Moreover, when it is applied to precondition the Krylov subspace methods, the preconditioning property of the product-type skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that the product-type skew-Hermitian triangular splitting iteration method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems

In this paper, we first present a local Hermitian and skew-Hermitian splitting (LHSS) iteration method for solving a class of generalized saddle point problems. The new method converges to the solution under suitable restrictions on the preconditioning matrix. Then we give a modified LHSS (MLHSS) iteration method, and further extend it to the generalized saddle point problems, obtaining the so-called generalized MLHSS (GMLHSS) iteration method. Numerical experiments for a model Navier-Stokes problem are given, and the results show that the new methods outperform the classical Uzawa method and the inexact parameterized Uzawa method.     

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.

     

A special Hermitian and skew-Hermitian splitting method for image restoration

In this paper, we consider an ill-posed image restoration problem with a noise contaminated observation, and a known convolution kernel. A special Hermitian and skew-Hermitian splitting (HSS) iterative method is established for solving the linear systems from image restoration. Our approach is based on an augmented system formulation. The convergence and operation cost of the special HSS iterative method for image restoration problems are discussed. The optimal parameter minimizing the spectral radius of the iteration matrix is derived. We present a detailed algorithm for image restoration problems. Numerical examples are given to demonstrate the performance of the presented method. Finally, the SOR acceleration scheme for the special HSS iterative method is discussed.     

A class of block alternating splitting implicit iteration methods for double saddle point linear systems

In the framework of a special block alternating splitting implicit (BASI) iteration scheme for generalized saddle point problems, we establish some new iteration methods for solving double saddle point problems by means of a suitable partitioning strategy. Convergence analysis of the corresponding BASI iteration methods indicates that they are convergent unconditionally under certain weak requirements for the related matrix splittings, which are satisfied directly for our specific application to double saddle point problems. Numerical examples for liquid crystal director and time-harmonic eddy current models are presented to demonstrate the efficiency of the proposed BASI preconditioners to accelerate the GMRES method.     

Optimal parameters in the HSS‐like methods for saddle‐point problems

For the Hermitian and skew‐Hermitian splitting iteration method and its accelerated variant for solving the large sparse saddle‐point problems, we compute their quasi‐optimal iteration parameters and the corresponding quasi‐optimal convergence factors for the more practical but more difficult case that the (1, 1)‐block of the saddle‐point matrix is not algebraically equivalent to the identity matrix. In addition, the algebraic behaviors and the clustering properties of the eigenvalues of the preconditioned matrices with respect to these two iterations are investigated in detail, and the formulas for computing good iteration parameters are given under certain principle for optimizing the distribution of the eigenvalues. Copyright © 2008 John Wiley & Sons, Ltd.     

On parameterized block triangular preconditioners for generalized saddle point problems

In this paper, we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems. These preconditioners generalize the common block diagonal and triangular preconditioners. We will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues. Numerical experiments of a model Stokes problem are presented.