Semi-regularized Hermitian and Skew-Hermitian Splitting Preconditioning for Saddle-Point Linear Systems

In this paper,a two-step semi-regularized Hermitian and skew-Hermitian splitting(SHSS)iteration method is constructed by introducing a regularization matrix in the(1,1)-block of the first iteration step,to solve the saddle-point linear system.By carefully selecting two different regularization matrices,two kinds of SHSS preconditioners are proposed to accelerate the convergence rates of the Krylov subspace iteration methods.Theoretical analysis about the eigenvalue distribution demonstrates that the proposed SHSS preconditioners can make the eigenvalues of the corresponding preconditioned matrices be clustered around 1 and uniformly bounded away from 0.The eigenvector distribution and the upper bound on the degree of the minimal polynomial of the SHSS-preconditioned matrices indicate that the SHSS-preconditioned Krylov subspace iterative methods can converge to the true solution within finite steps in exact arithmetic.In addition,the numerical example derived from the optimal control problem shows that the SHSS preconditioners can significantly improve the convergence speeds of the Krylov subspace iteration methods,and their convergence rates are independent of the discrete mesh size.

Accelerated RHSS Iteration Method for Stabilized Saddle-Point Problems

For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretical analysis shows that the ARHSS method converges unconditionally to the unique solution of the saddle point problem. Finally, we use a numerical example to confirm the effectiveness of the method.

复参数HSS迭代法求解非Hermitian正定线性方程组

将实参数的Hermitian/斜-Hermitian分裂(HSS)迭代法推广到复参数Hermitian/斜-Hermitian分裂(CHSS)迭代法,并证实CHSS迭代法是无条件收敛的。理论分析显示:CHSS迭代法的致缩因子的上界依赖系数矩阵Hermitian部分的谱,与矩阵的特征向量无关。数值例子显示方法的有效性。

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 con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.

预处理Hermitian和skew-Hermitian分裂迭代法

对于系数矩阵为大型稀疏非Hermitian正定线性方程组,白中治、Golub和Ng提出了Hermitian和skew-Hermitian分裂迭代法(HSS).该论文提出一种预处理Hermitian和skew-Hermitian分裂迭代法(PHSS).理论分析该法收敛于线性方程组的唯一解.

GENERALIZED PRECONDITIONED HERMITIAN AND SKEW-HERMITIAN SPLITTING METHODS FOR NON-HERMITIAN POSITIVE-DEFINITE LINEAR SYSTEMS

In this paper, a generalized preconditioned Hermitian and skew-Hermitian splitting （GPHSS） iteration method for a non-Hermitian positive-definite matrix is studied, which covers standard Hermitian and skew-Hermitian splitting （HSS） iteration and also many existing variants. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From practical point of view, we have analyzed and implemented inexact generalized preconditioned Hermitian and skew-Hermitian splitting （IGPHSS） iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments from three-dimensional convection-diffusion iterations are efficient and competitive with equation show that the GPHSS and IGPHSS standard HSS iteration and AHSS iteration.

非线性方程组的非交替Newton-PHSS迭代法

大型稀疏非Hermite正定Jacobi矩阵对应的非线性方程组的迭代求解历来受到重视.结合不精确Newton法和非交替PHSS迭代法,提出了迭代求解非线性方程组的NewtonNPHSS方法,给出了迭代法的局部收敛定理,并演算了数值例子,阐明了Newton-NPHSS是有效的迭代法.

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.

广义Lyapunov方程的HSS迭代法

提出了求解广义Lyapunov方程的HSS(Hermitian and skew-Hermitian splitting)迭代法,分析了该方法的收敛性,给出了收敛因子的上界.为了降低HSS迭代法的计算量,提出了求解广义Lyapunov方程的非精确HSS迭代法,并分析其收敛性.数值结果表明,求解广义Lyapunov方程的HSS迭代法及非精确HSS迭代法是有效的.

复奇异鞍点问题预条件修正AHSS法的半收敛性

提出了求解一类复奇异鞍点问题的预条件修正AHSS法。研究了所提出的新方法的半收敛性。对任意的正迭代参数,得到了所提出的新方法的半收敛定理。数值实验说明,新方法比HSS法求解鞍点问题时更有效。

一类复对称线性方程组的单步HSS迭代法

基于修正的埃尔米特和反埃尔米特分裂(MHSS)及预处理的MHSS(PMHSS)迭代法,提出了关于一类复对称线性方程组的单步MHSS(SMHSS)和单步PMHSS(SPMHSS)迭代法,进一步利用优化技巧给出了位移参数的动态选择格式,得出相应的带有灵活位移的SMHSS方法及SPMHSS迭代法.理论分析表明,迭代参数α在较弱的约束条件下,SMHSS迭代法收敛于复对称线性方程组的唯一解.同时,得到了SMHSS迭代矩阵的谱半径的上界,并且求得使上述上界最小的最优参数α~*.进一步给出了SPMHSS方法的收敛性分析.MHSS法和SMHSS迭代法之间的数值比较表明,在某些情况下,SMHSS迭代法比MHSS迭代法更优.

A Note on Parameterized Preconditioned Method for Singular Saddle Point Problems

Recently, some authors (Li, Yang and Wu, 2014) studied the parameterized preconditioned HSS (PPHSS) method for solving saddle point problems. In this short note, we further discuss the PPHSS method for solving singular saddle point problems. We prove the semi-convergence of the PPHSS method under some conditions. Numerical experiments are given to illustrate the efficiency of the method with appropriate parameters.

Several splittings for non-Hermitian linear systems

For large sparse non-Hermitian positive definite system of linear equations,we present several variants of the Hermitian and skew-Hermitian splitting(HSS)about the coefficient matrix and establish correspondingly several HSS-based iterative schemes.Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations,and they may show advantages on problems that the HSS method is ineffiective.

Accelerating the HS-type Richardson Iteration Method with Anderson Mixing

The Accelerated Hermitian/skew-Hermitian type Richardson(AHSR)iteration methods are presented for solving non-Hermitian positive definite linear systems with three schemes,by using Anderson mixing.The upper bounds of spectral radii of iteration matrices are studied,and then the convergence theories of the AHSR iteration methods are established.Furthermore,the optimal iteration parameters are provided,which can be computed exactly.In addition,the application to the model convection-diffusion equation is depicted and numerical experiments are conducted to exhibit the effectiveness and confirm the theoretical analysis of the AHSR iteration methods.

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.

求解特定鞍点问题的改进SOR-Like方法

鞍点问题广泛出现在众多的工程研究领域,如流体力学、电磁学、最优化问题、最小二乘问题、椭圆偏微分方程问题等.以SOR类方法为基础,结合HS分裂思想,将经典鞍点问题的求解方法推广到特殊鞍点问题的求解上.给出一种具有新型分裂迭代格式的MSOR-Like方法,用以求解一类含有非对称块的鞍点系统,给出了相应的收敛性分析以及最优松弛参数选取方法.数值算例验证了对于不同的预优矩阵,MSORLike方法只有收敛速度的分别,没有收敛性能的影响,且在相同计算精度下,该方法解决特殊鞍点问题的迭代效果优于常规方法解决经典鞍点问题.

线性方程组的迭代解法

线性方程组的数值求解常见于许多科学与工程计算领域,介绍了求解大型线性方程组的主要迭代算法。首先,对一些经典迭代法(Jacobi方法、Gauss-Seidel方法、SOR方法、SSOR方法和CG方法等)进行了详细的讨论,并从理论上对收敛性进行分析。其次,讨论了最新的Hermitian/Skew-Hermitian splitting(HSS)迭代理论,给出了迭代公式和收敛性定理。最后,通过数值实验对所有迭代法的有效性进行了验证。

关于非Hermitian正定线性代数方程组的超松弛HSS方法

本文针对求解大型稀疏非Hermitian正定线性方程组的IISS迭代方法,利用迭代法的松弛技术进行加速,提出了一种具有三个参数的超松弛HSS方法(SAHSS)和不精确的SAHSS方法(TSAHSS),它采用CG和一些Krylov子空间方法作为其内部过程,并研究了SAHSS和ISAHSS方法的收敛性.数值例子验证了新方法的有效性.

HSS METHOD WITH A COMPLEX PARAMETER FOR THE SOLUTION OF COMPLEX LINEAR SYSTEM

In this paper, a complex parameter is employed in the Hermitian and skew-Hermitian splitting （HSS） method （Bai, Golub and Ng： SIAM J. Matrix Anal. Appl., 24（2003）, 603-626） for solving the complex linear system Ax = f. The convergence of the resulting method is proved when the spectrum of the matrix A lie in the right upper （or lower） part of the complex plane. We also derive an upper bound of the spectral radius of the HSS iteration matrix, and a estimated optimal parameter a （denoted by a^st） of this upper bound is presented. Numerical experiments on two modified model problems show that the HSS method with a est has a smaller spectral radius than that with the real parameter which minimizes the corresponding upper hound. In particular, for the ＇dominant＇ imaginary part of the matrix A, this improvement is considerable. We also test the GMRES method preconditioned by the HSS preconditioning matrix with our parameter a est.

求解广义鞍点问题的一个双参数维数分裂方法

针对一类广义鞍点问题,利用HSS迭代方法的思想,将单参数维数分裂方法推广到双参数形式。先得到双参数维数分裂迭代法的迭代格式并得到相应的求解广义鞍点问题的双参数DS分裂迭代法,然后证明了该迭代方法是收敛的,改进和推广了求解广义鞍点问题的单参数维数分裂迭代算法。数值实验也验证了双参数DS分裂迭代法比单参数MDS分裂迭代法有效。