Bergou, El houcine. Numerical methods for least squares problems with application to data assimilation. PhD, Institut National Polytechnique de Toulouse, 2014

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Official URL: http://ethesis.inptoulouse.fr/archive/00002894/
Abstract
The LevenbergMarquardt algorithm (LM) 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 thesis the extension of the LM algorithm to the scenarios where the linearized least squares subproblems, of the form minAx  b ^2, 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 socalled ensemble Kalman smoother (EnKS). A convergence proof in L^p of EnKS in the limit for large ensembles to the Kalman smoother is given. We also show the convergence of LMEnKS approach, which is a variant of the LM algorithm with EnKS as a linear solver, to the classical LM algorithm where the linearized subproblem is solved exactly. The sensitivity of the trucated sigular value decomposition method to solve the linearized subprobems is studied. We formulate an explicit expression for the condition number of the truncated least squares solution. This expression is given in terms of the singular values of A and the Fourier coefficients of b.
Item Type:  PhD Thesis 

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Institution:  Université de Toulouse > Institut National Polytechnique de Toulouse  INPT (FRANCE) 
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Research Director:  Gratton, Serge 
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Deposited On:  27 Jan 2015 22:58 
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