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Levenberg-Marquardt methods based on probabilistic gradient models and inexact subproblem solution, with application to data assimilation

E. Bergou(bergou***at***cerfacs.fr)
S. Gratton(serge.gratton***at***enseeiht.fr)
L. N. Vicente(lnv***at***mat.uc.pt)

Abstract: The Levenberg-Marquardt algorithm is one of the most popular algorithms for the solution of nonlinear least squares problems. Motivated by the problem structure in data assimilation, we consider in this paper the extension of the classical Levenberg-Marquardt algorithm to the scenarios where the linearized least squares subproblems are solved inexactly and/or the gradient model is noisy and accurate only within a certain probability. Under appropriate assumptions, we show that the modified algorithm converges globally and almost surely to a first order stationary point. Our approach is applied to an instance in variational data assimilation where stochastic models of the gradient are computed by the so-called ensemble methods.

Keywords: Levenberg-Marquardt method, nonlinear least squares, regularization, random models, inexactness, variational data assimilation, Kalman filter/smoother, Ensemble Kalman filter/smoother

Category 1: Nonlinear Optimization (Nonlinear Systems and Least-Squares )

Category 2: Stochastic Programming

Category 3: Applications -- Science and Engineering

Citation: E. Bergou, S. Gratton, and L. N. Vicente, Levenberg-Marquardt methods based on probabilistic gradient models and inexact subproblem solution, with application to data assimilation, preprint 14-24, Dept. Mathematics, Univ. Coimbra, 2014.

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Entry Submitted: 06/27/2014
Entry Accepted: 06/27/2014
Entry Last Modified: 06/27/2014

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