基于Lanczos双A-正交的一种修正的QMR算法

本文研究了基于Lanczos双正交过程的拟极小残量法(QMR).将QMR算法中的Lanczos双正交过程用Lanczos双A-正交过程代替,利用该算法得到的近似解与最后一个基向量的线性组合来作为新的近似解,使新近似解的残差范数满足一个一维极小化问题,从而得到一种基于Lanczos双A-正交的修正的QMR算法.数值试验表明,对于某些大型线性稀疏方程组,新算法比QMR算法收敛快得多.     

广义特征值极小扰动问题的一类黎曼共轭梯度法

研究含参数$l$非方矩阵对广义特征值极小扰动问题所导出的一类复乘积流形约束矩阵最小二乘问题.与已有工作不同,本文直接针对复问题模型,结合复乘积流形的几何性质和欧式空间上的改进Fletcher-Reeves共轭梯度法,设计一类适用于问题模型的黎曼非线性共轭梯度求解算法,并给出全局收敛性分析.数值实验和数值比较表明该算法比参数$l=1$的已有算法收敛速度更快,与参数$l=n$的已有算法能得到相同精度的解.与部分其它流形优化相比与已有的黎曼Dai非线性共轭梯度法具有相当的迭代效率,与黎曼二阶算法相比单步迭代成本较低、总体迭代时间较少,与部分非流形优化算法相比在迭代效率上有明显优势.     

Armijo线性搜索下Hager-Zhang共轭梯度法的全局收敛性

Hager和Zhang^[4]提出了一种新的非线性共轭梯度法(简称 HZ 方法), 并证明了该方法在 Wolfe搜索和 Goldstein 搜索下求解强凸问题的全局收敛性.但是HZ方法在标准Armijo 搜索下求解非凸问题是否全局收敛尚不清楚.该文提出了一种保守的HZ共轭梯度法,并且证明了这种方法在 Armijo 线性搜索下求解非凸优化问题的全局收敛性.此外,作者给出了一些 数值结果以检验该方法的有效性.     

一个拟极小化左共轭梯度算法及其数值表现

左共轭梯度法是求解大型稀疏线性方程组的一种新兴的Krylov子空间方法.为克服该算法数值表现不稳定、迭代中断的缺点,本文对原方法进行等价变形,得到左共轭梯度方向的另一迭代格式,给出一个拟极小化左共轭梯度算法.数值结果证实了该变形算法与原算法的相关性.     

等几何分析的多重网格共轭梯度法

提高NURBS基函数阶数可以提高等几何分析的精度,同时也会降低多重网格迭代收敛速度.将共轭梯度法与多重网格方法相结合,提出了一种提高收敛速度的方法,该方法用共轭梯度法作为基础迭代算法,用多重网格进行预处理.对Poisson(泊松)方程分别用多重网格方法和多重网格共轭梯度法进行了求解,计算结果表明:等几何分析中采用高阶NURBS基函数处理三维问题时,多重网格共轭梯度法比多重网格法的收敛速度更快.     

建立在割线条件上的新Liu-Storey型共轭梯度法的全局收敛性(英文)

我们基于拟牛顿法的割线条件提出两种LS型共轭梯度法.有趣的是,我们提出的方法中对于βk的计算公式与戴和廖[3]提出的有相似的结构.但是,新方法能够在合理的假设下保证充分下降性,这一点是戴-廖方法所不具备的.在强Wolfe线搜索下,给出了新方法的全局收敛结果.数值结果论证了该方法的有效性.     

改进共轭梯度算法在矿井瓦斯含量预测中的应用

本为预测矿井瓦斯含量,根据影响矿井瓦斯含量的煤层开采深度、煤层厚度、瓦斯压力、煤的变质程度、煤层顶板岩性与煤层底板岩性等主要因素建立三层BP神经网络分析模型.针对标准BP算法存在的收敛速度慢、容易陷入局部极小等问题,从理论分析角度对共轭梯度算法和改进共轭梯度算法进行对比分析研究,并且分别用标准BP算法、共轭梯度算法和改进共轭梯度算法对BP神经网络分析模型进行训练和测试.结果表明,改进共轭梯度算法收敛速度快,预测结果相对误差保持在1%以内,并且误差波动相对平稳.因此,基于改进共轭梯度算法的BP神经网络分析模型,能够有效预测矿井瓦斯含量.     

预处理技术与PCG算法

1 共轭梯度法 1952年M·R·Hestenes和E.Stiefel从极小化的观点来讨论代数方程组Ax=b的解,给出了著名的共轭梯度法(Conjugate Gradient,简称CG).若A是N阶对称正定实矩阵,记向量x和y的内积为(x,y),定义x的二次函数     

求一类非凸规划问题全局解的确定性算法

本文对一类非凸规划问题(NP)给出一确定性全局优化算法.这类问题包括:在非凸的可行域上极小化有限个带指数的线性函数乘积的和与差,广义线性多乘积规划,多项式规划等.通过利用等价问题和线性化技巧提出的算法收敛到问题(NP)的全局极小.     

一类特殊矩阵方程的并行预处理变形共轭梯度算法

研究了求解一类矩阵方程AXB=C,提出了一种并行预处理变形共轭梯度法．该方法给出一种迭代法的预处理模式．首先给出的预处理矩阵是严格对角占优矩阵,构造并行迭代求解预处理矩阵方程的迭代格式,进而使用变形共轭梯度法并行求解．通过数值试验,预处理变形共轭梯度法与直接使用变形共轭梯度法相比较,该算法不仅有效提高了收敛速度,而且具有很高的并行性．     

By How Much Can Residual Minimization Accelerate the Convergence of Orthogonal Residual Methods?

We capitalize upon the known relationship between pairs of orthogonal and minimal residual methods (or, biorthogonal and quasi-minimal residual methods) in order to estimate how much smaller the residuals or quasi-residuals of the minimizing methods can be compared to those of the corresponding Galerkin or Petrov–Galerkin method. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRES) method, the CGNE and BiCG versions of applying CG to the normal equations, as well as the biconjugate gradient (BiCG) and the quasi-minimal residual (QMR) methods. Also the pairs consisting of the (bi)conjugate gradient squared (CGS) and the transpose-free QMR (TFQMR) methods can be added to this list if the residuals at half-steps are included, and further examples can be created easily.The analysis is more generally applicable to the minimal residual (MR) and quasi-minimal residual (QMR) smoothing processes, which are known to provide the transition from the results of the first method of such a pair to those of the second one. By an interpretation of these smoothing processes in coordinate space we deepen the understanding of some of the underlying relationships and introduce a unifying framework for minimal residual and quasi-minimal residual smoothing. This framework includes the general notion of QMR-type methods.     

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.     

关于具优势对称部分的不定线性代数方程组的分裂极小残量算法

１．引 言 考虑大型稀疏线性代数方程组 为利用系数矩阵的稀疏结构以尽可能减少存储空间和计算开销，Ｋｒｙｌｏｖ子空间迭代算法［１，１６，２３］及其预处理变型［６，８，１３，１８，１９］通常是求解（１）的有效而实用的方法．当系数矩阵对称正定时，共轭梯度法（ＣＧ（     

一种新型压缩预处理CGS算法

CGS算法是求解大型非对称线性方程组的常用算法，然而该算法无极小残差性质，因此它常因出现较大的中间剩余向量而出现典型的不规则收敛行为．本根据IRA方法提出了一种压缩预处理CGS方法，数值实验表明这种算法在一定程度上减小了迭代算法在收敛过程中的剩余问题，从而使得算法具有更好的稳定性，该法构造简单，减少了收敛次数，加快了收敛速度．     

On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling

Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix—vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.     

A QUASI-MINIMAL RESIDUAL VARIANT OF THE IOM (q) FOR LARGE UNSYMMETRIC LINEAR SYSTEMS

1　IntroductionThe solution of large N× N nonsingular unsymmetric( non-Hermitian) sparse sys-tems of linear equationsAx =b, ( 1 )is one of the most frequently encountered tasks in numerical computations.For example,such systems arise from finite difference or finite element approximations to partial differ-ential equationsA major class of methods for solving ( 1 ) is Krylov subspace or conjugate gradienttype methods.Most successful scheme of these methods are based on the orthogonal pro-jec…     

GPBi‐CGstab(L): A Lanczos‐type product method unifying Bi‐CGstab(L) and GPBi‐CG

Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab(L) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab(L) uses stabilizing polynomials of degree L, while GPBi‐CG uses polynomials given by a three‐term recurrence (or equivalently, a coupled two‐term recurrence) modeled after the Lanczos residual polynomials. Therefore, Bi‐CGstab(L) and GPBi‐CG have different aspects of generalization as a framework of LTPMs. In the present paper, we propose novel stabilizing polynomials, which combine the above two types of polynomials. The resulting method is referred to as GPBi‐CGstab(L). Numerical experiments demonstrate that our presented method is more effective than conventional LTPMs.     

A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems

We propose a new and more stable variant of the CGS method [27] for solving nonsymmetric linear systems. The method is based on squaring the Composite Step BCG method, introduced recently by Bank and Chan [1,2], which itself is a stabilized variant of BCG in that it skips over steps for which the BCG iterate is not defined and causes one kind of breakdown in BCG. By doing this, we obtain a method (Composite Step CGS or CSCGS) which not only handles the breakdowns described above, but does so with the advantages of CGS, namely, no multiplications by the transpose matrix and a faster convergence rate than BCG. Our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates. Numerical experiments show that the new method does produce improved performance over CGS on practical problems.Partially supported by the Office of Naval Research grant N00014-92-J-1890, the National Science Foundation grant ASC92-01266, and the Army Research Office grant DAAL03-91-G-150.     

Quasi-minimal abelian groups

An abelian group is said to be quasi-minimal (purely quasi-minimal, directly quasi-minimal) if it is isomorphic to all its subgroups (pure subgroups, direct summands, respectively) of the same cardinality as . Obviously quasi-minimality implies pure quasi-minimality which in turn implies direct quasi-minimality, but we show that neither converse implication holds. We obtain a complete characterisation of quasi-minimal groups. In the purely quasi-minimal case, assuming GCH, a complete characterisation is also established. An independence result is proved for directly quasi-minimal groups.

     

Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations

This article is concerned with solving the high order Stein tensor equation arising in control theory. The conjugate gradient squared (CGS) method and the biconjugate gradient stabilized (BiCGSTAB) method are attractive methods for solving linear systems. Compared with the large-scale matrix equation, the equivalent tensor equation needs less storage space and computational costs. Therefore, we present the tensor formats of CGS and BiCGSTAB methods for solving high order Stein tensor equations. Moreover, a nearest Kronecker product preconditioner is given and the preconditioned tensor format methods are studied. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples.