Dans une première partie théorique, nous nous penchons sur une analyse rigoureuse des performances de l'algorithme Sequential Monte Carlo (SMC) conduisant à des résultats de type bornes L^p et inégalités de concentration. Nous abordons notamment le cas particulier des SMC associés à des schémas de température, et analysons sur ce sujet un processus à schéma adaptatif.Dans une seconde partie appliquée, nous illustrons son utilisation par la résolution de problèmes inverses concrets en électromagnétisme. Le plus important d'entre eux consiste à estimer les propriétés radioélectriques de matériaux recouvrant un objet de géométrie connue, et cela à partir de mesures de champs rétrodiffusés. Nous montrons comment l'algorithme SMC, couplé à des calculs analytiques, permet une inversion bayésienne, et fournit des estimées robustes enrichies d'estimations des incertitudes. / Sequential and Quantum Monte Carlo methods, as well as genetic type search algorithms, can be interpreted as a mean field and interacting particle approximation of Feynman-Kac models in distribution spaces. The performance of these population Monte Carlo algorithms is strongly related to the stability properties of nonlinear Feynman-Kac semigroups. In a first theoretical part, we analyze these models in terms of Dobrushin ergodic coefficients of the reference Markov transitions and the oscillations of the potential functions. Sufficient conditions for uniform concentration inequalities w.r.t. time are expressed explicitly in terms of these two quantities. We provide an original perturbation analysis that applies to annealed and adaptive FK models, yielding what seems to be the first results of this kind for these type of models. Special attention is devoted to the particular case of Boltzmann-Gibbs measures' sampling. In this context, we design an explicit way of tuning the number of Markov Chain Monte Carlo iterations with temperature schedule. We also propose and analyze an alternative interacting particle method based on an adaptive strategy to define the temperature increments. In a second, applied part, we illustrate the use of these SMC algorithms in the field of inverse problems. Mainly, the following electromagnetism (EM) inverse problem is addressed. It consists in estimating local radioelectric properties of materials recovering an object from global EM scattering measurements, at various incidences and wave frequencies. This large scale ill-posed inverse problem is explored by an intensive exploitation of an efficient 2D Maxwell solver, distributed on high performance computing machines. Applied to a large training data set, a statistical analysis reduces the problem to a simpler probabilistic metamodel, on which Bayesian inference can be performed. Considering the radioelectric properties as a hidden dynamic stochastic process, that evolves in function of the frequency, it is shown how the Sequential Monte Carlo methods can take benefit of the structure and provide local EM property estimates.
| Identifer | oai:union.ndltd.org:theses.fr/2013BOR14787 |
| Date | 29 May 2013 |
| Creators | Giraud, François |
| Contributors | Bordeaux 1, Del Moral, Pierre, Minvielle, Pierre |
| Source Sets | Dépôt national des thèses électroniques françaises |
| Language | French |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text, StillImage |
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