关于正定可对称化线性代数方程组的预对称正则化算法

1引言考虑线性代数方程组A_x=b,A∈R~(n×n)非奇异,x,b∈R~n(1)的求解．当系数矩阵是大型稀疏的正定可对称化矩阵,文[1,2]讨论了一类预对称共轭梯度算法(LRSCG算法是其中之一),这类算法的实质是利用非对称的系数矩阵可对称化的性质,并结合共轭梯度法而构造的一种预处理的共轭梯度法[12,16,17]．但非对称的系数     

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．     

一种带参数的不完全Cholesky分解共轭梯度法

共轭梯度法在解高阶稀疏线性方程组方面有许多其它经典的迭代法所没有的优点,但当线性方程组相当病态、系数矩阵条件数很坏时,共轭梯度法的收敛速度很慢.因此,又产生了预条件处理共轭梯度法. 我们用预条件处理共轭梯度法求解线性方程组Ax=b(这里A是对称正定稀疏阵且条件数很大).预条件处理共轭梯度法旨在寻找一适当的正定矩阵C,C通常写成     

对称正定矩阵的多级迭代法

本文主要研究解对称正定矩阵的多级迭代法,并对其收敛性进行证明。最后用数值实验验证此方法的有效性．多级迭代法特别适用于并行计算,并且可以被理解为古典迭代法的扩展,或共轭梯度法的预处理子。     

一种求解鞍点问题的预处理并行算法

研究了一种求解鞍点问题的并行预处理变形共轭梯度算法.通过应用迭代法进行预处理后,再采用变形共轭梯度求解的模式.首先构造系数矩阵近似逆的多项式表达式,以此作为预处理矩阵的逆矩阵,对方程组进行预处理;然后采用变形共轭梯度法并行求解预处理后的线性方程组.为减少运算量,采用迭代方式并行计算多项式与向量的乘法运算.通过调整迭代次数,即调整多项式次数,检验各种次数的多项式进行预处理后的求解方程的效果.数值试验结果表明,该算法明显优于未预处理的变形共轭梯度法,且当预处理迭代次数取4时效果最好.     

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     

一种修正的HS共轭梯度法及全局收敛性

<正>1引言考虑无约束极小化问题:(?),(1)其中f(x)连续可微,其梯度函数用g(x)表示.共轭梯度法求解(1)的常用迭代格式为:x_(k+1)=x_k+α_kd_k,(2)(?)(3)其中g_k=▽f(x_k),α_k≥0是由某种线搜索得到的步长因子;d_k为搜索方向,β_k为标量,β_k的不同选择产生了不同的共轭梯度法.著名的β_k公式有:     

一种求解大型Lyapunov矩阵方程的预处理并行算法

研究了一种求解大型Lyapunov矩阵方程的并行预处理变形共轭梯度法.首先将处理小型矩阵方程的Smith预处理方法引入该问题的求解,将原矩阵方程转变为Stein方程,然后采用变形共轭梯度法并行求解预处理后的矩阵方程.其中遇到的难点是需要确定参数μ及求矩阵(A+μI)的逆.基于估计特征值的Gerschgorin圆定理给出了参数μ的估值,再采用变形共轭梯度法并行求得矩阵(A +μ l)的逆,从而形成预处理后的矩阵方程.通过数值试验,该算法与未预处理的变形共轭梯度法相比较,预处理算法明显优于未预处理的算法,而且其并行效率高达0.85.     

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.     

用块超松弛迭代法求解不定最小二乘问题

应用块对称超松弛(symmetric successive overrelaxation,SSOR)和块加速超松弛(accelerated overrelaxation,AOR)迭代法来解不定最小二乘问题,并分析两种算法的收敛性和最佳松弛因子.理论分析表明,尽管最佳的SSOR方法比最佳的AOR方收敛慢,但其最佳松弛因子取法更简单.数值算例验证了相应的理论结果.     

基于自适应遗传算法的逐次超松驰迭代法

确定逐次超松驰迭代法中的最佳松驰因子,迄今,人们还没有给出一可行实用的方法.利用自适应遗传算法全局搜索性能、并行性及其遗传操作,构造出近似确定最佳松驰因子的一种自适应进化方法,并由此得到一近似确定ω功能的自适应逐次超松驰迭代算法.数值算例表明,该算法在求解线性方程组中是可行的,实用和快捷的.     

Fuzziness index driven fuzzy relaxation algorithm and applications to image processing

An improved fuzzy relaxation algorithm for image contrast enhancement is introduced, the relationship between the convergence regions and the parameters in the transformations defined by the algorithm is shown, which is essential to the successful application of this algorithm. Furthermore, in order to measure the quality of an enhanced image, an index of fuzziness is used in this paper to evaluate the performance of the fuzzy relaxation scheme. This extended index of fuzziness is used as a criterion for automatically stopping the fuzzy relaxation process. The analytical result is tested by experiments of image contrast enhancement.     

带交易费用的离散多因素投资组合最优化

研究带有凹的交易费函数的离散多因素投资组合模型．与传统的投资组合模型不同的是,该模型中投资组合的决策变量是交易手数（整数）,其最优化模型是一个非线性整数规划问题．为此本文提出了一个基于拉格朗日松弛和连续松弛的混合分枝定界算法,为测试算法的有效性,我们分别采用美国股票市场真实数据和随机产生的数据,数值结果表明该算法是有效的．     

带交易费用的离散多因素投资组合最优化(英文)

研究带有凹的交易费函数的离散多因素投资组合模型.与传统的投资组合模型不同的是,该模型中投资组合的决策变量是交易手数(整数),其最优化模型是一个非线性整数规划问题.为此本文提出了一个基于拉格朗日松弛和连续松弛的混合分枝定界算法,为测试算法的有效性,我们分别采用美国股票市场真实数据和随机产生的数据,数值结果表明该算法是有效的.     

灰色正项几何规划研究

In the model of geometric programming, values of parameters cannot be gotten owing to data fluctuation and incompletion. But reasonable bounds of these parameters can be attained. This is to say, parameters of this model can be regarded as interval grey numbers. When the model contains grey numbers, it is hard for common programming method to solve them. By combining the common programming model with the grey system theory, and using some analysis strategies, a model of grey polynomial geometric programming, a model of θpositioned geometric programming and their quasi-optimum solution or optimum solution are put forward. At the same time, we also developed an algorithm for the problem. This approach brings a new way for the application research of geometric programming. An example at the end of this paper shows the rationality and feasibility of the algorithm.     

Optimal infrastructure design and expansion of broadband wireless access networks

This paper investigates the important infrastructure design and expansion problem for broadband wireless access networks subject to user demand constraints and system capacity constraints. For the problem, an integer program is derived and a heuristic solution procedure is proposed based on Lagrangean relaxation. In the computational experiments, our Lagrangean relaxation based algorithm can solve this complex design and expansion problem quickly and near optimally. Based on the test results, it is suggested that the proposed algorithm may be practically used for the infrastructure design and expansion problem for broadband wireless access networks.     

Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman-Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman-Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.     

离散单因素投资组合模型的对偶算法

本文研究金融优化中的离散单因素投资组合问题,该问题与传统投资组合模型的不同之处是决策变量为整数(交易手数),从而导致要求解一个二次整数规划问题．针对该模型的可分离性结构,我们提出了一种基于拉格朗日对偶和连续松弛的分枝定界算法。我们分别用美国股票市场的交易数据和随机产生的数据对算法进行了测试．数值结果表明该算法是有效的,可以求解多达150个风险证券的离散投资组合问题．     

Identification of the relaxation kernel in diffusion processes and viscoelasticity with memory via deconvolution

We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body. The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.     

求置换因子循环矩阵的逆阵及广义逆阵的快速算法

1 引 言 循环矩阵由于其应用非常广泛而成为一类重要的特殊矩阵,如在图象处理、编码理论、自回归滤波器设计等领域中经常会遇到以这类矩阵为系数的线性系统的求解问题.而对称循环组合系统也具有广泛的实际背景,例如造纸机的横向控制系统,具有平行结