A Newton-Kantorovich-SOR type theorem

In this paper we propose a new method for solving nonlinear systems of equations in finite dimensional spaces, combining the Newton-Raphson's method with the SOR idea. For the proof we adapt Kantorovich's demonstration given for the Newton-Raphson's method. As applications we reobtain the classical Newton-Raphson's method and the author's Newton-Kantorovich-Seidel type result.     

系统动力学模型结构分析中表函数的改进方法

分析了非线性系统动力学(SD)模型中表函数环节的传统表示方法,指出该方法对于进行SD模型的结构分析所造成的障碍,并提出了用牛顿插值的承袭性算法来改进表函数环节的表示方法的思路.接着分别介绍了用差商和差分牛顿插值法改进表函数的过程.最后,通过一个实例说明了表函数改进的方法,并展示了以此为基础进行结构分析的思路.     

Improvements of Some Projection Methods for Monotone Nonlinear Variational Inequalities

In this paper, we study the relationship of some projection-type methods for monotone nonlinear variational inequalities and investigate some improvements. If we refer to the Goldstein–Levitin–Polyak projection method as the explicit method, then the proximal point method is the corresponding implicit method. Consequently, the Korpelevich extragradient method can be viewed as a prediction-correction method, which uses the explicit method in the prediction step and the implicit method in the correction step. Based on the analysis in this paper, we propose a modified prediction-correction method by using better prediction and correction stepsizes. Preliminary numerical experiments indicate that the improvements are significant.     

An hybrid method that improves the accessibility of Steffensen’s method

Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.     

Matrix free MEBDF method for the solution of stiff systems of ODEs

In this paper, we present the details of a new method which is a hybrid method of MF-BDF and MEBDF. To obtain this new method which we call MF-MEBDF, we compose the matrix free properties of the first method and the accuracy of MEBDF, elaborately. Application of this new method to some important stiff problems show that MF-MEBDF is generally faster than MEBDF and more accurate than MF-BDF and MEBDF. Since in MEBDF, the LU factorization is used, we expect MF-MEBDF to be more efficient than the other two methods for large stiff systems of ODEs.     

Risk-averse dynamic programming for Markov decision processes

We introduce the concept of a Markov risk measure and we use it to formulate risk-averse control problems for two Markov decision models: a finite horizon model and a discounted infinite horizon model. For both models we derive risk-averse dynamic programming equations and a value iteration method. For the infinite horizon problem we develop a risk-averse policy iteration method and we prove its convergence. We also propose a version of the Newton method to solve a nonsmooth equation arising in the policy iteration method and we prove its global convergence. Finally, we discuss relations to min–max Markov decision models.     

A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing

In this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley & Sons, Ltd.     

The randomized Kaczmarz method with mismatched adjoint

This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact—a situation we refer to as “mismatched adjoint”. We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical examples.     

基于主客观信息综合判断的非线性拉开档次法

针对以最大程度的拉开被评价对象间差异的综合评价问题,通过分析线性拉开档次法存在的局限性,本文提出了一种基于主客观信息综合判断的非线性拉开档次法,旨在进一步丰富和完善拉开档次法的理论知识。首先对线性拉开档次法进行简单的介绍,并分析了线性拉开档次法的局限性;然后提出前提假设,给出非线性因子选取的原则以及确定指标序关系的方法,并给出排序后相邻指标重要程度比值的选取方法,根据评价原则利用规划模型求解各评价指标的权重系数,利用求得的指标权重采用非线性集结模型计算最终的评价结果;最后用一个算例验证了方法的有效性。     

椭圆外区域上Helmholtz问题的自然边界元法

本文研究椭圆外区域上Helmholtz方程边值问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式及自然积分方程,给出了自然积分方程的数值方法.由于计算的需要,我们详细地讨论了Mathieu函数的计算方法(当0相似文献   

A modified Lyapunov method and its applications to ODE

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations.     

求解PageRank问题的多步幂法修正的内外迭代法

引用两种加速计算PageRank的算法,分别为内外迭代法和两步分裂迭代算法.从这两种方法中,得到多步幂法修正的内外迭代方法.首先,详细介绍了算法实施过程.然后,对此算法的收敛性进行证明,并且将此算法的谱半径与两步分裂迭代算法的谱半径进行比较.最后,数值试验说明该算法的计算速度比两步分裂迭代法要快.     

On the Convergence of the Brent Method

1.IntroductionItiswellknownthattheBrentmethodforsolvingsystemsofnonlinearequati0nsistosolvethefo1lowingsystem:bymaldnguseoftheorthogonaltriangulaxfaCtoriz8tion.SupP0sethatwehaveanaPprokimationx(k)tox*,asoluti0nof(1.1).Thenthek-thiterativeprocedurecanbedescribedasfollows[1]:wherehk/Oisthedifferencestepcorrespondingtotheindexk(wewilldiscussthechoicesofhkinSecti0n4)-Constructanorthogonalmatrix(usuallybytheHouseholdtransf0rmation)Step4.Ifj相似文献   

系统聚类分析中应注意的两类问题

给出了选用九种相似性度量，用最短距离法聚类，结果互不相同的一个有趣的例子。对该例，用欧氏距离求出距离矩阵后，除用最短距离法聚类结果唯一外，用最长距离法、重心法、类平均法、离差平方和法聚类，结果均不唯一。     

Performance of two-disk partition data allocations

In this paper, we present a method, called the partition data allocation method, to allocate a set of data into a multi-disk system. Using the partition data allocation method, we shall show that, in a two-disk system, the performance is influenced by how we disperse similar data onto different disks.     

A Subspace Modified PRP Method for Large-scale Nonlinear Box-Constrained Optimization

In this article, we first propose a feasible steepest descent direction for box-constrained optimization. By the use of the direction and recently developed modified PRP method, we propose a subspace modified PRP method for box-constrained optimization. Under appropriate conditions, we show that the method is globally convergent. Numerical experiments are presented using box-constrained problems in the CUTEr test problem libraries.     

An algorithm for a selection procedure

When we have a selection problem that involves qualitative information, we can use the qualitative programming method and find the optimal choice by using integer programming. However, in practice after formulation of the problem we have a large scale model. In this note we introduce a very effective method for solving a qualitative programming problem. The only operations used in this method are scalar and matrix addition, and hence for problems of reasonable size, the method does not need high speed computers, and can be supplemented using hand calculators.     

On the Convergence of a Population-Based Global Optimization Algorithm

In global optimization, a typical population-based stochastic search method works on a set of sample points from the feasible region. In this paper, we study a recently proposed method of this sort. The method utilizes an attraction-repulsion mechanism to move sample points toward optimality and is thus referred to as electromagnetism-like method (EM). The computational results showed that EM is robust in practice, so we further investigate the theoretical structure. After reviewing the original method, we present some necessary modifications for the convergence proof. We show that in the limit, the modified method converges to the vicinity of global optimum with probability one.     

逐步删除法确定复杂系统主导结构及其应用

在流率基本入树建模法、反馈环行列式计算法和系统基模X-0-1行列式计算法的基础上,结合系统行为模拟分析的变化曲线,提出了确定系统关键变量主导结构的逐步删除法,为复杂系统的动态结构和行为模式的分析提供有效的方法.运用此方法,对"公司+农户"规模经营模式的系统关键变量主导结构进行了分析,并提出相应的管理方针,实现了用系统科学方法分析"三农"问题的目的.     

一种新的求总极值的水平值估计算法

提出了一种求解总极值问题的新水平值估计算法. 为此, 引入一类变差函数并研究它的性质; 给出基于变差函数的全局最优性条件, 并构造出一种求总极值的水平值估计算法. 为了实现这种算法, 采用了基于重点样本技术的Monte-Carlo方法来计算变差,并利用相对熵算法的主要思想更新取样密度.初步的数值实验说明了算法的有效性.