Determining an unknown source in the heat equation by a wavelet dual least squares method

We consider the problem of determining an unknown source, which depends only on the spatial variable, in a heat equation. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. For a reconstruction of the unknown source from measured data the dual least squares method generated by a family of Meyer wavelet subspaces is applied. An explicit relation between the truncation level of the wavelet expansion and the data error bound is found, under which the convergence result with the error estimate is obtained.     

A condition analysis of the weighted linear least squares problem using dual norms

In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise condition numbers derived in this paper can be computed at low cost when the dimension of the linear function is low due to dual operator theory. Moreover, we use the augmented system to perform a componentwise perturbation analysis of the solution and residual of the weighted linear least squares problems. We also propose two efficient condition number estimators. Our numerical experiments demonstrate that our condition numbers give accurate perturbation bounds and can reveal the conditioning of individual components of the solution. Our condition number estimators are accurate as well as efficient.     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

An iterative penalty method for the least squares solution of boundary value problems

This article is concerned with iterative techniques for linear systems of equations arising from a least squares formulation of boundary value problems. In its classical form, the solution of the least squares method is obtained by solving the traditional normal equation. However, for nonsmooth boundary conditions or in the case of refinement at a selected set of interior points, the matrix associated with the normal equation tends to be ill-conditioned. In this case, the least squares method may be formulated as a Powell multiplier method and the equations solved iteratively. Therein we use and compare two different iterative algorithms. The first algorithm is the preconditioned conjugate gradient method applied to the normal equation, while the second is a new algorithm based on the Powell method and formulated on the stabilized dual problem. The two algorithms are first compared on a one-dimensional problem with poorly conditioned matrices. Results show that, for such problems, the new algorithm gives more accurate results. The new algorithm is then applied to a two-dimensional steady state diffusion problem and a boundary layer problem. A comparison between the least squares method of Bramble and Schatz and the new algorithm demonstrates the ability of the new method to give highly accurate results on the boundary, or at a set of given interior collocation points without the deterioration of the condition number of the matrix. Conditions for convergence of the proposed algorithm are discussed. © 1997 John Wiley & Sons, Inc.     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解及其最佳逼近

基于共轭梯度法的思想,通过特殊的变形,建立了一类求矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解的迭代算法．对任意的初始双对称矩阵．在没有舍人误差的情况下,经过有限步迭代得到它的双对称最小二乘解;在选取特殊的初始双对称矩阵时,能得到它的的极小范数双对称最小二乘解．另外,给定任意矩阵,利用此方法可得到它的最佳逼近双对称解,数值例子表明,这种方法是有效的．     

A numerically stable least squares solution to the quadratic programming problem

The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.      

Strong underrelaxation in Kaczmarz's method for inconsistent systems

Summary We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.     

Solution of a Complex Least Squares Problem with Constrained Phase

The least squares solution of a complex linear equation is in general a complex vector with independent real and imaginary parts. In certain applications in magnetic resonance imaging, a solution is desired such that each element has the same phase. A direct method for obtaining the least squares solution to the phase constrained problem is described.     

线性子空间上求解AX=B的最小二乘问题的迭代算法

应用共轭梯度方法和线性投影算子,给出迭代算法求解了线性矩阵方程AX=B在任意线性子空间上的最小二乘解问题.在不考虑舍入误差的情况下,可以证明,所给迭代算法经过有限步迭代可得到矩阵方程AX=B的最小二乘解、极小范数最小二乘解及其最佳逼近.文中的数值例子证实了该算法的有效性.     

Some observations on local least squares

In previous work we introduced a construction to produce biorthogonal multiresolutions from given subdivisions. The approach involved estimating the solution to a least squares problem by means of a number of smaller least squares approximations on local portions of the data. In this work we use a result by Dahlquist, et al. on the method of averages to make observational comparisons between this local least squares estimation and full least squares approximation. We have explored examples in two problem domains: data reduction and data approximation. We observe that, particularly for design matrices with a repetitive pattern of column entries, the least squares solution is often well estimated by local least squares, that the estimation rapidly improves with the size of the local least squares problems, and that the quality of the estimate is largely independent of the size of the full problem. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 93E24     

Single projection Kaczmarz extended algorithms

In order to find the least squares solution of a very large and inconsistent system of equations, one can employ the extended Kaczmarz algorithm. This method simultaneously removes the error term, so that a consistent system is asymptotically obtained, and applies Kaczmarz iterations for the current approximation of this system. It has been shown that for random corrections of the right hand side and Kaczmarz updates selected at random, the algorithm converges to the least squares solution. In this work we consider deterministic strategies like the maximal-residual and the almost-cyclic control, and show convergence to a least squares solution.     

最小二乘法相关的一类特殊矩阵的代数运算

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A NEW PSEUDOSPECTRAL APPROXIMATION FOR THE FOWARD-BACKWARD HEAT EQUATION

ＡＮＥＷＰＳＥＵＤＯＳＰＥＣＴＲＡＬＡＰＰＲＯＸＩＭＡＴＩＯＮＦＯＲＴＨＥＦＯＷＡＲＤ－ＢＡＣＫＷＡＲＤＨＥＡＴＥＱＵＡＴＩＯＮ￥ＹｅＸｉｎｇｄｅ（叶兴德）ＪｉａｎｇＪｉｎｓｈｅｎｇ（江金生）（Ｄｅｐｔ．ｏｆＭａｔｈ．＆Ｉｎｆｏｒ．Ｓｃｉ．，Ｈａｎｇ...     

The structured sensitivity of Vandermonde-like systems

Summary We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.     

A dynamic meshless method for the least squares problem with some noisy subdomains

The interpolation method by radial basis functions is used widely for solving scattered data approximation. However, sometimes it makes more sense to approximate the solution by least squares fit. This is especially true when the data are contaminated with noise. A meshfree method namely, meshless dynamic weighted least squares (MDWLS) method, is presented in this paper to solve least squares problems with noise. The MDWLS method by Gaussian radial basis function is proposed to fit scattered data with some noisy areas in the problem’s domain. Existence and uniqueness of a solution is proved. This method has one parameter which can adjusts the accuracy according to the size of noises. Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains. The new approach is applied for some problems in two dimensions and the obtained results confirm the accuracy and efficiency of the proposed method. The numerical experiments illustrate that our MDWLS method has better performance than the traditional least squares method in case of noisy data.     

Multipole fast algorithm for the least‐squares approach of the method of fundamental solutions for three‐dimensional harmonic problems

In this article we describe an improvement in the speed of computation for the least‐squares method of fundamental solutions (MFS) by means of Greengard and Rokhlin's FMA. Iterative solution of the linear system of equations is performed for the equations given by the least‐squares formulation of the MFS. The results of applying the method to test problems from potential theory with a number of boundary points in the order of 80,000 show that the method can achieve fast solutions for the potential and its directional derivatives. The results show little loss of accuracy and a major reduction in the memory requirements compared to the direct solution method of the least squares problem with storage of the full MFS matrix. The method can be extended to the solution of overdetermined systems of equations arising from boundary integral methods with a large number of boundary integration points. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 828–845, 2003.     

基于邻域粗糙集属性约简的对偶约束式LS-SVM财务困境预测模型研究

为了提高财务困境预测的正确率,改善模型预测的效果,将邻域粗糙集和遗传算法应用于对偶约束式最小二乘支持向量机,提出了一种基于邻域粗糙集属性约简的对偶约束式最小二乘支持向量机预测模型.同时,给出了这一改进模型的实现步骤.实证结果表明,通过邻域粗糙集指标预处理和遗传算法参数优化后,不但提高了模型预测的正确率,还降低了模型运行的时间,证实了该模型应用于财务困境预测是有效的.     

Dual‐ and triple‐mode matrix approximation and regression modelling

We propose a dual‐ and triple‐mode least squares for matrix approximation. This technique applied to the singular value decomposition produces the classical solution with a new interpretation. Applied to regression modelling, this approach corresponds to a regularized objective and yields a new solution with properties of a ridge regression. The results for regression are robust and suggest a convenient tool for the analysis and interpretation of the model coefficients. Numerical results are given for a marketing research data set. Copyright © 2003 John Wiley & Sons, Ltd.     

四元数矩阵方程(AXB,CXD)=(E,F)的最小范数最小二乘Hermitian解

提出了研究四元数矩阵方程(AXB, CXD)=(E, F)的最小范数最小二乘Hermitian解的一个有效方法.首先应用四元数矩阵的实表示矩阵以及实表示矩阵的特殊结构,把四元数矩阵方程转化为相应的实矩阵方程,然后求出四元数矩阵方程(AXB, CXD)=(E, F)的最小二乘Hermitian解集,进而得到其最小范数最小二乘Hermitian解.所得到的结果只涉及实矩阵,相应的算法只涉及实运算,因此非常有效.最后的两个数值例子也说明了这一点.