In dimension one, optimal transportation is rather straightforward. The easiness with which a solution can be obtained in that setting has recently been used to tackle more general situations, each time thanks to the same method. First, disintegrate your problem to go back to the unidimensional case, and apply the available 1D methods to get a first result; then, improve it gradually using some evolution process.This dissertation explores that direction more thoroughly. Looking back at two problems only partially solved this way, I show how this viewpoint in fact allows to go even further.The first of these two problems concerns the computation of Yann Brenier's optimal map. Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio found a new way to obtain it, thanks to an differential equation for which an initial condition is given by the Knothe--Rosenblatt rearrangement. (The latter is precisely defined by a series of unidimensional transformations.) However, they only dealt with discrete target measures; I~generalize their approach to a continuous setting. By differentiation, the Monge--Ampère equation readily gives a PDE satisfied by the Kantorovich potential; but to get a proper initial condition, it is necessary to use the Nash--Moser version of the implicit function theorem.The basics of optimal transport are recalled in the first chapter, and the Nash--Moser theory is exposed in chapter 2. My results are presented in chapter 3, and numerical experiments in chapter 4.The last chapter deals with the IDT algorithm, devised by François Pitié, Anil C. Kokaram, and Rozenn Dahyot. It builds a transport map that seems close enough to the optimal map for most applications. A complete mathematical understanding of the procedure is, however, still lacking. An interpretation as a gradient flow in the space of probability measures is proposed, with the sliced Wasserstein distance as the functional. I also prove the equivalence between the sliced and usual Wasserstein distances.

This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

La première partie de cette thèse concerne l'inférence de modèles statistiques non normalisés. Nous étudions deux méthodes d'inférence basées sur de l'échantillonnage aléatoire : Monte-Carlo MLE (Geyer, 1994), et Noise Contrastive Estimation (Gutmann et Hyvarinen, 2010). Cette dernière méthode fut soutenue par une justification numérique d'une meilleure stabilité, mais aucun résultat théorique n'avait encore été prouvé. Nous prouvons que Noise Contrastive Estimation est plus robuste au choix de la distribution d'échantillonnage. Nous évaluons le gain de précision en fonction du budget computationnel. La deuxième partie de cette thèse concerne l'échantillonnage aléatoire approché pour les distributions de grande dimension. La performance de la plupart des méthodes d’échantillonnage se détériore rapidement lorsque la dimension augmente, mais plusieurs méthodes ont prouvé leur efficacité (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). Dans la continuité de certains travaux récents (Eberle et al., 2017 ; Cheng et al., 2018), nous étudions certaines discrétisations d’un processus connu sous le nom de kinetic Langevin diffusion. Nous établissons des vitesses de convergence explicites vers la distribution d'échantillonnage, qui ont une dépendance polynomiale en la dimension. Notre travail améliore et étend les résultats de Cheng et al. pour les densités log-concaves. / The first part of this thesis concerns the inference of un-normalized statistical models. We study two methods of inference based on sampling, known as Monte-Carlo MLE (Geyer, 1994), and Noise Contrastive Estimation (Gutmann and Hyvarinen, 2010). The latter method was supported by numerical evidence of improved stability, but no theoretical results had yet been proven. We prove that Noise Contrastive Estimation is more robust to the choice of the sampling distribution. We assess the gain of accuracy depending on the computational budget. The second part of this thesis concerns approximate sampling for high dimensional distributions. The performance of most samplers deteriorates fast when the dimension increases, but several methods have proven their effectiveness (e.g. Hamiltonian Monte Carlo, Langevin Monte Carlo). In the continuity of some recent works (Eberle et al., 2017; Cheng et al., 2018), we study some discretizations of the kinetic Langevin diffusion process and establish explicit rates of convergence towards the sampling distribution, that scales polynomially fast when the dimension increases. Our work improves and extends the results established by Cheng et al. for log-concave densities.

Los incendios en recintos confinados son un tipo de emergencia que involucra a bomberos cuyas vidas a veces se ponen en peligro. En cualquier incendio confinado, el equipo de emergencia puede encontrar dos tipos de ambientes de combustión, ventilados o infra-ventilados. El comportamiento cambiante de este escenario depende de múltiples factores como el tamaño del recinto, la ventilación o el combustible involucrado, entre otros. Sin embargo, la dificultad de manejar este tipo de situaciones junto con el potencial error humano sigue siendo un desafío sin resolver para los bomberos en la actualidad. En ocasiones si se dan las condiciones adecuadas, pueden aparecer los fenómenos, extremadamente peligrosos, que son estudio de este trabajo (flashover y backdraft). Por lo tanto, existe una gran demanda de nuevas técnicas y tecnologías para abordar este tipo de emergencias que amenazan la vida y puede causar graves daños estructurales. A lo anterior hay que añadir que la incorporación de cámaras térmicas en los servicios de extinción de incendios y salvamentos, supone un gran avance que puede ayudar a prevenir estos tipos de fenómenos en tiempo real utilizando técnicas de inteligencia artificial.

The purpose of this study was to describe how gender was portrayed and to determine how gender roles were depicted and defined in a selection of Modern and Postmodern American plays. This study was based on the symbolic interaction theory of gender that suggests that social roles are learned over time and are subject to constant reinforcement. The significance of this study was derived from the broad topic of gender because gender issues are relevant to a variety of fields and exploring the effects of gender in one field contributes to the understanding of gender in another field. The plays in this study were Votes for Women, Robins; Trifles, Glaspell; Our Town, Wilder; Moon for the Misbegotten, O'Neill; The Glass Menagerie, Williams; Death of a Salesman, Miller; A Raisin in the Sun, Hansberry; Funnyhouse of a Negro, Kennedy; Uncommon Women and Others, Wasserstein; Fefu and Her Friends, Fornes; spell #7, Shange; Fool for Love, Shepard; Fences, Wilson; Oleanna, Mamet; and How I Learned to Drive, Vogel. Two of the study's research questions explored the types of gender roles and behaviors that the characters presented. Two questions focused on considering if the time period or the sex of the playwright were factors in the presentations of gender. Gender behaviors were divided into four categories: Behavior Characteristics, Communication Patterns, Sources of Power, and Physical Appearance. Using narrative analysis techniques, the plays were analyzed for the specific traits in each category. The majority of the characters were assigned traditional gender roles and displayed traditional gender behavior traits. Based their gender roles and behavior in their roles, characters faced limitations that confined their actions and restricted their choices. Characters experienced consequences for their behaviors, and female characters received harsher punishments for deviant behaviors than male characters. Gender portrayal in Modern plays was more in keeping with traditional patterns than in Postmodern plays. Female playwrights presented more diverse roles for female characters and often explored gender as a major theme in their plays. Where applicable, race, in concert with gender, was an additional factor that governed characters' behaviors by further restricting behavior or possible actions.

The goal of optimal transport is to map a source probability measure to a destination one with the minimum possible cost. However, the optimal mapping might not be feasible under some practical constraints. One such example is to realize a transport mapping through an information bottleneck. As the optimal mapping may induce infinite mutual information between the source and the destination, the existence of an information bottleneck forces one to resort to some suboptimal mappings. Investigating this type of constrained optimal transport problems is clearly of both theoretical significance and practical interest. In this work, we substantiate a particular form of constrained optimal transport in the context of quantum-to-classical systems by establishing an Output-Constrained Rate-Distortion Theorem similar to the classical case introduced by Yuksel et al. This theorem develops a noiseless communication channel and finds the least required transmission rate R and common randomness Rc to transport a sufficiently large block of n i.i.d. source quantum states, to samples forming a perfectly i.i.d. classical destination distribution, while maintaining the distortion between them. The coding theorem provides operational meanings to the problem of Rate-Limited Optimal Transport, which finds the optimal transportation from source to destination subject to the rate constraints on transmission and common randomness. We further provide an analytical evaluation of the quantum-to-classical rate-limited optimal transportation cost for the case of qubit source state and Bernoulli output distributions with unlimited common randomness. The evaluation results in a transcendental system of equations whose solution provides the rate-distortion curve of the transportation protocol. We further extend this theorem to continuous-variable quantum systems by employing a clipping and quantization argument and using our discrete coding theorem. Moreover, we derive an analytical solution for rate-limited Wasserstein distance of 2nd order for Gaussian quantum systems with Gaussian output distribution. We also provide a Gaussian optimality theorem for the case of unlimited common randomness, showing that Gaussian measurement optimizes the rate in a system with Gaussian source and destination. / Thesis / Doctor of Philosophy (PhD) / We establish a coding theorem for rate-limited quantum-classical optimal transport systems with limited classical common randomness. The coding theorem, referred to as the output-constrained rate-distortion theorem, characterizes the rate region of measurement protocols on a product quantum source state for faithful construction of a given classical destination distribution while maintaining the source-destination distortion below a prescribed threshold with respect to a general distortion observable. This theorem provides a solution to the problem of rate-limited optimal transport, which aims to find the optimal cost of transforming a source quantum state to a destination distribution via a measurement channel with a limited classical communication rate. The coding theorem is further extended to cover Bosonic continuous-variable quantum systems. The analytical evaluation is provided for the case of a qubit measurement system with unlimited common randomness, as well as the case of Gaussian quantum systems.

<p dir="ltr">This thesis delves into the development and integration of energy-dissipative methods, with applications spanning numerical analysis, optimization, and deep neural networks, primarily targeting gradient flows and porous medium equations. In the realm of optimization, we introduce the element-wise relaxed scalar auxiliary variable (E-RSAV) algorithm, showcasing its robustness and convergence through extensive numerical experiments. Complementing this, we design an Energy-Dissipative Evolutionary Deep Operator Neural Network (DeepONet) to numerically address a suite of partial differential equations. By employing a dual-subnetwork structure and utilizing the Scalar Auxiliary Variable (SAV) method, the network achieves impeccable approximations of operators while upholding the Energy Dissipation Law, even when training data comprises only the initial state. Lastly, we formulate first-order schemes tailored for Wasserstein gradient flows. Our schemes demonstrate remarkable properties, including mass conservation, unique solvability, positivity preservation, and unconditional energy dissipation. Collectively, the innovations presented here offer promising pathways for efficient and accurate numerical solutions in both gradient flows and Wasserstein gradient flows, bridging the gap between traditional optimization techniques and modern neural network methodologies.</p>

Cette thèse comporte deux parties distinctes, toutes les deux liées à la théorie du transport optimal. Dans la première partie, nous considérons une variété riemannienne, deux mesures à densité régulière et un coût de transport, typiquement la distance géodésique quadratique et nous nous intéressons à la régularité de l’application de transport optimal. Le critère décisif à cette régularité s’avère être le signe du tenseur de Ma-Trudinger-Wang (MTW). Nous présentons tout d’abord une synthèse des travaux réalisés sur ce tenseur. Nous nous intéressons ensuite au lien entre la géométrie des lieux d’injectivité et le tenseur MTW. Nous montrons que dans de nombreux cas, la positivité du tenseur MTW implique la convexité des lieux d’injectivité. La deuxième partie de cette thèse est liée aux équations aux dérivées partielles. Certaines peuvent être considérées comme des flots gradients dans l’espace de Wasserstein W2. C’est le cas de l’équation de Keller-Segel en dimension 2. Pour cette équation nous nous intéressons au problème de quantification de la masse lors de l’explosion des solutions ; cette explosion apparaît lorsque la masse initiale est supérieure à un seuil critique Mc. Nous cherchons alors à montrer qu’elle consiste en la formation d’un Dirac de masse Mc. Nous considérons ici un modèle particulaire en dimension 1 ayant le même comportement que l’équation de Keller-Segel. Pour ce modèle nous exhibons des bassins d’attractions à l’intérieur desquels l’explosion se produit avec seulement le nombre critique de particules. Finalement nous nous intéressons au profil d’explosion : à l’aide d’un changement d’échelle parabolique nous montrons que la structure de l’explosion correspond aux points critiques d’une certaine fonctionnelle. / This thesis consists in two distinct parts both related to the optimal transport theory.The first part deals with the regularity of the optimal transport map. The key tool is the Ma-Trundinger-Wang tensor and especially its positivity. We first give a review of the known results about the MTW tensor. We then explore the geometrical consequences of the MTW tensor on the injectivity domain. We prove that in many cases the positivity of MTW implies the convexity of the injectivity domain. The second part is devoted to the behaviour of a Keller-Segel solution in the super critical case. In particular we are interested in the mass quantization problem: we wish to quantify the mass aggregated when the blow-up occurs. In order to study the behaviour of the solution we consider a particle approximation of a Keller-Segel type equation in dimension 1. We define this approximation using the gradient flow interpretation of the Keller-Segel equation and the particular structure of the Wasserstein space in dimension 1. We show two kinds of results; we first prove a stability theorem for the blow-up mechanism: we exhibit basins of attraction in which the solution blows up with only the critical number of particles. We then prove a rigidity theorem for the blow-up mechanism: thanks to a parabolic rescaling we prove that the structure of the blow-up is given by the critical points of a certain functional.

Dans cette thèse, notre but est de donner un cadre numérique général pour approcher les solutions des problèmes du transport optimal (TO). L’idée générale est d’introduire une régularisation entropique du problème initial. Le problème régularisé correspond à minimiser une entropie relative par rapport à une mesure de référence donnée. En effet, cela équivaut à trouver la projection d’un couplage par rapport à la divergence de Kullback-Leibler. Cela nous permet d’utiliser l’algorithme de Bregman/Dykstra et de résoudre plusieurs problèmes variationnels liés au TO. Nous nous intéressons particulièrement à la résolution des problèmes du transport optimal multi-marges (TOMM) qui apparaissent dans le cadre de la dynamique des fluides (équations d’Euler incompressible à la Brenier) et de la physique quantique (la théorie de fonctionnelle de la densité ). Dans ces cas, nous montrons que la régularisation entropique joue un rôle plus important que de la simple stabilisation numérique. De plus, nous donnons des résultats concernant l’existence des transports optimaux (par exemple des transports fractals) pour le problème TOMM. / In this thesis we aim at giving a general numerical framework to approximate solutions to optimal transport (OT) problems. The general idea is to introduce an entropic regularization of the initialproblems. The regularized problem corresponds to the minimization of a relative entropy with respect a given reference measure. Indeed, this is equivalent to find the projection of the joint coupling with respect the Kullback-Leibler divergence. This allows us to make use the Bregman/Dykstra’s algorithm and solve several variational problems related to OT. We are especially interested in solving multi-marginal optimal transport problems (MMOT) arising in Physics such as in Fluid Dynamics (e.g. incompressible Euler equations à la Brenier) and in Quantum Physics (e.g. Density Functional Theory). In these cases we show that the entropic regularization plays a more important role than a simple numerical stabilization. Moreover, we also give some important results concerning existence and characterization of optimal transport maps (e.g. fractal maps) for MMOT .

[pt] Uma medida de probabilidade com suporte compacto em um grupo de matrizes determina uma sequência de matrizes aleatórias i.i.d. Considere o processo multiplicativo correspondente e suas médias geométricas. O teorema de Furstenberg-Kesten, análogo da lei dos grandes números neste cenário, garante que as médias geométricas desse processo multiplicativo convergem quase certamente para uma constante, chamada de expoente de Lyapunov maximal da medida dada. Este conceito pode ser reformulado no contexto mais geral da teoria ergódica usando cociclos lineares aleatórios sobre o shift de Bernoulli. Uma questão natural diz respeito às propriedades de regularidade do expoente de Lyapunov como uma função dos seus dados. Sob uma condição de irredutibilidade e em um cenário específico (que foi posteriormente generalizado por vários autores) Le Page estabeleceu a continuidade de Holder do expoente de Lyapunov. Recentemente, Baraviera e Duarte obtiveram uma prova direta e elegante deste tipo de resultado. Seu argumento usa a fórmula de Furstenberg e as propriedades de regularidade da medida estacionária. Seguindo sua abordagem, neste trabalho obtemos um novo resultado mostrando que, sob a mesma hipótese de irredutibilidade, o expoente de Lyapunov depende Hölder continuamente da medida, relativamente à métrica de Wasserstein, generalizando assim o resultado de Baraviera e Duarte. / [en] A compactly supported probability measure on a group of matrices determines a sequence of i.i.d. random matrices. Consider the corresponding multiplicative process and its geometric averages. Furstenberg-Kesten s theorem, the analogue of the law of large numbers in this setting, ensures that the geometric averages of this multiplicative process converge almost surely to a constant, called the maximal Lyapunov exponent of the given measure. This concept can be reformulated in the more general context of ergodic theory using random linear cocycles over the Bernoulli shift. A natural question concerns the regularity properties of the Lyapunov exponent as a function of the data. Under an irreducibility condition and in a specific setting (which was later generalized by various authors) Le Page established the Holder continuity of the Lyapunov exponent. Recently, Baraviera and Duarte obtained a direct and elegant proof of this type of result. Their argument uses Furstenberg s formula and the regularity properties of the stationary measure. Following their approach, in this work we obtain a new result showing that under the same irreducibility hypothesis, the Lyapunov exponent depends Holder continuously on the measure, relative to the Wasserstein metric, thus generalizing the result of Baraviera and Duarte.