Le sujet de cette thèse est l'étude énumérative des cartes combinatoires et ses applications à l'énumération des autres objet s combinatoires.Les cartes combinatoires, aussi appelées simplement « cartes », sont un modèle combinatoire riche. Elles sont définies d'une manière intuitive et géométrique, mais elles sont aussi liées à des structures algébriques plus complexes. Par exemple, l'étude d'une famille de cartes appelées des « constellations » donne un cadre unifié à plusieurs problèmes d'énumération des factorisations dans le groupe symétrique. À la croisée des différents domaines, les cartes peuvent être analysées par une grande variété de méthodes, et leur énumération peut aussi nous aider à compter des autres objets combinatoires. Cette thèse présente un ensemble de résultats et de connexions très riches dans le domaine de l'énumération des cartes. Cette thèse se divise en quatre grandes parties. La première partie, qui correspond aux chapitres 1 et 2, est une introduction à l'étude énumérative des cartes. La deuxième partie, qui correspond aux chapitres 3 et 4, contient mes travaux sur l'énumération des constellations, qui sont des cartes particulières présentant un modèle unifié de certains types de factorisation de l'identité dans le groupe symétrique. La troisième partie, qui correspond aux chapitres 5 et 6, présente ma recherche sur le lien énumératif entre les cartes et des autres objets combinatoires, par exemple les généralisations du treillis de Tamari et les graphes aléatoires qui peuvent être plongés dans une surface donnée. La dernière partie correspond au chapitre 7, dé ns lequel je conclus cette thèse avec des perspectives et des directions de recherche dans l'étude énumérative des cartes. / This thesis deals with the enumerative study of combinatorial maps, and its application to the enumeration of other combinatorial objects. Combinatorial maps, or simply maps, form a rich combinatorial model. They have an intuitive and geometric definition, but are also related to some deep algebraic structures. For instance, a special type of maps called \emph{constellations} provides a unifying framework for some enumeration problems concerning factorizations in the symmetric group. Standing on a position where many domains meet, maps can be studied using a large variety of methods, and their enumeration can also help us count other combinatorial objects. This thesis is a sampling from the rich results and connections in the enumeration of maps.This thesis is structured into four major parts. The first part, including Chapter 1 and 2, consist of an introduction to the enumerative study of maps. The second part, Chapter 3 and 4, contains my work in the enumeration of constellations, which are a special type of maps that can serve as a unifying model of some factorizations of die identity in the symmetric group: The third part, composed by Chapter 5 and 6, shows my research on the enumerative link from maps to other combinatori al objects, such as generalizations of the Tamari lattice and random graphs embeddable onto surfaces. The last part is the closing chapter, in which the thesis concludes with some perspectives and future directions in the enumerative study of maps.

On considère dans cette thèse l'équation des ondes posée sur un domaine $\Omega$ supposé régulier. Si $\Gamma$ désigne une surface supposée observable, on peut définir la constante d'observabilité associée à $\Gamma$. L'intérêt de cette constante est de rendre compte de la qualité de la reconstruction dans le problème inverse qui consiste à reconstruire les données initiales à partir de la mesure de la solution sur $\Gamma$. Ainsi l'étude de cette constante s'applique entre autres à la détermination de la forme et du placement optimaux de capteurs, pour la mesure de toute sorte de phénomènes ondulatoires. Le but du premier chapire est de caractériser de manière théorique les domaines $\Gamma$ de surface prescrite qui maximisent cette constante d'observabilité, ou plus exactement une version "randomizée" de ce critère. Dans le second chapitre il s'agit d'appliquer les résulats obtenus au placement optimal de capteurs pour la tomographie photoacoustique. La tomographie photoacoustique est un procédé d'imagerie médicale ultra-sonore, non invasif encore peu développé qui est une alternative précise et plus économique à l'imagerie X. C'est dans ce cadre que l'on propose une modélisation de l'influence de la forme et de la disposition des capteurs dans le problème de reconstruction de la densité des tissus. Plus particulièrement, il s'agira de construire une fonctionnelle de la forme des capteurs, rendant compte de la qualité de l'image obtenue. / In the first part of this thesis, we consider the wave equation on a regular bounded domain $\Omega$. We investigate the problem of optimizing, in some appropriate sense, the shape and location of sensors spread on an arbitrary measurable subdomain $\Gamma$ of the boundary of $\Omega$. We introduce a spectral quantity called randomized observability constant, corresponding to the best constant in an average of the classical observability inequality, over random initial data. The pupose of the first chapter is to investigate optimal domains, maximizing the new objective function. The second part consists in applying the previous results to medical imaging, and more precisely to photoacoustic tomography. This imaging technique, constitutes a cutting-edge technology that has drawn considerable attention in the medical imaging area. Firstly because it is non-ionizing and non-invasive, and also because it constitutes a precise and cheap alternative to X imaging. In this framework, we propose here to model the influence of the shape and position of sensors in the inverse problem consisting in the reconstruction of the imaged body. In a nutshell, we build a functional of the shape of the sensors, providing an account for the reconstructed image quality.

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第22756号 / 工博第4755号 / 新制||工||1744(附属図書館) / 京都大学大学院工学研究科都市社会工学専攻 / (主査)教授 細田 尚, 教授 戸田 圭一, 准教授 音田 慎一郎 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22863号 / 情博第742号 / 新制||情||127(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)教授 矢ヶ崎 一幸, 教授 中村 佳正, 准教授 柴山 允瑠, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Cette thèse est consacrée principalement à l’étude de quelques problèmes elliptiques singuliers dans un domaine Ωɛ*, périodiquement perforé par des trous de taille ɛ. On montre l’existence et l’unicité d’une solution, pour tout ɛ fixé, ainsi que des résultats d’homogénéisation et correcteurs pour le problème singulier suivant :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Où l’on prescrit des conditions de Dirichlet homogènes sur la frontière extérieure Γɛ0 et des conditions de Robin non linéaires sur la frontière des trous Γɛ1. Le champ matriciel quasi linéaire A est elliptique, borné, périodique dans la primière variable et de Carathéodory. Le terme singulier non linéaire est le produit d’une fonction continue ζ (singulier en zéro) et de f, dont la sommabilité dépend de la croissance de ζ près de sa singularité. Le terme de bord non linéaire h est une fonction croissante de classe C1, ρ et g sont des fonctions périodiques non négatives avec sommabilité convenables. Pour étudier le comportement asymptotique du problème quand ɛ -> 0, on applique la méthode de l’éclatement périodique due à D. Cioranescu-A. Damlamian-G. Griso (cf. D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki pour les domaines perforés). Enfin, on montre l’existence et l’unicité de la solution faible pour la même équation, dans un domaine à deux composants Ω = Ω1 υ Ω2 υ Γ, étant Γ l’interface entre le composant connecté Ω1 et les inclusions Ω2. Plus précisément on considère{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Où λ est un réel non négatif et h représente le coefficient de proportionnalité entre le flux de chaleur et le saut de la solution, et il est supposé être borné et non négatif sur Γ. / This thesis is mainly devoted to the study of some singular elliptic problems posed in perforated domains. Denoting by Ωɛ* e domain perforated by ɛ-periodic holes of ɛ-size, we prove existence and uniqueness of the solution , for fixed ɛ, as well as homogenization and correctors results for the following singular problem :{█(-div (A (x/ɛ,uɛ)∇uɛ)=fζ(uɛ) dans Ωɛ*@uɛ=0 sur Γɛ0@@(A (x/ɛ,uɛ)∇uɛ)υ+ɛγρ (x/ɛ) h(uɛ)= ɛg (x/ɛ) sur Γɛ1@)┤Where homogeneous Dirichlet and nonlinear Robin conditions are prescribed on the exterior boundary Γɛ0 and on the boundary of the holles Γɛ1, respectively. The quasilinear matrix field A is elliptic, bounded, periodic in the first variable and Carathéodory. The nonlinear singular lower order ter mis the product of a continuous function ζ (singular in zero) and f whose summability depends on the growth of ζ near its singularity. The nonlinear boundary term h is a C1 increasing function, ρ and g are periodic nonnegative functions with prescribed summabilities. To investigate the asymptotic behaviour of the problem, as ɛ -> 0, we apply the Periodic Unfolding Method by D. Cioranescu-A. Damlamian-G. Griso, adapted to perforated domains by D. Cioranescu-A. Damlamian-P. Donato-G. Griso-R. Zaki. Finally, we show existence and uniqueness of a weak solution of the same equation in a two-component domain Ω = Ω1 υ Ω2 υ Γ, being Γ the interface between the connected component Ω1 and the inclusions Ω2. More precisely we consider{█(-div (A(x, u)∇u)+ λu=fζ(u) dans Ω\Γ,@u=0 sur δΩ@(A(x, u1)∇u1)υ1= (A(x, u2)∇u2)υ1 sur Γ,@(A(x, u1)∇u1)υ1= -h(u1-u2) sur Γ@)┤Where ν1 is the unit external vector to Ω1 and λ a nonnegative real number. Here h represents the proportionality coefficient between the continuous heat flux and the jump of the solution and it is assumed to be bounded and nonnegative on Γ.

This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers. The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q  + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.      In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities. / <p>QC 20130926</p>

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第19121号 / 情博第567号 / 新制||情||100(附属図書館) / 32072 / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 木上 淳, 教授 磯 祐介, 教授 西村 直志 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第20632号 / 工博第4370号 / 新制||工||1679(附属図書館) / 京都大学大学院工学研究科分子工学専攻 / (主査)教授 佐藤 啓文, 教授 白川 昌宏, 教授 山本 量一 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DGAM

The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscpoic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism - the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous function of the control parameters of the map. Consequently if the external perturbation is made to act on a control parameter of the map, we show that the booster distribution undergoes slight modifications as an effect of the weak external perturbation, thereby leading to a linear response of the mean value of the perturbed variable of the booster. This approach to linear response completely bypasses the criticism of van Kampen. The joint use of these two phenomena, diffusion and friction stemming from the interaction of the Brownian particle with the same booster, makes the microscopic derivation of a Fokker-Planck equation and Brownian motion, possible.

Wind power is a rapidly increasing resource of electrical power world-wide. With the increasing number of wind turbines installed one major concern is the noise they generate. Sometimes already built wind turbines have to be put down or down-regulated, when certain noise levels are exceeded, resulting in economical and environmental losses. Therefore, accurate sound propagation calculations would be beneficial already in a planning stage of a wind farm. A model that can account for varying wind speeds and complex terrains could therefore be of great importance when future wind farms are planned. In this report an extended version of the classical wave equation that allows for variations in wind speed and terrain is derived which can be used to solve complex terrain and wind settings. The equation are solved with the use of Fourier transforms and Chebyshev polynomials and a numerical code is developed. The numerical code is evaluated against test cases where analytical and simple solutions exist. Tests with no wind for both totally free propagation and with a ground surface is evaluated in both 2D and 3D settings. For these simple cases the developed code shows good agreement to analytical solutions if the computational domain is sufficiently large. More advanced test cases with wind and terrain is not evaluated in this report and needs further validation. If the sound pressure needs to be calculated for a large area, and if the frequency is high, the developed model has problems regarding computational time and memory. These problems could be solved by further development of the numerical code or by using other solution methods.