• Title, Summary, Keyword: least squares problem

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AN ITERATIVE ALGORITHM FOR SOLVING THE LEAST-SQUARES PROBLEM OF MATRIX EQUATION AXB+CYD=E

  • Shen, Kai-Juan;You, Chuan-Hua;Du, Yu-Xia
    • Journal of applied mathematics & informatics
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    • v.26 no.5_6
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    • pp.1233-1245
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    • 2008
  • In this paper, an iterative method is proposed to solve the least-squares problem of matrix equation AXB+CYD=E over unknown matrix pair [X, Y]. By this iterative method, for any initial matrix pair [$X_1,\;Y_1$], a solution pair or the least-norm least-squares solution pair of which can be obtained within finite iterative steps in the absence of roundoff errors. In addition, we also consider the optimal approximation problem for the given matrix pair [$X_0,\;Y_0$] in Frobenius norm. Given numerical examples show that the algorithm is efficient.

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ON THE PURE IMAGINARY QUATERNIONIC LEAST SQUARES SOLUTIONS OF MATRIX EQUATION

  • WANG, MINGHUI;ZHANG, JUNTAO
    • Journal of applied mathematics & informatics
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    • v.34 no.1_2
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    • pp.95-106
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    • 2016
  • In this paper, according to the classical LSQR algorithm forsolving least squares (LS) problem, an iterative method is proposed for finding the minimum-norm pure imaginary solution of the quaternionic least squares (QLS) problem. By means of real representation of quaternion matrix, the QLS's correspongding vector algorithm is rewrited back to the matrix-form algorthm without Kronecker product and long vectors. Finally, numerical examples are reported that show the favorable numerical properties of the method.

AN ITERATIVE ALGORITHM FOR THE LEAST SQUARES SOLUTIONS OF MATRIX EQUATIONS OVER SYMMETRIC ARROWHEAD MATRICES

  • Ali Beik, Fatemeh Panjeh;Salkuyeh, Davod Khojasteh
    • Journal of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.349-372
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    • 2015
  • This paper concerns with exploiting an oblique projection technique to solve a general class of large and sparse least squares problem over symmetric arrowhead matrices. As a matter of fact, we develop the conjugate gradient least squares (CGLS) algorithm to obtain the minimum norm symmetric arrowhead least squares solution of the general coupled matrix equations. Furthermore, an approach is offered for computing the optimal approximate symmetric arrowhead solution of the mentioned least squares problem corresponding to a given arbitrary matrix group. In addition, the minimization property of the proposed algorithm is established by utilizing the feature of approximate solutions derived by the projection method. Finally, some numerical experiments are examined which reveal the applicability and feasibility of the handled algorithm.

EFFICIENT ESTIMATION OF THE REGULARIZATION PARAMETERS VIA L-CURVE METHOD FOR TOTAL LEAST SQUARES PROBLEMS

  • Lee, Geunseop
    • Journal of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1557-1571
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    • 2017
  • The L-curve method is a parametric plot of interrelation between the residual norm of the least squares problem and the solution norm. However, the L-curve method may be hard to apply to the total least squares problem due to its no closed form solution of the regularized total least squares problems. Thus the sequence of the solution norm under the fixed regularization parameter and its corresponding residual need to be found with an efficient manner. In this paper, we suggest an efficient algorithm to find the sequence of the solutions and its residual in order to plot the L-curve for the total least squares problems. In the numerical experiments, we present that the proposed algorithm successfully and efficiently plots fairly 'L' like shape for some practical regularized total least squares problems.

Least Squares Approach for Structural Reanalysis

  • Kyung-Joon Cha;Ho-Jong Jang;Dal-Sun Yoon
    • Journal of the Korean Statistical Society
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    • v.25 no.3
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    • pp.369-379
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    • 1996
  • A study is made of approximate technique for structural reanalysis based on the force method. Perturbntion analysis of generalized least squares problem is adopted to reanalyze a damaged structure, and related results are presented.

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LEAST-SQUARES SPECTRAL COLLOCATION PARALLEL METHODS FOR PARABOLIC PROBLEMS

  • SEO, JEONG-KWEON;SHIN, BYEONG-CHUN
    • Honam Mathematical Journal
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    • v.37 no.3
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    • pp.299-315
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    • 2015
  • In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED ELEMENT METHOD FOR SOBOLEV EQUATIONS WITH A CONVECTION TERM

  • Ohm, Mi Ray;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.35 no.5
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    • pp.569-587
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    • 2019
  • In this paper, we consider a split least-squares characteristic mixed element method for Sobolev equations with a convection term. First, to manipulate both convection term and time derivative term efficiently, we apply a characteristic mixed element method to get the system of equations in the primal unknown and the flux unknown and then get a least-squares minimization problem and a least-squares characteristic mixed element scheme. Finally, we obtain a split least-squares characteristic mixed element scheme for the given problem whose system is uncoupled in the unknowns. We prove the optimal order in $L^2$ and $H^1$ normed spaces for the primal unknown and the suboptimal order in $L^2$ normed space for the flux unknown.

DUAL REGULARIZED TOTAL LEAST SQUARES SOLUTION FROM TWO-PARAMETER TRUST-REGION ALGORITHM

  • Lee, Geunseop
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.613-626
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    • 2017
  • For the overdetermined linear system, when both the data matrix and the observed data are contaminated by noise, Total Least Squares method is an appropriate approach. Since an ill-conditioned data matrix with noise causes a large perturbation in the solution, some kind of regularization technique is required to filter out such noise. In this paper, we consider a Dual regularized Total Least Squares problem. Unlike the Tikhonov regularization which constrains the size of the solution, a Dual regularized Total Least Squares problem considers two constraints; one constrains the size of the error in the data matrix, the other constrains the size of the error in the observed data. Our method derives two nonlinear equations to construct the iterative method. However, since the Jacobian matrix of two nonlinear equations is not guaranteed to be nonsingular, we adopt a trust-region based iteration method to obtain the solution.

ITERATIVE ALGORITHMS FOR THE LEAST-SQUARES SYMMETRIC SOLUTION OF AXB = C WITH A SUBMATRIX CONSTRAINT

  • Wang, Minghui;Feng, Yan
    • Journal of applied mathematics & informatics
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    • v.27 no.1_2
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    • pp.1-12
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    • 2009
  • Iterative algorithms are proposed for the least-squares symmetric solution of AXB = E with a submatrix constraint. We characterize the linear mappings from their independent element space to the constrained solution sets, study their properties and use these properties to propose two matrix iterative algorithms that can find the minimum and quasi-minimum norm solution based on the classical LSQR algorithm for solving the unconstrained LS problem. Numerical results are provided that show the efficiency of the proposed methods.

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