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Section: New Results

Estimation of non–linear dynamics under sparse constraints

Participant : Patrick Héas.

This is a collaboration with Cédéric Herzet (EPI FLUMINANCE, Inria Rennes–Bretagne Atlantique) and Angélique Drémeau (ENSTA Bretagne, Brest).

Following recent contributions in non–linear sparse representations, this work [19] , [18] focuses on a particular non–linear model, defined as the nested composition of functions. This family includes in particular discrete–time hidden Markov models. Recalling that most linear sparse representation algorithms can be straightforwardly extended to non–linear models, we emphasize that their performance highly relies on an efficient computation of the gradient of the objective function. In the particular case of interest, we propose to resort to a well–known technique from the theory of optimal control to evaluate the gradient. This computation is then implemented into the ${\ell }_1$–reweighted procedure proposed by Candès et al.  [24] , leading to a non–linear extension of it. As an example, we consider the problem of estimating the ocean state from satellite low–dimensional information by exploiting a geophysical dynamical model and a sparse decomposition of the initial condition in some redundant dictionary.

This work has also been presented at Congrès National d'Assimilation, a national event held in Toulouse in December 2014.