加权广义逆、加权最小二乘和约束最小二乘问题

本文采用如下记号:记C~m×n是具有复数域的m×n长方矩阵的集合,C~m=C~m×1是m维向量的集合.对A∈C~m×n称A~H∈C~m×n是A的共轭转置矩阵,rank(A)表示A的秩,R(A)和N(A)分别为A的值域和零空间,||·||=||·||2和||·||F分别为2-范数和Frobenius范数;I表示恒等矩阵.人们在研究数学规划、数值分析、数据处理,散射理论和电磁学等领域中都将问题归纳为如下的最小二乘问题:     

线性流形上次反对称矩阵逆特征值问题的最小二乘解

讨论了线性流形上次反对称矩阵逆特征值问题的最小二乘问题及其最佳逼近，给出了这些问题解的通式；并就这些问题的特殊情况进行了讨论，得到了一些结果。     

解非线性最小二乘问题的ABS方法

在本文中,基于解非线性方程组的ABS方法的思想,我们对非线性最小二乘问题建立了一类新的算法。在类似于Gauss-Newton法的收敛条件下,我们证明了算法的局部收敛性。此外,在对算法结构进行深入分析的基础上,我们将新算法转化为一种近似Gauss-Newton法。并建立了它的Kantorovich型收敛定理。数值结果表明ABS算法是有效的,且在一定程度上优越于Gauss-Newton法。     

双对称非负定阵一类逆特征值问题的最小二乘解

１．引言 逆特征值问题在工程中有广泛的应用,其研究已有一些很好的结果［１－５］．最近,文［６］还研究了双对称矩阵逆特征值问题,即研究了如下两个问题： 问题Ａ．已知Ｘ∈Ｒｎｘｍ,Ａ＝ｄｉａｇ（λ１…,λｍ）,求Ａ∈ＢＳＲｎｘｎ使 ＡＸ＝ＸＡ,其中 Ｒｎｘｍ表示全体 ｎ ｘ ｍ实矩阵集合, ＢＳＲｎｘｎ表示全体 ｎ ｘ ｎ双对称阵集合． 问题Ｂ．已知Ａ＊ＥＲｎｘｎ,求Ａ∈ＳＥ使 ｜｜Ａ＊－Ａ｜｜＝ ｉｎｆ ｜｜Ａ＊－Ａ｜｜ ＡＦＳＥ其中 ＳＥ是问题 Ａ的解集合,｜｜． ｜｜表示 Ｆｒｏｂｅｎｉｕｓ范数． 在实际问题中, …     

一类积—微分方程的正和逆问题

逆问题现在已在众多的科技领域得到了广泛的应用和发展,这里我们讨论以原子能技术的应用为背景的具非均匀介质板几何的定态迁移方程。     

对称约束平衡Procrustes问题的一个简单解法

本文提出一个非常简单的方法,解决对称约束的平衡Procrustes问题:给定两个同样大小的矩阵A,B∈R~m×n,求对称正交阵Q,使‖AQ—B‖_F达到最小.该方法同时具有较好的数值稳定性.     

关于一类连续最优控制问题的一种可实现的离散方法

本文对具有状态终端约束、控制受限的非线性连续最优控制问题给出一种新的可实现的离散方法，此方法通过求解非线最小二乘问题避免这类问题离散后出现的不可行现象，文中给出这种做法的理论证明和实现方案。     

广义线性最小二乘问题的一个精确解及其推广

本文在特定的约束条件下,给出了一维线性广义最小二乘问题解的精确表达式,并对解的范围进行了讨论。对n维线性广义最小二乘问题,给出了求解方法,迭代步骤和收敛性定理。     

对称的运输问题及其逆问题

本文对［１,２,６］中提出的运输问题进行了推广,并提出了一个强多项式算法,从而改进了原有的结果．同时对对称的运输问题的逆问题进行了研究,并借助于最小费用循环流技术得到了一个强多项式算法．     

Inverse Problem for a Class of Two-Dimensional Diffusion Equations with Piecewise Constant Coefficients

In this paper, we consider an inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients. This problem is studied using an explicit formula for the relevant spectral measures and an asymptotic expansion of the solution of the diffusion equations. A numerical method that reduces the inverse problem to a sequence of nonlinear least-square problems is proposed and tested on synthetic data.     

源项反演问题的条件稳定性

对于一维扩散方程的源项反演问题,探讨了反问题数据的相容性并应用积分恒等式方法建立了非线性源项反演的一种稳定性.     

半线性波方程的一个反问题

该文利用Banach不动点原理讨论了一个半线性波方程的反问题,文中给出了该问题解的存在性、唯一性和稳定性．     

广义对称矩阵反问题有解的条件

本利用矩阵的奇异值分解讨论了一类广义对矩阵阵反问题，得到了此类矩阵反问题有解的充分必要条件及通解的表达式。     

Inverse Matroid Intersection Problem

LetM ₁ andM ₂ be matroids onS,B be theirk-element common independent set, andw a weight function onS. Given two functionsb 0 andc 0 onS, the Inverse Matroid Intersection Problem (IMIP) is to determine a modified weight functionw such that (a)B becomes a maximum weight common independent set of cardinalityk underw, (b)c¦w — w¦ is minimum, and (c)¦w — w b. Many Inverse Combinatorial Optimization Problems can be considered as the special cases of the IMIP.In this paper we show that the IMIP can be solved in strongly polynomial time, and give a necessary and sufficient condition for the feasibility of the IMIP. Finally we extend the discussion to the version of the IMIP with Multiple Common Independent Sets.Research partially supported by the National Natural Science Foundation of China     

Inverse Boundary Value Problem for Maxwell Equations with Local Data

We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the boundary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov [4 Isakov , V. ( 2007 ). On uniqueness in the inverse conductivity problem with local data . Inverse Probl. Imaging 1 : 95 – 105 .[Crossref], [Web of Science ®] , [Google Scholar]] for the Schrödinger equation to Maxwell equations.     

Optimal Backward Perturbation Bounds for the LSS Problem

A computable expression is given for the backward error of an approximate solution to the problem of least squares over a sphere.This revised version was published online in October 2005 with corrections to the Cover Date.     

An Inverse Coefficient Problem for an Integro-differential Equation

In this article we consider the inverse coefficient problem of recovering the function { ( x ) system of partial differential equations that can be reduced to a second order integro-differential equation $ -u_{xx} + c(x)u_{x} + d\phi (x)u-\gamma d\phi (x)\int _{0}^{t}e^{-\gamma (t-\tau )}u(x,\tau )\, d\tau = 0 $ with boundary conditions. We prove the existence and uniqueness of solutions to the inverse problem and develop a numerical algorithm for solving this problem. Computational results for some examples are presented.     

一类双曲反问题的逼近算法及收敛性

该文考虑地球物理勘探中出现的间断特性阻抗的反演问题．利用样条插值理论，把无穷维空间上的反问题用有限维空间上的反问题来近似．利用半群理论，证明了近似反问题之解收敛于原反问题之解．据此可得到求解反问题的一种稳定的近似算法．     

On the Inverse Problem for Robust Capture Zone Construction

An interception problem with variable velocities and variable lateral acceleration boundaries is considered. Two problems, inverse to constructing the capture zone for a given linear strategy, are formulated. In the first (weak) formulation, a linear strategy is derived forcing its capture zone to contain a given set. In the second (strong) formulation, the capture zone of such a linear strategy is forced to coincide with a given set. A step-by-step algorithm, solving these problems, is derived. This algorithm is based on a detailed differential-geometrical analysis of the capture zone boundary. Illustrative examples are presented.