No description available.

Les travaux portent sur l'étude d'équations aux dérivées partielles à la charnière de la physique de la mécanique des milieux continus et de la géométrie différentielle, le point de départ étant le modèle d'électromagnétisme non-linéaire introduit par Max Born et Leopold Infeld en 1934 comme substitut aux traditionnelles équations linéaires de Maxwell. Ces équations sont remarquables par leurs liens avec la géométrie différentielle (surfaces extrémales dans l'espace de Minkowski) et ont connu un regain d'intérêt dans les années 90 en physique des hautes énergies (cordes et D-branes).Le travail se décompose en quatre chapitres.La théorie des systèmes paraboliques dégénérés d'EDP non-linéaires est fort peu développée, faute de pouvoir appliquer les principes de comparaison habituels (principe du maximum), malgré leur omniprésence dans de nombreuses applications (physique, mécanique, imagerie numérique, géométrie...). Dans le premier chapitre, on montre comment de tels systèmes peuvent être parfois dérivés, asymptotiquement, à partir de systèmes non-dissipatifs (typiquement des systèmes hyperboliques non-linéaires), par simple changement de variable en temps non-linéaire dégénéré à l'origine (où sont fixées les données initiales). L'avantage de ce point de vue est de pouvoir transférer certaines techniques hyperboliques vers les équations paraboliques, ce qui semble à première vue surprenant, puisque les équations paraboliques ont la réputation d'être plus facile à traiter (ce qui n'est pas vrai, en réalité, dans le cas de systèmes dégénérés). Le chapitre traite, comme prototype, du curve-shortening flow", qui est le plus simple des mouvements par courbure moyenne en co-dimension supérieure à un. Il est montré comment ce modèle peut être dérivé de la théorie des surfaces de dimension deux d'aire extrémale dans l'espace de Minkowski (correspondant aux cordes relativistes classiques) qui peut se ramener à un système hyperbolique. On obtient, presque automatiquement, l'équivalent parabolique des principes d'entropie relative et d'unicité fort-faible qu'il est, en fait, bien plus simple d'établir et de comprendre dans le cadre hyperbolique.Dans le second chapitre, la même méthode s'applique au système de Born-Infeld proprement dit, ce qui permet d'obtenir, à la limite, un modèle (non répertorié à notre connaissance) de Magnétohydrodynamique (MHD), où on retrouve à la fois une diffusivité non-linéaire dans l'équation d'induction magnétique et une loi de Darcy pour le champ de vitesse. Il est remarquable qu'un système d'apparence aussi lointaine des principes de base de la physique puisse être si directement déduit d'un modèle de physique aussi fondamental et géométrique que celui de Born-Infeld.Dans le troisième chapitre, un lien est établi entre des systèmes paraboliques et le concept de flot gradient de formes différentielles pour des métriques de transport. Dans le cas des formes volumes, ce concept a eu un succès extraordinaire dans le cadre de la théorie du transport optimal, en particulier après le travail fondateur de Felix Otto et de ses collaborateurs. Ce concept n'en est vraiment qu'à ses débuts: dans ce chapitre, on étudie une variante du «curve-shortening flow» étudié dans le premier chapitre, qui présente l'avantage d'être intégrable (en un certain sens) et de conduire à des résultats plus précis.Enfin, dans le quatrième chapitre, on retourne au domaine des EDP hyperboliques en considérant, dans le cas particulier des graphes, les surfaces extrémales de l'espace de Minkowski, de dimension et co-dimension quelconques. On parvient à montrer que les équations peuvent se reformuler sous forme d'un système élargi symétrique du premier ordre (ce qui assure automatiquement le caractère bien posé des équations) d'une structure remarquablement simple (très similaire à l'équation de Burgers) avec non linéarités quadratiques, dont le calcul n'a rien d'évident. / Our work concerns about the study of partial differential equations at the hinge of the continuum physics and differential geometry. The starting point is the model of non-linear electromagnetism introduced by Max Born and Leopold Infeld in 1934 as a substitute for the traditional linear Maxwell's equations. These equations are remarkable for their links with differential geometry (extremal surfaces in the Minkowski space) and have regained interest in the 90s in high-energy physics (strings and D-branches).The thesis is composed of four chapters.The theory of nonlinear degenerate parabolic systems of PDEs is not very developed because they can not apply the usual comparison principles (maximum principle), despite their omnipresence in many applications (physics, mechanics, digital imaging, geometry, etc.). In the first chapter, we show how such systems can sometimes be derived, asymptotically, from non-dissipative systems (typically non-linear hyperbolic systems), by simple non-linear change of the time variable degenerate at the origin (where the initial data are set). The advantage of this point of view is that it is possible to transfer some hyperbolic techniques to parabolic equations, which seems at first sight surprising, since parabolic equations have the reputation of being easier to treat (which is not true , in reality, in the case of degenerate systems). The chapter deals with the curve-shortening flow as a prototype, which is the simplest exemple of the mean curvature flows in co-dimension higher than 1. It is shown how this model can be derived from the two-dimensional extremal surface in the Minkowski space (corresponding to the classical relativistic strings), which can be reduced to a hyperbolic system. We obtain, almost automatically, the parabolic version of the relative entropy method and weak-strong uniqueness, which, in fact, is much simpler to establish and understand in the hyperbolic framework.In the second chapter, the same method applies to the Born-Infeld system itself, which makes it possible to obtain, in the limit, a model (not listed to our knowledge) of Magnetohydrodynamics (MHD) where we have non-linear diffusions in the magnetic induction equation and the Darcy's law for the velocity field. It is remarkable that a system of such distant appearance of the basic principles of physics can be so directly derived from a model of physics as fundamental and geometrical as that of Born-Infeld.In the third chapter, a link is established between the parabolic systems and the concept of gradient flow of differential forms with suitable transport metrics. In the case of volume forms, this concept has had an extraordinary success in the field of optimal transport theory, especially after the founding work of Felix Otto and his collaborators. This concept is really only on its beginnings: in this chapter, we study a variant of the curve-shortening flow studied in the first chapter, which has the advantage of being integrable (in a certain sense) and lead to more precise results.Finally, in the fourth chapter, we return to the domain of hyperbolic EDPs considering, in the particular case of graphs, the extremal surfaces of the Minkowski space of any dimension and co-dimension. We can show that the equations can be reformulated in the form of a symmetric first-order enlarged system (which automatically ensures the well-posedness of the equations) of a remarkably simple structure (very similar to the Burgers equation) with quadratic nonlinearities, whose calculation is not obvious.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第24263号 / 情博第807号 / 新制||情||136(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)准教授 加嶋 健司, 教授 太田 快人, 教授 山下 信雄 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DGAM

The main objective of this thesis is to develop novel misfit functions for full-waveform inversion such that (a) the estimation of the long-wavelength model will less likely stagnate in spurious local minima and (b) the inversion is immune to wavelet inaccuracy. First, I investigate the pros and cons of misfit functions based on optimal transport theory to indicate the traveltime discrepancy for seismic data. Even though the mathematically well-defined optimal transport theory is robust to highlight the traveltime difference between two probability distributions, it becomes restricted as applied to seismic data mainly because the seismic data are not probability distribution functions. We then develop a misfit function combining the local cross-correlation and dynamic time warping. This combination enables the proposed misfit automatically identify arrivals associated with a phase shift. Numerical and field data examples demonstrate its robustness for early arrivals and limitations for later arrivals.%, which means that a proper pre-processing step is still required. Next, we introduce differentiable dynamic time warping distance as the misfit function highlighting the traveltime discrepancy without non-trivial human intervention. Compared to the conventional warping distance, the differentiable version retains the property of representing the traveltime difference; moreover, it can eliminate abrupt changes in the adjoint source, which helps full-waveform inversion converge to geologically relevant estimates. Finally, we develop a misfit function entitled the deconvolutional double-difference measurement. The new misfit measures the first difference by deconvolution rather than cross-correlation. We also present the derivation of the adjoint source with the new misfit function. Numerical examples and mathematical proof demonstrate that this modification makes full-waveform inversion with the deconvolutional double-difference measurement immune to wavelet inaccuracy.

The use of aerial platforms such as unmanned aerial vehicles (UAVs), popularly known as drones, has emerged as a promising solution for providing reliable and cost-effective wireless communications. In particular, UAVs can be quickly and efficiently deployed to support cellular networks and enhance their quality-of-service (QoS) by establishing line-of-sight communication links. With their inherent attributes such as mobility, flexibility, and adaptive altitude, UAVs admit several key potential applications in wireless systems. Remarkably, despite these inherent advantages of UAVbased communications, little work has analyzed the performance tradeoffs associated with using UAVs as aerial wireless platforms. The key goal of this dissertation is to develop the analytical foundations for deployment, performance analysis, and optimization of UAV-enabled wireless networks. This dissertation makes a number of fundamental contributions to various areas of UAV communications that include: 1) Efficient deployment of UAVs, 2) Performance evaluation and optimization, and 3) Design of new flying, three-dimensional (3D) wireless systems. For deployment, using tools from optimization theory, holistic frameworks are developed for the optimal 3D placement of UAV base stations in uplink and downlink scenarios. The results show that the proposed deployment approaches significantly improve the downlink coverage for ground users, and enable ultra-reliable and energy-efficient uplink communications in Internet of Things (IoT) applications. For performance optimization, a novel framework is developed for maximizing the performance of a UAV-based wireless system, in terms of data service, under UAVs’ flight time constraints. To this end, using the mathematical framework of optimal transport theory, the optimal cell associations, that lead to a maximum data service to ground users within the limited UAVs’ hover duration, are analytically derived. The results shed light on the tradeoff between hover time and quality-of-service in UAV-based wireless networks. For performance evaluation, this dissertation provides a comprehensive analysis on the performance of a UAV-based communication system in coexistence with a terrestrial network. In particular, a tractable analytical framework is proposed for analyzing the coverage and rate performance of a network with a UAV base station and deviceto-device (D2D) users. The results reveal the fundamental tradeoffs in such a UAV-D2D network that allow adopting appropriate system design parameters. Then, this dissertation sheds light on the design of three new drone-enabled wireless systems. First, a novel framework for effective use of cache-enabled UAVs in wireless networks is developed. The results demonstrate how the users’ quality of experience substantially improves by exploiting UAVs’ mobility and user-centric information. Second, a new framework is proposed for deploying and operating a drone-based antenna array system that delivers wireless service to ground users within a minimum time. The results show significant enhancement in QoS, spectral and energy efficiency while levering the proposed drone antenna array system. Finally, to effectively incorporate various use cases of drones ranging from aerial users to base stations, the new concept of a fully-fledged 3D cellular network is introduced. For this new type of 3D wireless network, a unified framework for deployment, network planning, and performance optimization is developed that yields a maximum coverage and minimum latency in the network. In a nutshell, the analytical foundations and frameworks presented in this dissertation provide key guidelines for effective design and operation of UAV-based wireless communication systems. / Ph. D.

Every year, 250,000 people worldwide suffer a spinal cord injury (SCI) that leaves them with chronic paraplegia - permanent loss of ability to move their legs. SCI interrupts axons passing along the spinal cord, thereby isolating motor neurons from brain inputs. To date, there are no effective treatments that can reconnect these interrupted axons. In a recent breakthrough, .NeuroRestore developed the STIMO neuroprosthesis that can restore walking after paralyzing SCI using Epidural Electrical Stimulation (EES) of the lumbar spinal cord. Yet, the calibration of EES requires highly trained personnel and a vast amount of time, and the mechanism by which EES restores movement is not fully understood. In this master thesis, we propose to address this issue using modeling combined with Artificial Neural Networks (ANNs). To do so, we introduce the CyberSpine model to predict EES-induced motor response. The implementation of the model relies on the construction of a multipolar basis of solution of the Poisson equation which is then coupled to an ANN trained against actual data of an implanted STIMO user. Furthermore, we show that our CyberSpine model is particularly well adapted to extract biologically relevant information regarding the efficient connectivity of the patient’s spine. Finally, a user-friendly interactive visualization software is built. / Varje år drabbas 250 000 människor i hela världen av en ryggmärgsskada som ger dem kronisk paraplegi - permanent förlust av förmågan att röra benen. Vid en ryggmärgsskada bryts axonerna som passerar längs ryggmärgen, vilket isolerar de motoriska neuronpoolerna från hjärnans ingångar. Hittills finns det inga effektiva behandlingar som kan återansluta dessa avbrutna axoner. NeuroRestore utvecklade nyligen neuroprotesen STIMO som kan återställa gångförmågan efter förlamande ryggmärgsskada med hjälp av epidural elektrisk stimulering (EES) av ländryggmärgen. Kalibreringen av EES-stimuleringar kräver dock högutbildad personal och mycket tid, och den mekanism genom vilken EES återställer rörelse är inte helt klarlagd. I denna masteruppsats föreslår vi att vi tar itu med denna fråga med hjälp av modellering i kombination med artificiell intelligens. För att göra detta introducerar vi CyberSpine-modellen, en modell som kan förutsäga EES-inducerad motorisk respons. Implementeringen av modellen bygger på konstruktionen av en multipolär bas för lösning av Poisson-ekvationen som sedan kopplas till ett artificiellt neuralt nätverk som tränas mot faktiska data från en implanterad STIMO-deltagare. Dessutom visar vi att vår CyberSpine-modell är särskilt väl anpassad för att extrahera biologiskt relevant information om den effektiva anslutningen av patientens ryggrad. Slutligen bygger vi en användarvänlig interaktiv visualiseringsprogramvara.

Nous étudions dans ce travail les mouvements de foule intervenant dans les situa- tions d’urgence. Nous proposons un modèle macroscopique (la foule est représentée par une densité de personnes) obéissant à deux principes très simples. Tout d’abord, chaque personne possède une vitesse souhaitée (typiquement celle qui la mène vers la sortie), qu’elle adopterait en l’absence des autres. Ensuite, la foule doit respecter une contrainte de congestion, et la densité de personnes doit rester inférieure à une valeur fixée. Cette contrainte impose une vitesse de déplacement différente de la vitesse souhaitée. Nous choisissons de prendre comme vitesse réelle celle qui est la plus proche, au sens des moindres carrés, de la vitesse souhaitée, parmi les champs de vitesses admissibles, au sens où ils respectent la contrainte de densité maximale. Le modèle obtenu s’écrit sous la forme d’une équation de transport impliquant une vitesse peu régulière a priori, et qui ne peut être étudiée par des méthodes classiques. Nous démontrons un résultat d’existence grâce à la théorie du transport optimal, tout d’abord dans le cas d’une vitesse donnée comme le gradient d’une fonction, puis dans le cas général. Nous mettons également en œuvre un schéma numérique de type catching-up : à chaque pas de temps, la densité est déplacée selon le champ de vitesse souhaitée, puis est projetée sur l’ensemble des densités admissibles. Les résultats obtenus fournissent des temps d’évacuation dont l’ordre de grandeur est proche de la réalité. / In this work, we aim at modelling crowd motion in emergency situations. We propose a macroscopic model (where people are represented as a density) following two basic principles. First, each individual has a spontaneous velocity (typically, the one which leads to the nearest exit) which would be fulfilled in the absence of other people. On the other hand, the crowd has to respect a congestion constraint, and its density must remain underneath a critical density. This constraint prevents people from following their desired velocity. The actual velocity we consider is the closest, in a mean square sense, to the desired one, among the velocities which respect the maximal density constraint.The mathematical formulation writes as a transport equation which cannot be studied with classical methods, since the real velocity field has no a priori regularity, even if the desired velocity is smooth. Thanks to the optimal transport theory, we prove an existence result, first in the case where the desired velocity is the gradient of a given function, and then in the general framework. We also propose a numerical scheme which follows the catching-up principle: at each time step, we move the density according to the spontaneous velocity, and then project it onto the space of admissible densities. The numerical results we obtain reproduce qualitatively the experimental observations

Le problème du transport optimal, originellement introduit par Monge au 18ème siècle, consiste à minimiser l'énergie nécessaire au déplacement d'une masse dont la répartition est donnée vers une autre masse dont la répartition est elle aussi donnée; mathématiquement, cela se traduit par : trouver le minimiseur de l'intégrale de c(x,T(x)) (où c est le coût de transport de x vers T(x)) parmi toutes les applications T à mesure image prescrite.Cette thèse est consacrée à l'étude de problèmes variationnels similaires où l'on fait intervenir la matrice jacobienne de la fonction de transport, c'est-à-dire que le coût dépend de trois variables c(x,T(x),DT(x)) ; il s'agit typiquement de rajouter l'intégale de |DT(x)|^2 à la fonctionnelle afin d'obtenir une pénalisation Sobolev. Ce type de problème trouve ses motivations en mécanique des milieux continus, élasticité incompressible ou en analyse de forme et appelle d'un point de vue mathématique une approche totalement différente de celle du problème de transport usuel.Les questions suivantes sont envisagées :- bonne définition du problème, notamment de l'énergie de Dirichlet, via les espaces de Sobolev par rapport à une mesure, et résultats d'existence de minimiseurs ;- caractérisation de ces minimiseurs : optimalité du transport croissant sur la droite réelle, et approche du type équation d'Euler-Lagrange en dimension quelconque ;- sélection d'un minimiseur via une procédure de pénalisation du type Gamma-convergence (l'énergie de Dirichlet est mutipliée par un petit paramètre) lorsque le coût de transport est le coût de Monge donné par la distance, pour lequel l'application de transport optimale n'est pas unique ;- autres approches du problème et perspectives : formulation dynamique du type Benamou-Brenier, et formulation duale similaire à celle de Kantorovitch dans le cas du problème du transport optimal usuel. / The optimal transportation problem was originally introduced by Monge in the 18th century; it consists in minimizing the total energy of the displacement of a given repartition of mass onto another given repartition of mass. This is mathematically expressed by: find the minimizer of the integral of c(x,T(x)) (where c(x,T(x)) is the cost to send x onto T(x)) among the maps T with prescribed image measure.This thesis is devoted to similar variational problems, which involve the Jacobian matrix of the transport map, meaning that the cost depends on three variables c(x,T(x),DT(x)); we typically add the Dirichlet energy to the transport functional in view to obtain a Sobolev-type penalization. This kind of constraints finds its motivations in continuum mechanics, incompressible elasticity or shape analysis, and a quite different mathematical approach than in the usual theory of optimal transportation is needed.We consider the following questions:- proper definition of the problem, in particular of the Dirichlet energy, thanks to the theory of Sobolev spaces with respect to a measure, and existence results;- characterizations of these minimizers: optimality of the monotone transport map on the real line, and Euler-Lagrange-like approach in any dimension;- selection of a minimizer via a Gamma-convergence-like penalization procedure (we multiply the Dirihlet energy with a vanishing positive parameter) where the transport cost is the Monge cost given by the distance (for which the optimal transport map is not unique);- other related problems and perspectives: dynamic Benamou-Brenier-like formulation, and dual Kantorovich-like formulation.

Dans cette thèse, nous nous intéressons à la résolution de nombreux problèmes d’optique anidolique. Plus précisément, il s’agit de construire des composants optiques qui satisfont des contraintes d’illumination à savoir que l’on veut que la lumière réfléchie(ou réfractée) par ce composant corresponde à une distribution fixée en avance. Comme applications, nous pouvons citer la conception de phares de voitures ou de caustiques. Nous montrons que ces problèmes de conception de composants optiques peuvent être vus comme des problèmes de transport optimal et nous expliquons en quoi cette formulation permet d’étudier l’existence et la régularité des solutions. Nous montrons aussi comment, en utilisant des outils de géométrie algorithmique, nous pouvons utiliser une méthode numérique efficace, la méthode de Newton amortie, pour résoudre tous ces problèmes. Nous obtenons un algorithme générique capable de construire efficacement un composant optique qui réfléchit (ou réfracte)une distribution de lumière prescrite. Nous montrons aussi la convergence de l’algorithme de Newton pour résoudre le problème de transport optimal dans le cas où le support de la mesure source est une union finie de simplexes. Nous décrivons également la relation commune qui existe entre huit différents problèmes de conception de composants optiques et montrons qu’ils peuvent tous être vus comme des équations de Monge-Ampère discrètes. Nous appliquons aussi la méthode de Newton à de nombreux problèmes de conception de composants optiques sur différents exemples simulés ainsi que sur des prototypes physiques. Enfin, nous nous intéressons à un problème apparaissant en transport optimal numérique à savoir le choix du point initial. Nous développons trois méthodes simples pour trouver de “bons” points initiaux qui peuvent être ensuite utilisés comme point de départ dans des algorithmes de résolution de transport optimal. / In this thesis, we are interested in solving many inverse problems arising inoptics. More precisely, we are interested in designing optical components such as mirrors andlenses that satisfy some light conservation constraints meaning that we want to control thereflected (or refracted) light in order match a prescribed intensity. This has applications incar headlight design or caustic design for example. We show that optical component designproblems can be recast as optimal transport ones for different cost functions and we explainhow this allows to study the existence and the regularity of the solutions of such problems. Wealso show how, using computational geometry, we can use an efficient numerical method namelythe damped Newton’s algorithm to solve all these problems. We will end up with a singlegeneric algorithm able to efficiently build an optical component with a prescribed reflected(or refracted) illumination. We show the convergence of the Newton’s algorithm to solve theoptimal transport problem when the source measure is supported on a finite union of simplices.We then describe the common relation between eight optical component design problemsand show that they can all be seen as discrete Monge-Ampère equations. We also apply theNewton’s method to optical component design and show numerous simulated and fabricatedexamples. Finally, we look at a problem arising in computational optimal transport namelythe choice of the initial weights. We develop three simple procedures to find “good” initialweights which can be used as a starting point in computational optimal transport algorithms.

Optimal transport has emerged as a promising and effective tool for supporting modern image processing, geometric processing, and even machine learning. Indeed, the optimal transport theory enables great flexibility in modeling problems, as different optimization resources can be successfully employed while preserving a context relevant property that can be interpreted as mass. In this research, we introduce a novel automatic technique for eye fundus image registration which is based on optimal transport theory, image processing filters, graph matching, and geometric transformations into a concise and unified framework. Given two ocular fundus images, we construct representative graphs which embed in their structures spatial and topological information from the eyes blood vessels. The graphs produced are then used as input by our optimal transport model in order to establish a correspondence between their sets of nodes. We also proposed a new measure that estimates the register quality and an extension of an outlier removal technique called DeSAC. Finally, the best geometric transformation is performed on the image to properly accomplish the registration task. Our method relies on a solid mathematical foundation, is easy-to-implement and performs well when dealing with outliers created during the matching stage, producing deterministic and accurate solutions. We demonstrate the accuracy and effectiveness of the proposed methodology through a comprehensive set of qualitative and quantitative comparisons against various representative state-of-the-art methods on different fundus image databases. / O transporte ótimo se tornou uma ferramenta promissora e eficaz para apoiar o processamento de imagens moderno, processamento geométrico e até aprendizado de máquina. De fato, a teoria do transporte ótimo permite uma grande flexibilidade na modelagem de problemas, pois diferentes recursos de otimização podem ser empregados enquanto se preserva uma propriedade relevante ao contexto que pode ser interpretada como massa. Nesta pesquisa, nós introduzimos uma nova técnica automática para o registro da imagem do fundo do olho que é baseada na teoria óptima do transporte, filtros de processamento de imagem, correspondência de grafos e transformações geométricas em uma estrutura concisa e unificada . Dadas duas imagens de fundo ocular, construímos grafos representativos que incorporam em suas estruturas informações espaciais e topológicas dos vasos sanguíneos do olho. Os grafos produzidos são usados como entrada pelos nossos modelo de transporte ótimo, a fim de estabelecer uma correspondência entre seus conjuntos de nós. Propomos também uma nova medida que estima a qualidade do registro e uma extensão de uma tecnica de removeção de outliers chamada DeSAC. Finalmente, transformações geométricas são realizadas entre as imagens para realizar adequadamente a tarefa de registro. Nosso método baseia-se em uma sólida base matemática, é fácil de implementar e funciona bem lidando com outliers criados durante o estágio de correspondência, produzindo soluções determinísticas e precisas. Demonstramos a exatidão e eficácia da metodologia proposta por meio de uma abordagem abrangente de comparações qualitativas e quantitativas contra vários métodos representativos do estado da arte em diferentes bases de dados de imagens de fundo de olho.