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.     

外推的MHSS迭代法求解一类大型稀疏的复对称线性系统

本文对改良的Hermitian和反Hermitian分裂迭代方法 (MHSS)使用了外推技术,构造了外推的MHSS(EMHSS)迭代法.从理论上给出了EMHSS迭代方法的迭代矩阵与MHSS迭代方法的迭代矩阵之间的关系,并讨论了EMHSS迭代方法的收敛条件.最后用数值实验验证了所提方法的有效性.     

一类弱非线性方程组的Picard-MHSS迭代方法

修正的Hermite/反Hermite分裂(MHSS)迭代方法是一类求解大型稀疏复对称线性代数方程组的无条件收敛的迭代算法.基于非线性代数方程组的特殊结构和性质,我们选取Picard迭代为外迭代方法,MHSS迭代作为内迭代方法,构造了求解大型稀疏弱非线性代数方程组的Picard-MHSS和非线性MHSS-like方法.这两类方法的优点是不需要在每次迭代时均精确计算和存储Jacobi矩阵,仅需要在迭代过程中求解两个常系数实对称正定子线性方程组.除此之外,在一定条件下,给出了两类方法的局部收敛性定理.数值结果证明了这两类方法是可行、有效和稳健的.     

带常系数的Cauchy型奇异积分方程的快速方法

Petrov-Galerkin 方法是研究Cauchy型奇异积分方程的最基本的数值方法. 用此方法离散积分方程可得一系数矩阵是稠密的线性方程组. 如果方程组的阶比较大, 则求解此方程组所需要的计算复杂度则会变得很大. 因此, 发展此类方程的快速数值算法就变成了必然. 该文将就对带常系数的Cauchy型奇异积分方程给出一种快速数值方法. 首先用一稀疏矩阵来代替稠密系数矩阵, 其次用数值积分公式离散上述方程组得到其完全离散的形式,然后用多层扩充方法求解此完全离散的线性方程组. 证明经过上述过程得到方程组的逼进解仍然保持了最优阶, 并且整个过程所需要的计算复杂度是拟线性的. 最后通过数值实验证明结论.     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.     

基于多层增量未知元方法的一类三维对流扩散方程的研究

对于一类一般形式的三维对流扩散方程, 运用有限差分方法, 在增量未知元方法(IU)下, 可以得到一个IU型正定但非对称的线性方程组.其系数矩阵条件数要远远优于不用IU方法的情形^[1]. 考虑到IU方法的这一优点, 作者在文中将IU方法与几种经典的迭代方法相结合, 来求解上述系统. 作者从理论上对该系统的IU型系数矩阵条件数进行了估计, 并通过数值试验验证了这几种IU型迭代方法的有效性.     

一个无约束优化的非单调拟牛顿型ODE方法

给出一个求解无约束优化的非单调拟牛顿型ODE方法.它的主要特点是:在每次迭代时,搜索方向仅需计算矩阵和向量的乘积就能获得,从而避免求解线性方程组系统,减少算法的计算量.然后采用一个改进的非单调线搜索以获得下一个新迭代点.在适当的条件下,该方法还是整体收敛和局部超线性收敛的.初步的数值试验结果表明了其有效性.     

求解一类分块二阶线性方程组的QHSS迭代方法

本文将QHSS迭代方法运用于求解一类分块二阶线性方程组.通过适当地放宽QHSS迭代方法的收敛性条件,我们给出了用QHSS迭代方法求解一类分块二阶线性方程组的具体迭代格式,并证明了当系数矩阵中的(1,1)块对称半正定时该QHSS迭代方法的收敛性.我们还用数值实验验证了QHSS迭代方法的可行性和有效性.     

一种求解分布控制问题的含参预处理最小残量法(英文)

近年来,提出了一些求解约束优化分布控制问题的方法,其中最常用的方法是先离散偏微分方程,然后求解离散得到的线性方程组.文献中提出了一些Krylov子空间预处理方法用来求解该线性方程组.通过分析张晓莹等提出的块对角预处理矩阵(Zhang X Y,Yan H Y.Huang Y M.On preconditionedMINRES method for solving the distributed control problems.Commun Appl Math Comput,2014,28:128-132.),构造了一个含参数的块对角预处理线性方程组,并运用含参数预处理最小残量方法求解该线性方程组.预处理矩阵的谱分析表明当参数大于1时,含参预处理线性方程组的谱分布更加集中.数值实验结果验证了含参数的预处理最小残量方法对于求解分布式控制问题是有效的.     

三阶线性常微分方程Sinc方程组的结构预处理方法

三阶线性常微分方程在天文学和流体力学等学科的研究中有着广泛的应用.本文介绍求解三阶线性常微分方程由Sinc方法离散所得到的线性方程组的结构预处理方法.首先, 我们利用Sinc方法对三阶线性常微分方程进行离散,证明了离散解以指数阶收敛到原问题的精确解.针对离散后线性方程组的系数矩阵的特殊结构, 提出了结构化的带状预处理子,并证明了预处理矩阵的特征值位于复平面上的一个矩形区域之内.然后, 我们引入新的变量将三阶线性常微分方程等价地转化为由两个二阶线性常微分方程构成的常微分方程组, 并利用Sinc方法对降阶后的常微分方程组进行离散.离散后线性方程组的系数矩阵是分块2×2的, 且每一块都是Toeplitz矩阵与对角矩阵的组合.为了利用Krylov子空间方法有效地求解离散后的线性方程组,我们给出了块对角预处理子, 并分析了预处理矩阵的性质.最后, 我们对降阶后二阶线性常微分方程组进行了一些比较研究.数值结果证实了Sinc方法能够有效地求解三阶线性常微分方程.     

Two-step modulus-based matrix splitting iteration method for linear complementarity problems

Bai has recently presented a modulus-based matrix splitting iteration method, which is a powerful alternative for solving the large sparse linear complementarity problems. In this paper, we further present a two-step modulus-based matrix splitting iteration method, which consists of a forward and a backward sweep. Its convergence theory is proved when the system matrix is an H ₊-matrix. Moreover, for the two-step modulus-based relaxation iteration methods, more exact convergence domains are obtained without restriction on the Jacobi matrix associated with the system matrix, which improve the existing convergence theory. Numerical results show that the two-step modulus-based relaxation iteration methods are superior to the modulus-based relaxation iteration methods for solving the large sparse linear complementarity problems.     

Two-Stage Multisplitting Iteration Methods Using Modulus-Based Matrix Splitting as Inner Iteration for Linear Complementarity Problems

The matrix multisplitting iteration method is an effective tool for solving large sparse linear complementarity problems. However, at each iteration step we have to solve a sequence of linear complementarity sub-problems exactly. In this paper, we present a two-stage multisplitting iteration method, in which the modulus-based matrix splitting iteration and its relaxed variants are employed as inner iterations to solve the linear complementarity sub-problems approximately. The convergence theorems of these two-stage multisplitting iteration methods are established. Numerical experiments show that the two-stage multisplitting relaxation methods are superior to the matrix multisplitting iteration methods in computing time, and can achieve a satisfactory parallel efficiency.     

Respectively scaled HSS iteration methods for solving discretized spatial fractional diffusion equations

For the discrete linear systems resulted from the discretization of the one‐dimensional anisotropic spatial fractional diffusion equations of variable coefficients with the shifted finite‐difference formulas of the Grünwald–Letnikov type, we propose a class of respectively scaled Hermitian and skew‐Hermitian splitting iteration method and establish its asymptotic convergence theory. The corresponding induced matrix splitting preconditioner, through further replacements of the involved Toeplitz matrices with certain circulant matrices, leads to an economic variant that can be executed by fast Fourier transforms. Both theoretical analysis and numerical implementations show that this fast respectively scaled Hermitian and skew‐Hermitian splitting preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods employed as effective linear solvers for the target discrete linear systems.     

On regularized Hermitian splitting iteration methods for solving discretized almost‐isotropic spatial fractional diffusion equations

The shifted finite‐difference discretization of the one‐dimensional almost‐isotropic spatial fractional diffusion equation results in a discrete linear system whose coefficient matrix is a sum of two diagonal‐times‐Toeplitz matrices. For this kind of linear systems, we propose a class of regularized Hermitian splitting iteration methods and prove its asymptotic convergence under mild conditions. For appropriate circulant‐based approximation to the corresponding regularized Hermitian splitting preconditioner, we demonstrate that the induced fast regularized Hermitian splitting preconditioner possesses a favorable preconditioning property. Numerical results show that, when used to precondition Krylov subspace iteration methods such as generalized minimal residual and biconjugate gradient stabilized methods, the fast preconditioner significantly outperforms several existing ones.     

A single-step HSS method for non-Hermitian positive definite linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of the linear system. Furthermore, we derive an upper bound for the spectral radius of the SHSS iteration matrix, and the quasi-optimal parameter is obtained to minimize the above upper bound. Numerical experiments are reported to the efficiency of the SHSS method; numerical comparisons show that the proposed SHSS method is superior to the HSS method under certain conditions.     

On sinc discretization and banded preconditioning for linear third‐order ordinary differential equations

Some draining or coating fluid‐flow problems and problems concerning the flow of thin films of viscous fluid with a free surface can be described by third‐order ordinary differential equations (ODEs). In this paper, we solve the boundary value problems of such equations by sinc discretization and prove that the discrete solutions converge to the true solutions of the ODEs exponentially. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods, such as GMRES, preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples are given to illustrate the effective performance of our method. Copyright © 2010 John Wiley & Sons, Ltd.     

An efficient numerical method for preconditioned saddle point problems

In this paper, we consider the solution of linear systems of saddle point type by a preconditioned numerical method. We first transform the original linear system into two sub-systems with small size by a preconditioning strategy, then employ the conjugate gradient (CG) method to solve the linear system with a SPD coefficient matrix, and a splitting iteration method to solve the other sub-system, respectively. Numerical experiments show that the new method can achieve faster convergence than several effective preconditioners published in the recent literature in terms of total runtime and iteration steps.     

A preconditioned general modulus-based matrix splitting iteration method for linear complementarity problems of <Emphasis Type="Italic">H</Emphasis>-matrices

In this paper, we propose a preconditioned general modulus-based matrix splitting iteration method for solving modulus equations arising from linear complementarity problems. Its convergence theory is proved when the system matrix is an H₊-matrix, from which some new convergence conditions can be derived for the (general) modulus-based matrix splitting iteration methods. Numerical results further show that the proposed methods are superior to the existing methods.     

On generalized successive overrelaxation methods for augmented linear systems

For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71–85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method. Subsidized by The Special Funds For Major State Basic Research Projects (No. G1999032803) and The National Natural Science Foundation (No. 10471146), P.R. China