A Regularization Homotopy Iterative Method for Ill-Posed Nonlinear Least Squares Problem and its Application

2011 ◽  
Vol 90-93 ◽  
pp. 3268-3273 ◽  
Author(s):  
Li Min Tang
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Regularization Parameter ◽  
Nonlinear Least Squares ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Nonlinear Least Squares Problem ◽  
Calculation Results ◽  
Ill Posed ◽  
Defor Mation

A regularization homotopy iterative method established for ill-posed nonlinear least squares problem. Two new regularization parameter selecting strategies are proposed, which are called direct search method and interval division method. The calculation results of nonlinear least squares problems show that the regularization homotopy iterative method and parameter selecting strategies proposed in this paper are correctly and applicable. And also calculation results of nonlinear adjustment of free networks with rank deficiency of Jianglong bridge pier displacement defor-mation monitoring control network show that the method not only decrease the iterative matrix con-dition number, but also make the condition number small fluctuation in full iterative process.

scholarly journals A Variable Projection Method for Solving Separable Nonlinear Least Squares Problems

1974 ◽  
Vol 3 (27) ◽  
Author(s):  
Linda Kaufman
Keyword(s):  
Numerical Analysis ◽  
Least Squares ◽  
Nonlinear Least Squares ◽  
Orthogonal Matrix ◽  
Linear Least Squares ◽  
Least Squares Problems ◽  
Least Squares Problem ◽  
Linear Least Squares Problem ◽  
Nonlinear Least Squares Problem ◽  
Trapezoidal Decomposition

<p>Consider the separable nonlinear least squares problem of finding ~a in R^n and ~alpha in R^k which, for given data (y_i, t_i) i=1,...,m and functions varphi_j(~alpha,t) j=1,2,...n (m&gt;n), minimize the functional</p><p>r(~a,~alpha) = ||~y - Phi(~alpha)~a||_(2)^(2)</p><p>where Phi(~alpha)_(i,j) = varphi_(j)(~alpha,t_j). This problem can be reduced to a nonlinear least squares problem involving $\mathovd{\mathop{\alpha}\limits_{\textstyle\tilde{}}}$ only and a linear least squares problem involving ~a only. the reduction is based on the results of Colub and Pereyra, <em>SIAM J. Numerical Analysis</em>, April 1973, and on the trapezoidal decomposition of Phi, in which an orthogonal matrix Q and a permutation matrix P are found such that</p><p>\begin{displaymath} Q Phi R = R &amp; S 0 &amp; 0 \end{array}\right) \begin{array}{l} \rbrace\, r \\ \mbox{} \end{array} \end{displaymath}</p><p>where R is nonsingular and upper trianular. To develop an algorithm to solve the nonlinear least squares probelm a formula is proposed for the Frechet derivation D(Phi_(2) (~alpha)) where Q i partioned into</p>

scholarly journals Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 158
Author(s):  
Ioannis K. Argyros ◽  
Stepan Shakhno ◽  
Roman Iakymchuk ◽  
Halyna Yarmola ◽  
Michael I. Argyros
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Local Convergence ◽  
Nonlinear Least Squares ◽  
Computational Effort ◽  
Least Squares Problems ◽  
Restricted Region ◽  
Convergence Orders ◽  
Operator Decomposition ◽  
Lipschitz Conditions

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort.

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