Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation

1971 ◽  
Vol 18 (4) ◽  
pp. 289-297 ◽  
Author(s):  
Kenneth M. Brown ◽  
J. E. Dennis
Keyword(s):  
Least Squares ◽  
Nonlinear Least Squares ◽  
Least Squares Approximation ◽  
Derivative Free ◽  
Levenberg Marquardt

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 SENSOP: A derivative-free solver for nonlinear least squares with sensitivity scaling

1993 ◽  
Vol 21 (6) ◽  
pp. 621-631 ◽  
Author(s):  
I. S. Chan ◽  
A. A. Goldstein ◽  
J. B. Bassingthwaighte
Keyword(s):  
Least Squares ◽  
Nonlinear Least Squares ◽  
Derivative Free

scholarly journals Hybrid Levenberg–Marquardt and weak-constraint ensemble Kalman smoother method

2016 ◽  
Vol 23 (2) ◽  
pp. 59-73 ◽  
Author(s):  
J. Mandel ◽  
E. Bergou ◽  
S. Gürol ◽  
S. Gratton ◽  
I. Kasanický
Keyword(s):  
Least Squares ◽  
Newton Method ◽  
Nonlinear Least Squares ◽  
Additional Observation ◽  
Linear Least Squares ◽  
Regularization Term ◽  
Marquardt Method ◽  
Kalman Smoother ◽  
Levenberg Marquardt ◽  
Adjoint Operators

Abstract. The ensemble Kalman smoother (EnKS) is used as a linear least-squares solver in the Gauss–Newton method for the large nonlinear least-squares system in incremental 4DVAR. The ensemble approach is naturally parallel over the ensemble members and no tangent or adjoint operators are needed. Furthermore, adding a regularization term results in replacing the Gauss–Newton method, which may diverge, by the Levenberg–Marquardt method, which is known to be convergent. The regularization is implemented efficiently as an additional observation in the EnKS. The method is illustrated on the Lorenz 63 model and a two-level quasi-geostrophic model.

scholarly journals Hybrid Levenberg–Marquardt and weak constraint ensemble Kalman smoother method

2015 ◽  
Vol 2 (3) ◽  
pp. 865-902 ◽  
Author(s):  
J. Mandel ◽  
E. Bergou ◽  
S. Gürol ◽  
S. Gratton
Keyword(s):  
Least Squares ◽  
Newton Method ◽  
Nonlinear Least Squares ◽  
Additional Observation ◽  
Linear Least Squares ◽  
Regularization Term ◽  
Marquardt Method ◽  
Kalman Smoother ◽  
Levenberg Marquardt ◽  
Adjoint Operators

Abstract. We propose to use the ensemble Kalman smoother (EnKS) as the linear least squares solver in the Gauss–Newton method for the large nonlinear least squares in incremental 4DVAR. The ensemble approach is naturally parallel over the ensemble members and no tangent or adjoint operators are needed. Further, adding a regularization term results in replacing the Gauss–Newton method, which may diverge, by the Levenberg–Marquardt method, which is known to be convergent. The regularization is implemented efficiently as an additional observation in the EnKS. The method is illustrated on the Lorenz 63 and the two-level quasi-geostrophic model problems.

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