On the Extrema of Linear Least-Squares Problems

We use the concept of generalized inverses to show that the extrema of the function are minima, A being a rectangular matrix not necessarily of full rank     

A Linear Least-Squares MFS for Certain Elliptic Problems

The Method of Fundamental Solutions (MFS) is a boundary-type meshless method for the solution of certain elliptic boundary value problems. In this work, we propose an efficient algorithm for the linear least-squares version of the MFS, when applied to the Dirichlet problem for certain second order elliptic equations in a disk. Various aspects of the method are discussed and a comparison with the standard MFS is carried out. Numerical results are presented.     

Weighted Generalized Inverses,Oblique Projections,and Least-Squares Problems

A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.     

Weighted Least-Squares Polynomial Deflation

We present a unified analysis of several methods of polynomialdeflation, including methods commonly in use. We also discussthe suitability of "minimum-norm" deflations and present theresults of numerical tests on a new method.     

A Note on Backward Error Analysis for Generalized Linear Complementarity Problems

This note extends the classical Oettli–Prager theorem to generalized linear complementarity problems.     

Adaptive Wavelet Methods for Linear and Nonlinear Least-Squares Problems

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27–75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations $G(u)=0$ , where $G:H \rightarrow K'$ , is studied. For formulations of partial differential equations as first-order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient.     

QR Factorization for Linear Least-Squares Problems on a Hypercube Multiprocessor

Least squares problems occur in many branches of science. Typicallythere may be a large number of data points or observations andonly a small to moderate number of variables. On sequentialmachines these problems can be time-consuming and thereforethe use of parallel machines to solve large least-squares problemsmay well yield substantial savings. The solution of least-squaresproblems by a QR factorization using Givens rotations seemsto be particularly suitable for a parallel machine, becausethere is much choice in the order of the Givens rotations andmany Givens rotations can be carried out in parallel. In this paper, an implementation of a QR factorization on theIntel hypercube is described. Each row of the least-squaresmatrix is assigned to a processor and most of the rotationsinvolve rows within one processor in the usual case when eachprocessor receives several rows. However, it is also necessaryto carry out rotations involving rows in different processorsand we call these rotations merges. Two ways of implementingthe merges are described and they are compared on the groundsof load balance and the number of communications required. Onefeature of the implementations is that processors can continueto do Givens rotations on rows within the processor while waitingfor messages that are required for merges. There is also someflexibility in the order of the merges and this can be incorporatedinto the algorithm. For each column, the merges are carriedout according to a tree structure and the choices of trees andtheir roots are discussed. Numerical results are given to showthe usefulness and efficiency of the proposed algorithms.     

Weighted Tensor Product Algorithms for Linear Multivariate Problems

We study the ε-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λ^all consists of all linear functionals while Λ^std consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1/ε and d. For Λ^all we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. ForΛ^std we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences.     

线性流形上反对称正交对称矩阵反问题的最小二乘解

该文讨论了线性流形上矩阵方程AX=B反对称正交对称反问题的最小二乘解及其最佳逼近问题．给出了最小二乘问题解集合的表达式，得到了给定矩阵的最佳逼近问题的解，最后给出计算任意矩阵的最佳逼近解的数值方法及算例．     

线性流形上反对称正交对称矩阵反问题的最小二乘解

该文讨论了线性流形上矩阵方程AX=B反对称正交对称反问题的最小二乘解及其最佳逼近问题. 给出了最小二乘问题解集合的表达式, 得到了给定矩阵的最佳逼近问题的解, 最后给出计算任意矩阵的最佳逼近解的数值方法及算例.     

求解加权线性最小二乘问题的预处理迭代方法

给出了求解一类加权线性最小二乘问题的预处理迭代方法,也就是预处理的广义加速超松弛方法（GAOR）,得到了一些收敛和比较结果．比较结果表明当原来的迭代方法收敛时,预处理迭代方法会比原来的方法具有更好的收敛率．而且,通过数值算例也验证了新预处理迭代方法的有效性．     

关于左移加权移位的循环向量

设{w_n}₁^∞是一复的有界序列。l²上由T(x₀,x₁,x₂,…)=(w₁x₁,w₂x₂,…)定义的算子T称为以{w_n}₁^∞为权的左移加权移位。本文证明了T为循环算子的充要条件是{w_n}₁^∞至多只有一项为零;讨论了某些特殊加权移位的循环向量;并指出[1]的有误之处。所得结果是[1]中结果的推广。     

Computation of Optimal Backward Perturbation Bounds for Large Sparse Linear Least Squares Problems

In this note we propose an algorithm based on the Lanczos bidiagonalization to approximate the backward perturbation bound for the large sparse linear squares problem. The algorithm requires ((m + n)l) operations where m and n are the size of the matrix under consideration and l <#60;<#60; min(m,n). The import of the proposed algorithm is illustrated by some examples coming from the Harwell-Boeing collection of test matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

The Method of Fundamental Solutions: A Weighted Least-Squares Approach

We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In particular, we study the case in which the number of collocation points exceeds the number of singularities, which leads to an over-determined linear system. In such a case, the resulting linear system is over-determined and the proposed algorithm chooses the approximate solution for which the error, when restricted to the boundary, minimizes a suitably defined discrete Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence of the method in the case of the Laplace’s equation with Dirichlet boundary data in the disk. We develop an alternative way of implementing the numerical algorithm, which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present numerical experiments suggesting that introduction of Sobolev weights improves the approximation. AMS subject classification (2000) 35E05, 35J25, 65N12, 65N15, 65N35, 65T50     

Backward Stability of Clenshaw's Algorithm

We study numerical properties of Clenshaw's algorithm for summing the series w = _{n = 0} ^N b _n p _n where p _n satisfy the linear three-term recurrence relation. We prove that under natural assumptions Clenshaw's algorithm is backward stable with respect to the data b _n, n = 0,N.     

A Weighted Least-Squares Mixed Finite Element Formulation for Quasi-Incompressible Elastodynamics

The purpose of this work is the application of the least-squares finite element method to an elastodynamic, quasi-incompressible problem under small strain assumptions. Therefore a mixed finite element based on a weighted least-squares formulation is developed. The L₂-norm minimization of the time-discretized residuals of the given first-order system of partial differential equations leads to a functional depending on displacements and stresses. In the numerical example the proposed mixed element is compared to an alternative approach, which is based on a least-squares mixed finite element with improved momentum balance, see [1]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.     

Stability and Regularization of Vector Problems of Integer Linear Programming

In this article we consider the problem of finding the Pareto set, and also the problem of lexicographic optimization. We study several types of stability, understood as preservation of certain properties of the efficient solution set under "small" changes of input data. The borders of such changes are ascertained. Necessary and sufficient conditions of stability are specified. A regularizing operator is proposed for transferring a probably unstable problem to a series of stable ones.     

Backward Extensions of Recursively Generated Weighted Shifts and Quadratic Hyponormality

Given the weight sequence for a subnormal recursively generated weighted shift on Hilbert space, one approach to the study of classes of operators weaker than subnormal has been to form a backward extension of the shift by prefixing weights to the sequence. We characterize positive quadratic hyponormality and revisit quadratic hyponormality of certain such backward extensions of arbitrary length, generalizing earlier results, and also show that a function apparently introduced as a matter of convenience for quadratic hyponormality actually captures considerable information about positive quadratic hyponormality.     

Linear Stability Analysis for a Korteweg Fluid

Linear stability analysis of the homogeneous equilibrium solution of Euler equations for an isothermal, inviscid, compressible fluid endowed with a Korteweg-type stress tensor, in a harmonic potential field, is performed. We show that, by perturbing the fluid at rest, the transition from stability to instability takes place via a marginal state exhibiting a stationary cellular pattern of motions. By analyzing the disturbances in normal modes we obtain threshold values of the harmonic angular frequency that, in correspondence of a given temperature, could trigger fluid’s fragmentation. In the picture of initial stage of a volcano eruption, this model could describe the transition from the two-phase system magma-dissolved gas, at supersaturation pressure in the chamber, to the rising foam at conduit’s base induced by an external stress of elastic-type.