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.

scholarly journals Least Squares Estimator for Vasicek Model Driven by Fractional Levy Processes

2018 ◽  
Vol 14 (2) ◽  
pp. 8013-8024
Author(s):  
Qingbo Wang ◽  
Xiuwei Yin
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Asymptotic Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Estimation Problem ◽  
Least Squares Estimator ◽  
Parameter Estimation Problem ◽  
Vasicek Model ◽  
Model Driven

In this paper, we consider parameter estimation problem for Vasicek model driven by fractional lévy processes defined We construct least squares estimator for drift parameters based on time?continuous observations, the consistency and asymptotic distribution of these estimators are studied in the non?ergodic case. In contrast to the fractional Vasicek model, it can be regarded as a Lévy generalization of fractional Vasicek model.

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>

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

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