Local Convergence Analysis of the Levenberg–Marquardt Framework for Nonzero-Residue Nonlinear Least-Squares Problems Under an Error Bound Condition

2019 ◽  
Vol 183 (3) ◽  
pp. 1099-1122 ◽  
Author(s):  
Roger Behling ◽  
Douglas S. Gonçalves ◽  
Sandra A. Santos
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Error Bound ◽  
Local Convergence ◽  
Nonlinear Least Squares ◽  
Least Squares Problems ◽  
Levenberg Marquardt ◽  
Local Convergence Analysis

scholarly journals Local convergence of the Gauss-Newton-Kurchatov method under generalized Lipschitz conditions

2021 ◽  
Vol 13 (2) ◽  
pp. 305-314
Author(s):  
S.M. Shakhno ◽  
H.P. Yarmola
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Local Convergence ◽  
Nonlinear Least Squares ◽  
Convergence Order ◽  
Least Squares Problems ◽  
Convergence Radius ◽  
Numerical Examples ◽  
Lipschitz Conditions ◽  
Theoretical Results

We investigate the local convergence of the Gauss-Newton-Kurchatov method for solving nonlinear least squares problems. This method is a combination of Gauss-Newton and Kurchatov methods and it is used for problems with the decomposition of the operator. The convergence analysis of the method is performed under the generalized Lipshitz conditions. The conditions of convergence, radius and the convergence order of the considered method are established. Given numerical examples confirm the theoretical results.

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.

Modified inexact Levenberg–Marquardt methods for solving nonlinear least squares problems

2019 ◽  
Vol 74 (2) ◽  
pp. 547-582 ◽  
Author(s):  
Jifeng Bao ◽  
Carisa Kwok Wai Yu ◽  
Jinhua Wang ◽  
Yaohua Hu ◽  
Jen-Chih Yao
Keyword(s):  
Least Squares ◽  
Nonlinear Least Squares ◽  
Least Squares Problems ◽  
Levenberg Marquardt

scholarly journals Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 99 ◽  
Author(s):  
Ioannis Argyros ◽  
Stepan Shakhno ◽  
Yurii Shunkin
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Difference Method ◽  
Divided Difference ◽  
Nonlinear Least Squares ◽  
Convergence Order ◽  
Least Squares Problems ◽  
Convergence Radius ◽  
Numerical Examples ◽  
Theoretical Results

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results.

scholarly journals Extended convergence of Gauss-Newton’s method and uniqueness of the solution

2018 ◽  
Vol 34 (2) ◽  
pp. 135-142
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
YEOL JE CHO ◽  
SANTHOSH GEORGE ◽  
◽  
...  
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Least Squares ◽  
Lipschitz Functions ◽  
New Technique ◽  
Least Squares Problems ◽  
Uniqueness Of The Solution ◽  
Ball Convergence

The aim of this paper is to extend the applicability of the Gauss-Newton’s method for solving nonlinear least squares problems using our new idea of restricted convergence domains. The new technique uses tighter Lipschitz functions than in earlier papers leading to a tighter ball convergence analysis.

scholarly journals A variant of the Levenberg-Marquardt method with adaptive parameters for systems of nonlinear equations

2021 ◽  
Vol 7 (1) ◽  
pp. 1241-1256
Author(s):  
Lin Zheng ◽  
◽  
Liang Chen ◽  
Yanfang Ma ◽  
◽  
...  
Keyword(s):  
Least Squares ◽  
Error Bound ◽  
Nonlinear Equations ◽  
Quadratic Convergence ◽  
Nonlinear Least Squares ◽  
Systems Of Nonlinear Equations ◽  
Local Error ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Wolfe Line Search

<abstract><p>The Levenberg-Marquardt method is one of the most important methods for solving systems of nonlinear equations and nonlinear least-squares problems. It enjoys a quadratic convergence rate under the local error bound condition. Recently, to solve nonzero-residue nonlinear least-squares problem, Behling et al. propose a modified Levenberg-Marquardt method with at least superlinearly convergence under a new error bound condtion <sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>. To extend their results for systems of nonlinear equations, by choosing the LM parameters adaptively, we propose an efficient variant of the Levenberg-Marquardt method and prove its quadratic convergence under the new error bound condition. We also investigate its global convergence by using the Wolfe line search. The effectiveness of the new method is validated by some numerical experiments.</p></abstract>

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