We introduce new methods based on the L2 Optimal Transport (OT) problem and the Navier-Stokes equations to approximate a fluid velocity field from images obtained with Particle Image Velocimetry (PIV) measurements. The main idea is to consider two successive images as the initial and final densities in the OT problem, and to use the associated OT flow as an estimate of the underlying physical flow. We build a simple but realistic model for PIV data, and use it to analyze the behavior of the transport map in this situation. We then design and implement a series of post-processing filters created to improve the quality of the numerical results, and we establish comparisons with traditional cross-correlation algorithms. These results indicate that the OT-PIV procedure performs well on low to medium seeding densities, and that it gives better results than typical cross-correlation algorithms in some cases. Finally, we use a variational method to project the OT velocity field on the space of solutions of the Navier-Stokes equations, and extend it to the rest of the fluid domain, outside the particle locations. This extension provides an effective filtering of the OT solution beyond the local post-processing filters, as demonstrated by several numerical experiments. / Graduate

One of the main research questions in financial mathematics is that of portfolio construction: how should one systematically invest their wealth in a financial market? This problem has been tackled in numerous ways, typically through the modeling of market prices and the optimization of an investment objective. A recent approach to portfolio construction is that offered by Stochastic Portfolio Theory, in which a relatively general market model is assumed, and the portfolio selection criterion is to outperform a benchmark with probability one. In order to achieve this, Robert Fernholz developed the method of functional generation, which allows one to explicitly construct and study portfolios that depend deterministically on the currently observable prices. The typical example of such a strategy is the diversity-weighted portfolio, which we extend in the first chapter of this work with a negative-parameter variation. We show that several modifications of this portfolio outperform the market index in theory, under certain assumptions on the market, and we perform an empirical study that confirms this. In our second chapter, we develop a data-driven portfolio construction method that goes beyond functional generation, allowing for the inclusion of factors other than current prices. We empirically show that this Bayesian nonparametric approach, which utilizes Gaussian processes, leads to drastically improved performance compared to benchmark portfolios. Next, we establish a formal equivalence between the method of functional generation and the mathematical field of optimal transport. Our results fortify known relations between the two, and extend this connection to additive functional generation, a recent variation of the method. In Chapter 4, we apply our results to derive new properties and characterizations of functionally-generated wealth processes in very general market models. Finally, we develop methods for incorporating defaults into functional generation, improving its real-world implementability.

Le transport optimal trouve un nombre grandissant d’applications, dont celle qui nous intéresse dans ce travail, l'interpolation d’images. Malgré cet essor, la résolution numérique de ce transport soulève des difficultés et le développement d’algorithmes efficaces reste un problème d'actualité, en particulier pour des images de grande taille, comme on en trouve dans certains domaines (météorologie,...).Nous nous intéressons dans ce travail à la formulation de Benamou et Brenier, qui ont placé le problème dans un contexte de mécanique des milieux continus en ajoutant une dimension temporelle. Leur formulation consiste en la minimisation d’une fonctionnelle sur un espace des contraintes contenant une condition de divergence nulle, et les algorithmes existants utilisent une projection sur cet espace.A l'opposé, dans cette thèse, nous définissons et mettons en oeuvre des algorithmes travaillant directement dans cet espace.En effet, nous montrons que la fonctionnelle a de meilleures propriétés de convexité sur celui-ci.Pour travailler dans cet espace, nous considérons trois représentations des champs de vecteurs à divergence nulle. La première est une base d’ondelettes à divergence nulle. Cette formulation a été implémentée numériquement dans le cas des ondelettes périodiques à l'aide d'une descente de gradient, menant à un algorithme de convergence lente mais validant la faisabilité de la méthode. La deuxième approche consiste à représenter les vecteurs à divergence nulle par leur fonction de courant munie d'un relèvement des conditions au bord et la troisième à utiliser la décomposition de Helmholtz-Hodge.Nous montrons de plus que dans le cas unidimensionnel en espace, en utilisant l’une ou l'autre de ces deux dernières représentations, nous nous ramenons à la résolution d’une équation de type courbure minimale sur chaque ligne de niveau du potentiel, munie des conditions de Dirichlet appropriées.La minimisation de la fonctionnelle est alors assurée par un algorithme primal-dual pour problèmes convexes de Chambolle-Pock, qui peut aisément être adapté à nos différentes formulations et est facilement parallèlisable, menant à une implémentation performante et simple.En outre, nous démontrons les gains significatifs de nos algorithmes par rapport à l’état de l’art et leur application sur des images de taille réelle. / Optimal transport has an increasing number of applications, including image interpolation, which we study in this work. Yet, numerical resolution is still challenging, especially for real size images found in applications.We are interested in the Benamou and Brenier formulation, which rephrases the problem in the context of fluid mechanics by adding a time dimension.It is based on the minimization of a functional on a constraint space, containing a divergence free constraint and the existing algorithms require a projection onto the divergence-free constraint at each iteration.In this thesis, we propose to work directly in the space of constraints for the functional to minimize.Indeed, we prove that the functional we consider has better convexity properties on the set of constraints.To work in this space, we use three different divergence-free vector decompositions. The first in which we got interested is a divergence-free wavelet base. This formulation has been implemented numerically using periodic wavelets and a gradient descent, which lead to an algorithm with a slow convergence but validating the practicability of the method.First, we represented the divergence-free vector fields by their stream function, then we studied the Helmholtz-Hodge decompositions. We prove that both these representations lead to a new formulation of the problem, which in 1D + time, is equivalent to the resolution of a minimal surface equation on every level set of the potential, equipped with appropriate Dirichlet boundary conditions.We use a primal dual algorithm for convex problems developed by Chambolle and Pock, which can be easily adapted to our formulations and can be easily sped up on parallel architectures. Therefore our method will also provide a fast algorithm, simple to implement.Moreover, we show numerical experiments which demonstrate that our algorithms are faster than state of the art methods and efficient with real-sized images.

Au début des années 2000, J. D. Benamou et Y. Brenier ont proposé une formulation dynamique du transport optimal basée sur la recherche en espace-temps d'une densité et d'une quantité de mouvement minimisant une énergie de déplacement entre deux densités. Ils ont alors proposé, pour la résolution numérique de ce problème, d'écrire ce dernier sous la forme d'une recherche de point selle d'un certain lagrangien via un algorithme de lagrangien augmenté. Nous étudierons, à l'aide de la théorie des opérateurs non-expansifs, la convergence de cet algorithme vers un point selle du lagrangien introduit, et ceci dans les conditions les plus générales possibles, en particulier dans les cas où les densités de départ et d'arrivée s'annulent sur certaines zones du domaine de transport. La principale difficulté de notre étude consistera en la preuve de l'existence d'un point selle, et surtout de l'unicité de la composante densité-quantité de mouvement dans de telles conditions. En effet, celles-ci impliquent de devoir traiter avec des plans de transport optimaux non réguliers : c'est pourquoi une importante partie de nos travaux aura pour objet une étude approfondie de la régularité d'un champ de vitesse associé à de tels plans de transport. Nous tenterons également de caractériser les propriétés d'un champ de vitesse associé à un plan de transport optimal dans l'espace quadratique. Pour finir, nous explorerons différentes approches relatives à l'introduction de contraintes physiques dans la formulation dynamique du transport optimal, basées sur une pénalisation du domaine de transport ou du champ de vitesse. / In the beginning of the 2000 years, J. D. Benamou and Y. Brenier have proposed a dynamical formulation of the optimal transport problem, corresponding to the time-space search of a density and a momentum minimizing a transport energy between two densities. They proposed, in order to solve this problem in practice, to deal with it by looking for a saddle point of some Lagrangian by an augmented Lagrangian algorithm. Using the theory of non-expansive operators, we will study the convergence of this algorithm to a saddle point of the Lagrangian introduced, in the most general feasible conditions, particularly in cases where initial and final densities are canceling on some areas of the transportation domain. The principal difficulty of our study will consist of the proof, in these conditions, of the existence of a saddle point, and especially in the uniqueness of the density-momentum component. Indeed, these conditions imply to have to deal with non-regular optimal transportation maps: that is why an important part of our works will have for object a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space. To finish, we will explore different approaches for introducing physical priors in the dynamical formulation of optimal transport, based on penalization of the transportation domain or of the velocity field.

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Neural Processes (NPs) are a class of regression models that learn a map from a set of input-output pairs to a distribution over functions. NPs are computationally tractable and provide a number of benefits over traditional nonlinear regression models. Despite these benefits, there are two main domains where NPs fail. This thesis is focused on presenting extensions of the Neural Process to these two areas. The first of these is the extension of Neural Processes graph and network data which we call Graph Neural Processes (GNP). A Graph Neural Process is defined as a Neural Process that operates on graph data. It takes spectral information from the graph Laplacian as inputs and then outputs a distribution over values. We demonstrate Graph Neural Processes in edge value imputation and discuss benefits and drawbacks of the method for other application areas. The second extension of Neural Processes comes in the fundamental training mechanism. NPs are traditionally trained using maximum likelihood, a probabilistic technique. We show that there are desirable classes of problems where NPs fail to learn. We also show that this drawback is solved by using approximations of the Wasserstein distance. We give experimental justification for our method and demonstrate its performance. These Wasserstein Neural Processes (WNPs) maintain the benefits of traditional NPs while being able to approximate new classes of function mappings.

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京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23314号 / 情博第750号 / 新制||情||128(附属図書館) / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 山本 章博, 教授 鹿島 久嗣, 教授 河原 達也 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

The central objects of study in this thesis are metric measure spaces. These are metric spaces which are endowed with a reference measure and enriched with basic topological, geometric and measure theoretical properties. The objective of the first part of the work is to characterize metric measure spaces whose symmetry groups admit a differential structure making them Lie groups. The second part is concerned with the analysis of the induced geometry of spaces admitting non-trivial symmetries. More in detail, it is shown that in many cases synthetic notions of Ricci curvature lower bounds are inherited by quotient spaces.

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