Stochastic sampling methods are ubiquitous in statistical mechanics, Bayesian statistics, and theoretical computer science. However, when the distribution that is being sampled is multimodal, many of these techniques converge slowly, so that a great deal of computing time is necessary to obtain reliable answers. Parallel and simulated tempering are sampling methods that are designed to converge quickly even for multimodal distributions. In this thesis, we assess the extent to which this goal is acheived.We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, meaning that it converges quickly. These conditions are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models in R^M as M increases, and for the mean-field Ising model.We also obtain upper bounds on the convergence rates of parallel and simulated tempering, yielding a single set of sufficient conditions for torpid mixing of both techniques. These conditions imply torpid mixing of parallel and simulated tempering on a normal mixture model with unequal covariances in $\R^M$ as $M$ increases and on the mean-field Potts model with $q \geq 3$, regardless of the number and choice of temperatures, as well as on the mean-field Ising model if an insufficient (fixed) set of temperatures is used. The latter result is in contrast to the rapid mixing of parallel and simulated tempering on the mean-field Ising model with a linearly increasing set of temperatures. / Dissertation

Ce travail concerne le développement, la mise au point et la modélisation d’un procédé de précipitation de nanoparticules. Le précipité « modèle » étudié est la magnétite (Fe3O4). La méthode chimique de Massart est choisie pour fabriquer les nanoparticules de magnétite, car elle est déjà bien étudiée. Un procédé de précipitation est conçu en réacteur semi-fermé et à recirculation du fluide de la cuve, permettant ainsi de réaliser un mélange intensif des fluides réactifs par des mélangeurs rapides (un tube en T et deux mélangeurs Hartridge-Roughton de tailles différentes). Différents paramètres opératoires sont testés pour déterminer leur influence sur la qualité du précipité. De nombreuses techniques analytiques sont mises en œuvre pour déterminer les propriétés des nanoparticules obtenues. Les résultats montrent que, malgré une chimie inchangée, le type de microréacteur choisi influence sensiblement la qualité des nanoparticules élémentaires et des agglomérats de magnétite. Le potentiel de nano-adsorption de la magnétite est aussi étudié et se révèle prometteur. Enfin, la modélisation hydrodynamique des mélangeurs rapides est réalisée par CFD / The present work is focused on developing and modeling a precipitation process for the production of magnetite (Fe3O4) nanoparticles. The Massart chemical method is chosen to obtain the magnetite nanoparticles owing to its detailed study on the reaction parameters. A semi-batch reactor with a recirculation system is chosen to realize this precipitation process and rapid mixers (T mixer and Hartridge-Roughton mixers of different dimensions) are used to provide an intensive mixing of reagent fluids. Different operating parameters are tested to determine their influences on the precipitate quality. Many analytic techniques are employed to determine the properties of obtained nanoparticles. The results indicate that, without changing of chemical parameters, the quality of magnetite elementary nanoparticles and agglomerates depend sensibly on the type of microreactors utilized. In addition, the magnetite nanoparticles are considered to be a hopeful nanoadsorbent and the related tests are studied. Finally, the CFD technique is used to model the hydrodynamic behaviors of the rapid mixers