Bochner-Lebesgue空间内的最佳同时逼近（英文）

在本文中,我们研究了Bochner-Lebesgue空间内的相对于欧氏空间的Minkowski范数的最佳同时逼近.首先,给出了由距离函数表示的最佳同时逼近的刻画.然后,利用可测选择定理证明其函数取值于一个闭的可分子空间的Bochner-Lebesgue空间,其同时可逼近性等价于此闭的可分子空间的同时可逼近性.最后,指出子空间的可分性是同时可逼近性等价的必要条件.     

关于Lagrange插值逼近中几个问题研究的新进展

本文将综述 L agrange插值逼近中几个问题研究的新进展 ,并提出几个新问题 .主要的论题是 Lagrange插值多项式序列的收敛与发散 ,用 L agrange插值多项式同时逼近可微分函数及其导数 ,以及修改的 L agrange插值多项式对函数的逼近     

关于用Bernstein型插值同时逼近的注记

本文给出了用 Bernstein 型插值同时逼近函数及其导数的相当简单的证明方法,并建立了一般的逼近定理。     

一个二元周期函数类用三角多项式的逼近

得到了二元周期可微函数类△^rp在空间Lq尺度下用三角多项式的最佳逼近精确阶。     

关于Szász-Durrmeyer-Bézier算子的点态逼近

Bézier型算子是一些著名算子的推广,已有研究成果主要是对有界变差函数的逼近,而对于应用光滑模研究其中心逼近定理的结果很少.本文利用一阶Ditzian-Totik模得到了Szász-Durrmeyer-Bézier算子点态逼近的正、逆定理及等价定理这一完美的逼近结果.     

关于Bernstein多项式的同时逼近

本文研究函数f的Bernstein多项式Bn（f，X）对函数及其导数的同时逼近，对于fP≥1，给出的渐近展开式．     

向量值函数的最佳Lp逼近

本文主要讨论向量值函数空间L,(μ,x)内的元素用凸子集逼近时最佳逼近的特征。并指出它在联合最佳逼近上应用。     

一类神经网络逼近可积函数

用连续模刻画了实轴上Cardaliguet-Eurrard型神经网络算子逼近连续函数速度的上界估计,同时,对于Lebesgue可积函数的逼近,构造相应的神经网络算子,并且给出其逼近速度的Jackson型估计.     

函数最佳平方逼近多项式是否唯一

本文对最佳一致逼近多项式与最佳平方逼近多项式相互比较,给出了相同的内积和基函数最佳平方逼近多项式是唯一,并且验证了勒让德展开部分和得到的最佳平方逼近多项式与Hn中的最佳平方逼近多项式是一致的.     

代数双曲空间中拟Legendre基的应用

鉴于Legendre基等正交基在代数多项式空间中的广泛应用,论文在深入研究代数双曲空间的拟Legendre基性质的基础上,给出了其在反函数逼近和等距曲线逼近上的应用。利用多项式和双曲函数的混合多项式序列来逼近反函数,并通过实例证明给出方法的有效性;对基曲线的法矢曲线进行逼近,构造H-Bézier曲线的等距曲线的最佳逼近,这种方法直接求得逼近曲线的控制顶点,计算简单,截断误差小。     

On the limitations of low-rank approximations in contact mechanics problems

Typical strategies for reducing the computational cost of contact mechanics models use low-rank approximations. The underlying hypothesis is the existence of a low-dimensional subspace for the displacement field and a non-negative low-dimensional subcone for the contact pressure. However, given the local nature of contact, it seems natural to wonder whether low-rank approximations are a good fit for contact mechanics or not. In this article, we investigate some of their limitations and provide numerical evidence showing that contact pressure is linearly inseparable in many practical cases. To this end, we consider various mechanical problems involving nonadhesive frictionless contacts and analyze the performance of the low-rank models in terms of three different criteria, namely, compactness, generalization, and specificity.     

Non‐convex dual forms based on exponential intervening variables,with application to weight minimization

We study the weight minimization problem in a dual setting. We propose new dual formulations for non‐linear multipoint approximations with diagonal approximate Hessian matrices, which derive from separable series expansions in terms of exponential intervening variables. These, generally, nonconvex approximations are formulated in terms of intervening variables with negative exponents, and are therefore applicable to the solution of the weight minimization problem in a sequential approximate optimization (SAO) framework. Problems in structural optimization are traditionally solved using SAO algorithms, like the method of moving asymptotes, which require the approximate subproblems to be strictly convex. Hence, during solution, the nonconvex problems are approximated using convex functions, and this process may in general be inefficient. We argue, based on Falk's definition of the dual, that it is possible to base the dual formulation on nonconvex approximations. To this end we reintroduce a nonconvex approach to the weight minimization problem originally due to Fleury, and we explore certain convex and nonconvex forms for subproblems derived from the exponential approximations by the application of various methods of mixed variables. We show in each case that the dual is well defined for the form concerned, which may consequently be of use to the future code developers. Copyright © 2009 John Wiley & Sons, Ltd.     

RESPONSE TO ALI S. HADI

In many large-dimensional multivariate problems, it is useful to reduce the number of variates. One method of reducing the number of dimensions is to project the original data onto a subspace. The statistical analysis is then carried out in this subspace. Principal-component regression is an example of such a technique. For these applications it is useful to have a measure of the distance between subspaces and to study the sampling stability of such subspaces. To solve these problems, we use a metric on subspaces and bootstrap techniques. The techniques are applied to seven-dimensional vectors of upwelling radiances from the current meteorological satellites. We study the subspaces spanned by the principal components based on a sample categorized by location and surface type.     

子空间预测控制算法在主动噪声振动中的应用

为解决主动噪声和振动控制中的执行器饱和约束条件问题,本文在子空间系统辨识的基础上,研究了一种新颖的子空间预测控制方法。该控制方法联合了系统辨识和控制器设计,直接由输入-输出数据得到将来时刻的输出预测值,自动校正系统中的参数,克服了传统的模型预测控制中繁琐的系统辨识环节。同时子空间预测控制允许执行器机构出现饱和现象,在考虑由饱和现象导致的约束条件时,利用线性矩阵不等式将约束优化问题转化为无约束优化问题。采用椭球优化算法迭代地产生一系列体积逐渐减小的椭球序列,该序列最终能收敛到一个最优解。在椭球算法的基础上推导了该算法达到收敛时所需要迭代次数的一个上界。这在智能优化算法中是很难求得到的。最后以直升机悬停状态时发生的颤振为例,利用本文中的子空间预测控制和椭球优化算法设计闭环系统的反馈控制器,验证闭环系统的输出响应能较好地跟踪期望值,从而得出本文方法的有效性。     

一种基于子空间划分的LCD显示器光谱特征化模型

目的为实现LCD显示器的光谱特征化,提出一种基于子空间划分的BP神经网络结合PCA的光谱特征化模型。方法对显示器色空间进行子空间划分,在各子空间中进行模型的训练与检测。结果子空间划分后模型的色度和光谱精度,较未进行子空间划分的模型有明显提高,PCA在不影响模型精度的同时,降低了光谱维度,提高了算法的运行效率。结论该模型是一种高精度显示器特征化模型。     

Structural optimization of uncertain dynamical systems considering mixed-design variables

In this paper attention is directed to the reliability-based optimization of uncertain structural systems under stochastic excitation involving discrete-continuous sizing type of design variables. The reliability-based optimization problem is formulated as the minimization of an objective function subject to multiple reliability constraints. The probability that design conditions are satisfied within a given time interval is used as a measure of system reliability. The problem is solved by a sequential approximate optimization strategy cast into the framework of conservative convex and separable approximations. To this end, the objective function and the reliability constraints are approximated by using a hybrid form of linear, reciprocal and quadratic approximations. The approximations are combined with an effective sensitivity analysis of the reliability constraints in order to generate explicit expressions of the constraints in terms of the design variables. The explicit approximate sub-optimization problems are solved by an appropriate discrete optimization technique. The optimization scheme exhibits monotonic convergence properties. Two numerical examples showing the effectiveness of the approach reported herein are presented.     

Procedures for approximate eigenproblem reanalysis of structures

In vibration mode superposition analysis, the main computational effort is spent in the solution of the eigenproblem. In reanalysis procedures this solution must be repeated for each change in the design. It is shown in this study how the combined approximations method can be used to improve the efficiency of some common iterative procedures. These procedures can then be used for efficient eigenproblem reanalysis. Solution of the reduced eigenproblem by the combined approximations method is first demonstrated. Procedures based on inverse iteration, inverse iteration with shifts, and subspace iteration are then considered. Some numerical examples show that high accuracy is achieved for very large changes in the design. In cases where higher accuracy is required, additional terms of the approximations may be considered. Copyright © 2004 John Wiley & Sons, Ltd.     

On the Exact Solution of Boundary Value Problems with Non-separable Geometries,Like a Rough Surface

A Dirichlet problem is considered for the Helmholtz equation and for a class of geometries for which the Helmholts operator does not separate (e.g. a rough surface). It is shown that, contrary to the widely held view, it is possible to obtain the solution of this problem in a closed form which resembles closely the solutions obtained for separable geometries—expansions generated by Sturm-Liouville theory. As with separable geometries, we show in particular that the expansion coefficients can be written explicitly as integrals containing a priori known functions—that matrix inversion is not required for the determination of the expansion coefficients. The problem includes as a special case the scattering of electromagnetic waves from a rough cylindrical surface which is the boundary of a perfect conductor. The method is very general and can be used for much more complicated boundary value problems, such as scattering by dielectric interface.     

Computer algorithms for calculating efficient initial vectors for subspace iteration method

In this paper, the effects of selecting initial vectors on computation efficiency for a subspace iteration method are investigated. Four algorithms are used for selecting the initial vectors. First, arbitrary starting iteration vectors are chosen according to Bathe and Wilson's algorithm.¹ In the other algorithms, the initial vectors are the retrieved eigenvectors from the Guyan and quadratic reduction methods. Improvement of the eigenvalue approximations of the subspace iteration method over reduction methods is presented. The computation effort is examined for the various algorithms used for initial iteration vectors.