Vitesse de convergence de l'échantillonneur de Gibbs appliqué à des modèles de la physique statistique

Abstract : Monte Carlo Markov chain methods MCMC are mathematical tools used to simulate probability measures π defined on state spaces of high dimensions. The speed of convergence of this Markov chain X to its invariant state π is a natural question to study in this context.To measure the convergence rate of a Markov chain we use the total variation distance. It is well known that the convergence rate of a reversible Markov chain depends on its second largest eigenvalue in absolute value denoted by β!. An important part in the estimation of β! is the estimation of the second largest eigenvalue which is denoted by β1.Diaconis and Stroock (1991) introduced a method based on Poincaré inequality to obtain a bound for β1 for general finite state reversible Markov chains.In this thesis we use the Chen and Shiu approach to study the case of the Gibbs sampler for the 1−D Ising model with three and more states which is also called Potts model. Then, we generalize the result of Shiu and Chen (2015) to the case of the 2−D Ising model with two states.The results we obtain improve the ones obtained by Ingrassia (1994). Then, we introduce some method to disrupt the Gibbs sampler in order to improve its convergence rate to equilibrium.
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Amine Helali. Vitesse de convergence de l'échantillonneur de Gibbs appliqué à des modèles de la physique statistique. Analyse numérique [math.NA]. Université de Bretagne occidentale - Brest; Université de Sfax (Tunisie), 2019. Français. ⟨NNT : 2019BRES0002⟩. ⟨tel-02292870⟩

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