COMPACT DIFFERENCE SCHEMES FOR THE DIFFUSION AND SCHRODINGER EQUATIONS. APPROXIMATION,STABILITY, CONVERGENCE,EFFECTIVENESS, MONOTONY

Various compact difference schemes （both old and new, explicit and implicit, one-level and two-level）, which approximate the diffusion equation and SchrSdinger equation with periodical boundary conditions are constructed by means of the general approach. The results of numerical experiments for various initial data and right hand side are presented. We evaluate the real order of their convergence, as well as their stability, effectiveness, and various kinds of monotony. The optimal Courant number depends on the number of grid knots and on the smoothness of solutions. The competition of various schemes should be organized for the fixed number of arithmetic operations, which are necessary for numerical integration of a given Cauchy problem. This approach to the construction of compact schemes can be developed for numerical solution of various problems of mathematical physics.     

随机延迟微分方程的数值方法的收敛性和稳定性

本文研究了随机延迟微分方程的平衡方法的收敛性和均方稳定性.利用半鞅收敛定理,给出了真解的渐进稳定和均方稳定的一个更弱的条件.平衡方法下随机延迟微分方程的真解的均方稳定性.     

具有线性收获项的Nicholson's Blowflies差分模型正概周期解的存在唯一性与指数收敛性(英文)

本文研究具有线性收获项的Nicholson's Blowflies差分模型,运用压缩映射不动点定理,获得存在唯一的正概周期解的充分条件.此外,通过利用Liapunov泛函研究正概周期解的指数收敛性,解决了L.Berezansky 2010年提出的一个公开问题.     

关于渐近非扩张映射的Cesàro意义下的修正Ishikawa迭代(英文)

设E是一致凸Banach空间,且具有一致Gteaux可微范数,C是E的一个非空闭凸子集,T是渐近非扩张映射.对于任意x∈C,本文引入Cesàro意义上的修正Ishikawa迭代:x0=x∈C,yn=γnun+δnxn+(1-γn-δ)n+11∑j=0nTjxn,xn+1=μnvn+αnγf(xn)+βnxn+[(1-μn-βn)I-αnA]n+11∑j=0nTjyn,n≥0在适当的条件下证明此迭代序列的强(弱)收敛性.     

带参数共轭梯度法簇的全局收敛性

共轭梯度法是最优化中最常用的方法之一,广泛地应用于求解大规模优化问题,其中参数β_k的不同选取可以构成不同的共轭梯度法.给出了一类含有三个参数的共轭梯度算法,这种算法能够在给定的条件下证明选定的β_k在每一步都能产生一个下降方向,同时在强Wolfe线搜索下,这种算法具有全局收敛性.     

矩阵方程X~s+A~*X~(-t)A=Q的无逆迭代解法

文中给出了求解矩阵方程Xs+A*X-tA=Q的最小极值正定解的无逆迭代法,证明了算法的收敛性,并给出了说明算法有效性的数值例子.     

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.     

基于一阶Takagi-Sugeno系统的Pi-Sigma网络的改进神经-模糊梯度学习算法

This paper presents a Pi-Sigma network to identify first-order Tagaki-Sugeno（T-S） fuzzy inference system and proposes a simplified gradient-based neuro-fuzzy learning algorithm.A comprehensive study on the weak and strong convergence for the learning method is made,which indicates that the sequence of error function goes to a fixed value,and the gradient of the error function goes to zero,respectively.     

具有变量自由参数的分形插值曲面的构造与性质

在矩形区域上使用具有变量自由参数的迭代函数系,构造了一类新的分形插值曲面,并研究了这类曲面的若干性质.证明了这类变参数的迭代函数系能生成连续的二元分形插值曲面,指出该类曲面可用作任意数据点上的自仿射和非自仿射的分形插值模型.在两种度量意义下,导出了能刻画这类分形曲面敏感性的一些不等式.给出了相应的分形插值函数与数据生成函数之间的误差估计.最后,在一定条件下,证明了这类分形插值函数序列一致收敛于数据生成函数.     

求解非线性最小二乘问题的迭代法

基于解线性方程组的Jacobi迭代法的思想，本文提出了一种求解非线性最小二乘问题的迭代法，大量的数值试验表明该迭代法具有较好的收敛性，另外，还将所设计的迭代法进行了并行化。