广义次对称矩阵反问题的最小二乘解

讨论了广义次对称矩阵反问题的最小二乘解,得到了解的一般表达式,并就该问题的特殊情形:矩阵反问题,得到了可解的充分必要条件及解的通式.此外,证明了最佳逼近问题解的存在唯一性,并给出了其解的具体表达式.     

Preconditioned GMRES methods with incomplete Givens orthogonalization method for large sparse least-squares problems

We propose to precondition the GMRES method by using the incomplete Givens orthogonalization (IGO) method for the solution of large sparse linear least-squares problems. Theoretical analysis shows that the preconditioner satisfies the sufficient condition that can guarantee that the preconditioned GMRES method will never break down and always give the least-squares solution of the original problem. Numerical experiments further confirm that the new preconditioner is efficient. We also find that the IGO preconditioned BA-GMRES method is superior to the corresponding CGLS method for ill-conditioned and singular least-squares problems.     

Nonconvergence results for the application of least-squares estimation to Ill-posed problems

One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in , where corresponds to a parametric representation providing sufficiently good approximation to the true solutionx*. Call the minimizerx=A( ). Take = _N for a sequence { _N } of subspaces becoming dense, and so determine an approximating sequences {x _N A ( _N )}. It is shown, withf linear and one-to-one, that one need not havex _Nx* iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).     

Preconditioned conjugate gradient method for generalized least squares problems

A variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented. If the problem is min (Ax − b)^TW⁻¹(Ax − b) with A ∈ R^m×n and W ∈ R^m×m symmetric and positive definite, the method needs only a preconditioner A₁ ∈ R^n×n, but not the inverse of matrix W or of any of its submatrices. Freund's comparison result for regular least squares problems is extended to generalized least squares problems. An error bound is also given.     

A reduced Newton method for constrained linear least-squares problems

We propose an iterative method that solves constrained linear least-squares problems by formulating them as nonlinear systems of equations and applying the Newton scheme. The method reduces the size of the linear system to be solved at each iteration by considering only a subset of the unknown variables. Hence the linear system can be solved more efficiently. We prove that the method is locally quadratic convergent. Applications to image deblurring problems show that our method gives better restored images than those obtained by projecting or scaling the solution into the dynamic range.     

On convex modified output least-squares for elliptic inverse problems: stability,regularization, applications,and numerics

Abstract

Inverse problems of identifying parameters in partial differential equations constitute an important class of problems with diverse real-world applications. These identification problems are commonly explored in an optimization framework and there are many optimization formulations having their own advantages and disadvantages. Although a non-convex output least-squares (OLS) objective is commonly used, a convex-modified output least-squares (MOLS) has shown encouraging results in recent years. In this work, we focus on various aspects of the MOLS approach. We devise a rigorous (quadratic and non-quadratic) regularization framework for the identification of smooth as well as discontinuous coefficients. This framework subsumes the total variation regularization that has attracted a great deal of attention in identifying sharply varying coefficients and also in image processing. We give new existence results for the regularized optimization problems for OLS and MOLS. Restricting to the Tikhonov (quadratic) regularization, we carry out a detailed study of various stability aspects of the inverse problem under data perturbation and give new stability estimates for general inverse problems using OLS and MOLS formulations. We give a discretization scheme for the continuous inverse problem and prove the convergence of the discrete inverse problem to the continuous one. We collect discrete formulas for OLS and MOLS and compute their gradients and Hessians. We present applications of our theoretical results. To show the feasibility of the MOLS framework, we also provide computational results for the inverse problem of identifying parameters in three different classes of partial differential equations .     

DIRECT ITERATIVE METHODS FOR RANK DEFICIENT GENERALIZED LEAST SQUARES PROBLEMS

1. IntroductionThe generalized LS problemis frequently found in solving problems from statistics, engineering, economics, imageand signal processing. Here A e Rmxn with m 2 n, b E Re and W E Rmxm issymmetric positive definite. The large sparse rank deficient generalized LS problemsappeal in computational genetics when we consider mited linear model for tree oranimal genetics [2], [31, [5].Recentlyg Yuan [9] and [10], Yuan and lusem [11] considered direct iterative methodsfor the problem …     

GSOR方法最优迭代参数的简化推导

对于增广线性系统,Bai等研究了广义SOR方法（Bai Z Z,Parlett B,Wang Z Q.On generaliged successive overrelaxation methods for augmented linear systems.Numerische Mathematik,2005,102（1）：1-38）,并得到其最优迭代参数.给出了另外一种推导最优迭代参数的简化方法,这种方法对于求解其他参数加速定常迭代方法的最优迭代参数非常有意义.     

Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method. The second author was supported in part by the US National Science Foundation under Grant DMS-0612908.     

Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

We employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix _An−ρ_Bn$A_n-\rho B_n$ involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem _Anx=λ_Bnx$A_{n}x=\lambda B_{n}x$, where _An$A_n$ and _Bn$B_n$ are given matrices of suitable sizes. The sine transform-based preconditioner can improve the spectral distribution of the shifted Toeplitz matrix and, hence, can speed up the convergence rate of the preconditioned Lanczos method. The sine transform-based preconditioner can be implemented efficiently by the fast transform algorithm. A convergence analysis shows that the preconditioned Lanczos method converges sufficiently fast, and numerical results show that this method is highly effective for a large matrix.     

Solving shortest length least-squares problems via dynamic programming

If the matrixA is not of full rank, there may be many solutions to the problem of minimizing Ax–b overx. Among such vectorsx, the unique one for which x is minimum is of importance in applications. This vector may be represented asx=A ⁺ b. In this paper, the functional equation technique of dynamic programming is used to find the shortest solution to the least-squares problem in a sequential fashion. The algorithm is illustrated with an example.Our debt to the late Professor Richard Bellman is clear, and we wish to thank Professor Harriet Kagiwada for many stimulating conversations concerning least-squares problems over a long period of years.     

Improved trial methods for a class of generalized Bernoulli problems

The aim of this article is to develop improved trial methods for the solution of a generalized exterior Bernoulli free boundary problem. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. Appropriate update rules are obtained by expanding the state's Dirichlet data at the actual boundary via a Taylor expansion of first and second order. The resulting trial methods converge linearly for both cases, although the trial method based on the second order Taylor expansion is much more robust. Nevertheless, via results of shape sensitivity analysis, we are able to modify the update rules such that their convergence is improved. The feasibility of the proposed trial methods and their performance is demonstrated by numerical results.     

Convergence criteria for generalized gradient methods of solving locally Lipschitz feasibility problems

We prove the convergence of a class of iterative algorithms for solving locally Lipschitz feasibility problems, that is, finite systems of inequalities f _i (x)0, (i I), where each f _i is a locally Lipschitz functional on ⁿ . We also obtain a new convergence criterion for the so-called block-iterative projection methods of finding common points of finite families of convex closed subsets of ⁿ as defined by Aharoni and Censor ([3]).The work of Dan Butnariu was done while visiting the Department of Mathematics of the University of Texas at Arlington.     

Viscosity approximation methods for generalized equilibrium problems and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of solutions of a generalized equilibrium problem (for short, GEP) and the set of fixed points of a nonexpansive mapping in the setting of Hilbert spaces. By using well-known Fan-KKM lemma, we derive the existence and uniqueness of a solution of the auxiliary problem for GEP. On account of this result and Nadler’s theorem, we propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping. Furthermore, it is proven that the sequences generated by this iterative scheme converge strongly to a common element of the set of solutions of GEP and the set of fixed points of a nonexpansive mapping.     

On preconditioned Uzawa methods and SOR methods for saddle-point problems

This paper studies convergence analysis of a preconditioned inexact Uzawa method for nondifferentiable saddle-point problems. The SOR-Newton method and the SOR-BFGS method are special cases of this method. We relax the Bramble-Pasciak-Vassilev condition on preconditioners for convergence of the inexact Uzawa method for linear saddle-point problems. The relaxed condition is used to determine the relaxation parameters in the SOR-Newton method and the SOR-BFGS method. Furthermore, we study global convergence of the multistep inexact Uzawa method for nondifferentiable saddle-point problems.     

求解分布控制问题的预处理最小残量方法(英文)

通过分析Bai(Bai Z Z.Block preconditioners for elliptic PDE-constrained optimization problems.Computing,2011,91:379-395)给出的离散分布控制问题的块反对角预处理线性系统,提出了该问题的一个等价线性系统,并且运用带有预处理子的最小残量方法对该系统进行求解.理论分析和数值实验结果表明,所提出的预处理最小残量方法对于求解该类椭圆型偏微分方程约束最优分布控制问题非常有效,尤其当正则参数适当小的时候.     

Recession methods for generalized vector equilibrium problems

By using the recession method, we give some necessary and/or sufficient condition of solutions of generalized vector equilibrium problems.     

Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions.     

Equivalence of generalized mixed complementarity and generalized mixed least element problems in ordered spaces

In this article, we derive some equivalences of generalized mixed non-linear programs, generalized mixed least-element problems, generalized mixed complementarity problems and generalized mixed variational inequality problems under certain regularity and growth conditions. We also generalize the notion of a Z-type map for point-to-set maps. Our results improve and extend recent results in the literature.     

Factorized quasi-Newton methods for nonlinear least squares problems

This paper provides a modification to the Gauss—Newton method for nonlinear least squares problems. The new method is based on structured quasi-Newton methods which yield a good approximation to the second derivative matrix of the objective function. In particular, we propose BFGS-like and DFP-like updates in a factorized form which give descent search directions for the objective function. We prove local and q-superlinear convergence of our methods, and give results of computational experiments for the BFGS-like and DFP-like updates.This work was supported in part by the Grant-in-Aid for Encouragement of Young Scientists of the Japanese Ministry of Education: (A)61740133 and (A)62740137.