<div><div>In this thesis, we focus on two challenges arising in kinetic equations, high dimensions and uncertainties. To reduce the dimensions, we proposed efficient methods for linear Boltzmann and full Boltzmann equations based on dynamic low-rank frameworks. For linear Boltzmann equation, we proposed a method that is based on macro-micro decomposition of the equation; the low-rank approximation is only used for the micro part of the solution. The time and spatial discretizations are done properly so that the overall scheme is second-order accurate (in both the fully kinetic and the limit regime) and asymptotic-preserving (AP). That is, in the diffusive regime, the scheme becomes a macroscopic solver for the limiting diffusion equation that automatically captures the low-rank structure of the solution. Moreover, the method can be implemented in a fully explicit way and is thus significantly more efficient compared to the previous state of the art. We demonstrate the accuracy and efficiency of the proposed low-rank method by a number of four-dimensional (two dimensions in physical space and two dimensions in velocity space) simulations. We further study the adaptivity of low-rank methods in full Boltzmann equation. We proposed a highly efficient adaptive low- rank method in Boltzmann equation for computations of steady state solutions. The main novelties of this approach are: On one hand, to the best of our knowledge, the dynamic low- rank integrator hasn’t been applied to full Boltzmann equation till date. The full collision operator is local in spatial variable while the convection part is local in velocity variable. This separated nature is well-suited for low-rank methods. Compared with full grid method (finite difference, finite volume,...), the dynamic low-rank method can avoid the full computations of collision operators in each spatial grid/elements. Resultingly, it can achieve much better efficiency especially for some low rank flows (e.g. normal shock wave). On the other hand, our adaptive low-rank method uses a novel dynamic thresholding strategy to adaptively control the computational rank to achieve better efficiency especially for steady state solutions. We demonstrate the accuracy and efficiency of the proposed adaptive low rank method by a number of 1D/2D Maxwell molecule benchmark tests. On the other hand, for kinetic equations with uncertainties, we focus on non-intrusive sampling methods where we are able to inherit good properties (AP, positivity preserving) from existing deterministic solvers. We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization prob- lems. Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical multilevel Monte Carlo method especially for problems with dis- continuities<br></div></div>

Applied statisticians are often faced with the problem of dealing with high dimensional data sets when attempting to describe the variability of a single set of variables, or trying to predict the variation of one set of variables from another. In this study, two data reduction methods are described: Redundancy Analysis and Partial Least Squares. A hybrid approach developed by Bougeard et al., (2007) and called Continuum Redundancy-Partial Least Squares, is described. All three methods are extended to the frequency domain in order to allow the lower dimensional subspace used to describe the variability to change with frequency. To illustrate and compare the three methods, and their frequency dependent generalizations, an idealized coupled atmosphere-ocean model is introduced in state space form. This model provides explicit expressions for the covariance and cross spectral matrices required by the various methods; this allows the strengths and weaknesses of the methods to be identified.

Cette thèse s'inscrit dans le cadre de l'analyse statistique de données en grande dimension. Nous avons en effet aujourd'hui accès à un nombre toujours plus important d'information. L'enjeu majeur repose alors sur notre capacité à explorer de vastes quantités de données et à en inférer notamment les structures de dépendance. L'objet de cette thèse est d'étudier et d'apporter des garanties théoriques à certaines méthodes d'estimation de structures de dépendance de données en grande dimension.La première partie de la thèse est consacrée à l'étude de modèles parcimonieux et aux méthodes de type Lasso. Après avoir présenté les résultats importants sur ce sujet dans le chapitre 1, nous généralisons le cas gaussien à des modèles exponentiels généraux. La contribution majeure à cette partie est présentée dans le chapitre 2 et consiste en l'établissement d'inégalités oracles pour une procédure Group Lasso appliquée aux modèles linéaires généralisés. Ces résultats montrent les bonnes performances de cet estimateur sous certaines conditions sur le modèle et sont illustrés dans le cas du modèle Poissonien. Dans la deuxième partie de la thèse, nous revenons au modèle de régression linéaire, toujours en grande dimension mais l'hypothèse de parcimonie est cette fois remplacée par l'existence d'une structure de faible dimension sous-jacente aux données. Nous nous penchons dans cette partie plus particulièrement sur la méthode PLS qui cherche à trouver une décomposition optimale des prédicteurs étant donné un vecteur réponse. Nous rappelons les fondements de la méthode dans le chapitre 3. La contribution majeure à cette partie consiste en l'établissement pour la PLS d'une expression analytique explicite de la structure de dépendance liant les prédicteurs à la réponse. Les deux chapitres suivants illustrent la puissance de cette formule aux travers de nouveaux résultats théoriques sur la PLS . Dans une troisième et dernière partie, nous nous intéressons à la modélisation de structures au travers de graphes et plus particulièrement à la détection de communautés. Après avoir dressé un état de l'art du sujet, nous portons notre attention sur une méthode en particulier connue sous le nom de spectral clustering et qui permet de partitionner les noeuds d'un graphe en se basant sur une matrice de similarité. Nous proposons dans cette thèse une adaptation de cette méthode basée sur l'utilisation d'une pénalité de type l1. Nous illustrons notre méthode sur des simulations. / This thesis falls within the context of high-dimensional data analysis. Nowadays we have access to an increasing amount of information. The major challenge relies on our ability to explore a huge amount of data and to infer their dependency structures.The purpose of this thesis is to study and provide theoretical guarantees to some specific methods that aim at estimating dependency structures for high-dimensional data. The first part of the thesis is devoted to the study of sparse models through Lasso-type methods. In Chapter 1, we present the main results on this topic and then we generalize the Gaussian case to any distribution from the exponential family. The major contribution to this field is presented in Chapter 2 and consists in oracle inequalities for a Group Lasso procedure applied to generalized linear models. These results show that this estimator achieves good performances under some specific conditions on the model. We illustrate this part by considering the case of the Poisson model. The second part concerns linear regression in high dimension but the sparsity assumptions is replaced by a low dimensional structure underlying the data. We focus in particular on the PLS method that attempts to find an optimal decomposition of the predictors given a response. We recall the main idea in Chapter 3. The major contribution to this part consists in a new explicit analytical expression of the dependency structure that links the predictors to the response. The next two chapters illustrate the power of this formula by emphasising new theoretical results for PLS. The third and last part is dedicated to graphs modelling and especially to community detection. After presenting the main trends on this topic, we draw our attention to Spectral Clustering that allows to cluster nodes of a graph with respect to a similarity matrix. In this thesis, we suggest an alternative to this method by considering a $l_1$ penalty. We illustrate this method through simulations.