求解具有多个右端项线性方程组的总体CGS算法

1 引言 在电磁场散射[1]等一些应用问题中,人们需要计算如下具有多个右端项大型非对称线性方程组的解     

一种自适应预处理的BiCRSTAB算法

提出一种自适应预处理的BiCRSTAB方法,该预处理可以看作一个隐式构造多项式的预处理方法,由BiCRSTAB算法中嵌入几步GMRES迭代自适应构造而成.数值算例表明,该方法能有效减少迭代步数,从而减少计算过程中的贮存量和运算量.     

叠压缩型不动点迭代算法的收敛速度

本文给出了叠压缩型映照不动点迭代算法的三种收敛速度,作为应用,给出了多元非线方程组解的存在性定量的一个推广。     

一种灵活的混合GMRES算法

1　引　　言考虑线性方程组Ax =b (1 .1 )其中 A∈RN× N是非奇异的 .求解方程组 (1 .1 )的很多迭代方法都可归类于多项式法 ,即满足x(n) =x(0 ) +qn- 1 (A) r(0 ) ,degqn- 1 ≤ n -1这里 x(n) ,n≥ 0为第 n步迭代解 ,r(n) =b-Ax(n) 是对应的迭代残量 .等价地 ,r(n) =pn(A) r(0 ) ,degpn≤ n;pn(0 ) =1 (1 .2 )其中 pn(z) =1 -zqn- 1 (z)称为残量多项式 .或有r(n) -r(0 ) ∈ AKn(r(0 ) ,A)其中 Kn(v,A)≡span{ Aiv} n- 1 i=0 是对应于 v,A的 Krylov子空间 .对于非对称问题 ,可以用正交性条件r(n)⊥ AKn(r(0 ) ,A)来确定 (1 .2 )中的…     

二重奇异积分近似计算的一种优化算法

本文对于矩形区域上某一内点为奇点的奇异积分的近似计算给出了优化中心数值算法，它在迭代计算过程中避免了函数值的重复计算．采用外推法减少迭代次数．     

一种生成迭代数列的新方法

围绕两个典型迭代数列的构造问题,以问题为驱动,提出一种生成迭代数列的新方法,并通过数值实验或理论证明验证迭代数列的收敛性.     

无约束广义几何规划的一种最新算法

1　引　　言近十几年来 ,几何规划新的有效数值求解方法成果很少 ,但几何规划在工程中的应用却十分广泛 ,随着线性、二次规划和非线性规划的各种新的数值方法的出现 ,必将把几何规划推向新阶段 .本文充分利用广义几何规划的特点 ,根据目标函数的梯度及 Hessian阵具有简单的特殊表达式 ,再结合信赖域算法构造了一种特殊算法 ,每次迭代只需解一类特殊的线性方程组 ,并在相对弱的条件下证明了全局收敛性和局部二次收敛性 ,具有比采用一般非线性规划求解速度快、精度高、占用内存少等优点 .考虑如下无约束的广义几何规划问题minh(t) = mj=1cj n…     

求解M-张量方程的两种新型算法

多重线性系统在当今的工程计算和数据挖掘等领域有很多实际应用,许多问题可以转化为多重线性系统求解问题.在本文中,我们首先提出了一种新的迭代算法来求解系数张量为M-张量的多重线性系统,在此基础上又提出了一种新的改进算法,并对两种算法的收敛性进行了分析.数值算例的结果表明,本文提出的两种算法是有效的并且改进算法的迭代时间更少.     

关于拟非扩张映像有限族的一种新的杂交投影算法

在Hilbert空间中针对拟非扩张映像的有限族,我们提出了一种新的杂交投影算法,使用新的分析技巧证明了算法所生成的序列强收敛于拟非扩张映像族的公共不动点,最后我们给出数值实验表明所提出的算法的有效性.     

基于信赖域技术的非单调超记忆梯度算法

基于信赖域技术和修正拟牛顿方程,结合Zhang H.C.非单调策略,设计了新的求解无约束最优化问题的非单调超记忆梯度算法,分析了算法的收敛性和收敛速度.数值实验表明算法是有效的,适于求解大规模问题.     

CGS/GMRES(k): AN ADAPTIVE PRECONDITIONED CGS ALGORITHM FOR NONSYMMETRIC LINEAR SYSTEMS

Recently Y. Saad proposed a flexible inner-outer preconditioned GMRES algorithm for nonsymmetric linear systems [4]. Following their ideas, we suggest an adaptive preconditioned CGS method, called CGS/GMRES (k), in which the preconditioner is constructed in the iteration step of CGS, by several steps of GMRES(k). Numerical experiments show that the residual of the outer iteration decreases rapidly. We also found the interesting residual behaviour of GMRES for the skewsymmetric linear system Ax = b, which gives a convergence result for restarted GMRES (k). For convenience, we discuss real systems.     

Implicitly restarted and deflated GMRES

We introduce a deflation method that takes advantage of the IRA method, by extracting a GMRES solution from the Krylov basis computed within the Arnoldi process of the IRA method itself. The deflation is well-suited because it is done with eigenvectors associated to the eigenvalues that are closest to zero, which are approximated by IRA very quickly. By a slight modification, we adapt it to the FOM algorithm, and then to GMRES enhanced by imposing constraints within the minimization condition. The use of IRA enables us to reduce the number of matrix-vector products, while keeping a low storage. This revised version was published online in June 2006 with corrections to the Cover Date.     

CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm

The Generalized Minimal Residual (GMRES) method and the Quasi-Minimal Residual (QMR) method are two Krylov methods for solving linear systems. The main difference between these methods is the generation of the basis vectors for the Krylov subspace. The GMRES method uses the Arnoldi process while QMR uses the Lanczos algorithm for constructing a basis of the Krylov subspace. In this paper we give a new method similar to QMR but based on the Hessenberg process instead of the Lanczos process. We call the new method the CMRH method. The CMRH method is less expensive and requires slightly less storage than GMRES. Numerical experiments suggest that it has behaviour similar to GMRES. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the convergence behavior of the restarted GMRES algorithm for solving nonsymmetric linear systems

The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s>0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented.     

A new adaptive GMRES algorithm for achieving high accuracy

GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications. © 1998 John Wiley & Sons, Ltd.     

基于GMRES的多项式预处理广义极小残差法

求解大型稀疏线性方程组一般采用迭代法,其中GMRES(m)算法是一种非常有效的算法,然而该算法在解方程组时,可能发生停滞．为了克服算法GMRES(m)解线性系统Ax=b过程中可能出现的收敛缓慢或不收敛,本文利用GMRES本身构造出一种有效的多项式预处理因子pk(z),该多项式预处理因子非常简单且易于实现．数值试验表明,新算法POLYGMRES(m)较好地克服了GMRES(m)的缺陷．     

A note on simultaneous preconditioning and symmetrization of non‐symmetric linear systems

Motivated by the theory of self‐duality that provides a variational formulation and resolution for non‐self‐adjoint partial differential equations (Ann. Inst. Henri Poincaré (C) Anal Non Linéaire 2007; 24 :171–205; Selfdual Partial Differential Systems and Their Variational Principles. Springer: New York, 2008), we propose new templates for solving large non‐symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well‐known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill‐conditioned, and highly non‐symmetric systems. The numerical and theoretical results are provided to show the efficiency of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Krylov subspace methods with deflation and balancing preconditioners for least squares problems

For solving least squares problems, the CGLS method is a typical method in the point of view of iterative methods. When the least squares problems are ill-conditioned, the convergence behavior of the CGLS method will present a deteriorated result. We expect to select other iterative Krylov subspace methods to overcome the disadvantage of CGLS. Here the GMRES method is a suitable algorithm for the reason that it is derived from the minimal residual norm approach, which coincides with least squares problems. Ken Hayami proposed BAGMRES for solving least squares problems in [\emph{GMRES Methods for Least Squares Problems, SIAM J. Matrix Anal. Appl., 31(2010)}, pp.2400-2430]. The deflation and balancing preconditioners can optimize the convergence rate through modulating spectral distribution. Hence, in this paper we utilize preconditioned iterative Krylov subspace methods with deflation and balancing preconditioners in order to solve ill-conditioned least squares problems. Numerical experiments show that the methods proposed in this paper are better than the CGLS method.     

DQGMRES: a Direct Quasi-minimal Residual Algorithm Based on Incomplete Orthogonalization

We describe a Krylov subspace technique, based on incomplete orthogonalization of the Krylov vectors, which can be considered as a truncated version of GMRES. Unlike GMRES(m), the restarted version of GMRES, the new method does not require restarting. Like GMRES, it does not break down. Numerical experiments show that DQGMRES(k) often performs as well as the restarted GMRES using a subspace of dimension m=2k. In addition, the algorithm is flexible to variable preconditioning, i.e., it can accommodate variations in the preconditioner at every step. In particular, this feature allows the use of any iterative solver as a right-preconditioner for DQGMRES(k). This inner-outer iterative combination often results in a robust approach for solving indefinite non-Hermitian linear systems.     

Dynamic block GMRES: an iterative method for block linear systems

We present variants of the block-GMRES() algorithms due to Vital and the block-LGMRES(,) by Baker, Dennis and Jessup, obtained with replacing the standard QR factorization by a rank-revealing QR factorization in the Arnoldi process. The resulting algorithm allows for dynamic block deflation whenever there is a linear dependency between the Krylov vectors or the convergence of a right-hand-side occurs. implementations of the algorithms were tested on a number of test matrices and the results show that in some cases a substantial reduction of the execution time is obtained. Also a parallel implementation of our variant of the block-GMRES() algorithm, using and was tested on parallel computer, showing good parallel efficiency. This work was carried out while the author was at IM/UFRGS.