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.     

关于PageRank的广义二级分裂迭代方法

本文研究计算PageRank的迭代法,在Gleich等人提出的内/外迭代方法的基础上,提出了具有三个参数的广义二级分裂迭代法,该方法包含了内/外迭代法和幂迭代法,并研究了该方法的收敛性.基于该方法的收缩因子的计算公式,讨论了迭代参数可能的选择,通过参数的选择能有效提高内/外迭代法的收敛效率.     

关于Katz指标的二级分裂迭代方法

本文研究复杂网络中计算Katz指标的迭代法,基于网络拓扑结构,在快速Katz指标算法的基础上,运用二级分裂迭代思想,提出了具有两个参数的二级分裂迭代法,并研究了该方法的收敛性.基于该方法的收缩因子的计算公式,讨论了迭代参数可能的选择,通过参数的选择能有效提高二级迭代法的收敛效率.最后通过数值实例验证了此方法的有效性.     

对称正定Toeplitz方程组的多级迭代求解

二级迭代法亦称内外迭代法. 多级迭代法由多个二级迭代嵌套而成.这些方法特别适合于并行计算,同时可以理解为古典迭代法的延伸或共轭梯度法的预处理子.本文讨论了对称正定Toeplitz线性方程组多级迭代法. 首先,基于Toeplitz矩阵的结构, 我们给出了多级块Jacobi分裂,然后证明了每一级分裂均为P-正则分裂, 并证明了当每一级内迭代次数均为偶数时,迭代法的收敛性. 最后通过数值实例验证了此方法的有效性.     

一类H矩阵线性互补问题的预处理二步模基矩阵分裂迭代方法

本文我们利用预处理技术推广了求解线性互补问题的二步模基矩阵分裂迭代法,并针对H-矩阵类给出了新方法的收敛性分析,得到的理论结果推广了已有的一些方法.     

H-矩阵线性方程组的一类预条件并行多分裂SOR迭代法

基于并行多分裂算法的思想及SOR迭代格式, 本文提出一种求解H-矩阵线性方程组新的并行多分裂SOR迭代法, 新方法某种程度上避免了SOR迭代法中选取最优参数的困难. 同时, 选取Kohno等(1997)提出的预条件子$P=I+S_{\alpha}$对原始线性方程组进行预处理, 进而给出了一种实用的预条件并行多分裂SOR迭代法. 理论分析和数值实验均表明, 新算法是实用而有效的.     

水平线性互补问题投影迭代法和模系矩阵分裂迭代法的等价性

水平线性互补问题(HLCP)是著名线性互补问题(LCP)的重要推广形式之一,投影迭代法和模系矩阵分裂迭代法是最近提出的求解HLCP两类非常有效的热点方法.本文研究表明,尽管这两类方法导出原理不同,但在一定条件下是等价的.特别地,当模系矩阵分裂迭代法中参数矩阵Ω取为特定的正对角矩阵时,投影Jacobi法、投影Gauss-Seidel法和投影SOR法分别等价于模系Jacobi迭代法、加速的模系Gauss-Seidel迭代法和加速的模系SOR迭代法.此外,对一般的正对角矩阵Ω,本文也研究了两类方法的等价性.最后,通过数值算例验证了本文的理论结果.     

关于Stokes和线性Navier-Stokes方程的广义维数分裂迭代方法

本文研究了鞍点问题的迭代法.在Benzi等人提出的维数分裂(DS)迭代方法的基础上,提出了具有三个参数的广义维数分裂(GDS)迭代法,该方法包含了DS迭代法,理论分析表明该方法是无条件收敛的.通过对有限差分法和有限元法离散的Stokes问题及有限元法离散的Oseen问题的数值结果表明,本文所给方法是有效的.     

一种求解鞍点问题的广义对称超松弛迭代法

本文研究了鞍点问题的迭代算法.利用新的待定参数加速迭代格式并结合SSOR分裂的方法,获得了有两个参数的广义对称超松弛迭代法及其收敛性条件.数值例子表明选择适当的参数值可以提高算法的收敛效率,推广和改进了SOR-like迭代法.     

求解时谐涡流模型鞍点问题的分块交替分裂隐式迭代算法的改进

分块交替分裂隐式迭代方法是求解具有鞍点结构的复线性代数方程组的一类高效迭代法.本文通过预处理技巧得到原方法的一种加速改进方法,称之为预处理分块交替分裂隐式迭代方法·理论分析给出了新方法的收敛性结果.对于一类时谐涡旋电流模型问题,我们给出了若干满足收敛条件的迭代格式.数值实验验证了新型算法是对原方法的有效改进.     

On a type of matrix splitting preconditioners for a class of block two-by-two linear systems

Recently, Bai et al. (2013) proposed an effective and efficient matrix splitting iterative method, called preconditioned modified Hermitian/skew-Hermitian splitting (PMHSS) iteration method, for two-by-two block linear systems of equations. The eigenvalue distribution of the iterative matrix suggests that the splitting matrix could be advantageously used as a preconditioner. In this study, the CGNR method is utilized for solving the PMHSS preconditioned linear systems, and the performance of the method is considered by estimating the condition number of the normal equations. Furthermore, the proposed method is compared with other PMHSS preconditioned Krylov subspace methods by solving linear systems arising in complex partial differential equations and a distributed control problem. The numerical results demonstrate the difference in the performance of the methods under consideration.     

Convergences of splitting iterative methods for symmetric indefinite linear systems

In this paper, we generalize the saddle point problem to general symmetric indefinite systems, we also present a kind of convergent splitting iterative methods for the symmetric indefinite systems. A special divergent splitting is introduced. The sufficient condition is discussed that the eigenvalues of the iteration matrix are real. The spectral radius of the iteration matrix is discussed in detail, the convergence theories of the splitting iterative methods for the symmetric indefinite systems are obtained. Finally, we present a preconditioner and discuss the eigenvalues of preconditioned matrix.     

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

Based on the HSS (Hermitian and skew-Hermitian splitting) and preconditioned HSS methods, we will present a generalized preconditioned HSS method for the large sparse non-Hermitian positive definite linear system. Our method is essentially a two-parameter iteration which can extend the possibility to optimize the iterative process. The iterative sequence produced by our generalized preconditioned HSS method can be proven to be convergent to the unique solution of the linear system. An exact parameter region of convergence for the method is strictly proved. A minimum value for the upper bound of the iterative spectrum is derived, which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting. An efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method. The optimality and efficiency are effectively testified by some comparisons with numerical results.     

求解2×2块对称不定线性方程组迭代法的收敛性分析(英文)

本文研究求解系数矩阵为2×2块对称不定矩阵时的线性方程组,提出了一种新的分裂迭代法,并通过研究迭代矩阵的谱半径,详细讨论了新方法的收敛性.最后,我们也讨论了预条件矩阵特征根的几条性质.     

一类线性抛物型方程组的交替方向多步法及其理论分析

Efficient multistep procedure for time-stepping Galerkin method in which we use an alternating direction preconditioned iterative methods for approximately solving the linear equations arising at each timestep in a discrete Galerkin method for a class of linear parabolic systems is derived and analyzed. The optimal order error estimate is obtained. Numerical experiments show that the method has the characteristics of high efficiency and high accuracy.     

Sine transform based preconditioning techniques for space fractional diffusion equations

We study the preconditioned iterative methods for the linear systems arising from the numerical solution of the multi-dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed according to the symmetric and skew-symmetric splitting of the Toeplitz factor in the resulting coefficient matrix. Theoretical analyses show that the upper bound of relative residual norm of the GMRES method when applied to the preconditioned linear system is mesh-independent which implies the linear convergence. Numerical experiments are carried out to illustrate the correctness of the theoretical results and the effectiveness of the proposed preconditioning technique.     

PAHSS-PTS alternating splitting iterative methods for nonsingular saddle point problems

In this paper, we propose the PAHSS-PTS alternating splitting iterative methods for nonsingular saddle point problems. Convergence properties of the proposed methods are studied and corresponding convergence results are given under some suitable conditions. Numerical experiments are presented to confirm the theoretical results, which impliy that PAHSS-PTS iterative methods are effective and feasible.     

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.     

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.