Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient. For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods. We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference. The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice. / Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt. Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden. Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz. Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert.

Dans ce travail de recherche, nous sommes intéresses par la modélisation de l'hématopoïèse. Les cellules souches hématopoïétiques (CSH) sont des cellules indifférenciées de la moelle osseuse, possédant la capacité de se renouveler et de se différencier (pour la production des globules rouges, globules blancs et les plaquettes). Le processus de l'hématopoïèse souvent révèle des irrégularités qui causent les maladies hématologiques. En modélisant la leucémie myéloide chronique (LMC), une maladie hématologique fréquente, nous représentons l'hématopoïèse des cellules normales et cancéreuses par un système d'équations différentielles ordinaires (EDO). L'homéostasie des cellules normales et différente de l'homéostasie des cellules cancéreuses, et dépend de quelques lignées des cellules normales et cancéreuses. Nous analysons la dynamique globale du modèle pour obtenir les conditions de régénération de l'hématopoïèse ou bien la persistance de la LMC. Nous démontrons aussi que la coexistence des cellules normales et cancéreuses ne peut avoir lieu pour longtemps. Imatinib est un traitement de base de la LMC, avec un dosage variant de 400 à 1000 mg par jour. Certains patients présentent des réponses différentes à la thérapie, pouvant être hématologique, cytogénétique et moléculaire. La thérapie échoue dans deux cas: le patient demande un temps plus long pour réagir, alors il s'agit d'une réponse suboptimale; ou bien le patient résiste après une bonne réponse initiale. Pour déterminer le dosage optimal, nécessaire à la réduction des cellules cancéreuses, nous représentons les effets de la thérapie par un problème de contrôle optimal. Notre but est de minimiser le cout du traitement et le nombre des cellules cancéreuses. La réponse suboptimale, la résistance et le rétablissement sont alors obtenus suivant l'influence de l'imatinib sur les taux de division et de mortalité des cellules cancéreuses. Nous étudions par ailleurs l'hématopoïèse selon un modèle structuré en age, décrivant l'évolution des CSH normales et cancéreuses. Nous démontrons que le taux de division des CSH cancéreuses joue un rôle important dans la détermination du contrôle optimal. En contrôlant la croissance des cellules normales et cancéreuses avec compétition inter spécifique, nous démontrons que le dosage optimal dépend de l'homéostasie des CSH cancéreuses. / Modelling hematopoiesis represents a feature of our research. Hematopoietic stem cells (HSC) are undifferentiated cells, located in bone marrow, with unique abilities of self-renewal and differentiation (production of white cells, red blood cells and platelets).The process of hematopoiesis often exhibits abnormalities causing hematological diseases. In modelling Chronic Myeloid Leukemia (CML), a frequent hematological disease, we represent hematopoiesis of normal and leukemic cells by means of ordinary differential equations (ODE). Homeostasis of normal and leukemic cells are supposed to be different and depend on some lines of normal and leukemic HSC. We analyze the global dynamics of the model to obtain the conditions for regeneration of hematopoiesis and persistence of CML. We prove as well that normal and leukemic cells can not coexist for a long time. Imatinib is the main treatment of CML, with posology varying from 400 to 1000 mg per day. Some affected individuals respond to therapy with various levels being hematologic, cytogenetic and molecular. Therapy fails in two cases: the patient takes a long time to react, then suboptimal response occurs; or the patient resists after an initial response. Determining the optimal dosage required to reduce leukemic cells is another challenge. We approach therapy effects as an optimal control problem to minimize the cost of treatment and the level of leukemic cells. Suboptimal response, resistance and recovery forms are obtained through the influence of imatinib onto the division and mortality rates of leukemic cells. Hematopoiesis can be investigated according to age of cells. An age-structured system, describing the evolution of normal and leukemic HSC shows that the division rate of leukemic HSC plays a crucial role when determining the optimal control. When controlling the growth of cells under interspecific competition within normal and leukemic HSC, we prove that optimal dosage is related to homeostasis of leukemic HSC.

Due to failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the direct simulation Monte Carlo method (DSMC) to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. To gain a better understanding of evaporation physics, a non-steady simulation for slow evaporation in a microscopic system, based on the Navier-Stokes-Fourier equations, is conducted. The one-dimensional problem consists of a liquid and vapor layer (both pure water) with respective heights of 0.1mm and a corresponding Knudsen number of Kn=0.01, where vapor is pumped out. The simulation allows for calculation of the evaporation rate within both the transient process and in steady state. The main contribution of this work is the derivation of new evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with proven applicability in the transition regime. The approach for deriving the boundary conditions is based on an entropy balance, which is integrated around the liquid-vapor interface. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier-Stokes-Fourier solutions for two steady-state, one-dimensional problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement to DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to Navier-Stokes-Fourier (NSF) solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed which suggest continuation of this work. / Graduate

Ce document traite premièrement des diverses tentatives de modélisation et de simulation de la nage anguilliforme puis élabore une nouvelle technique, basée sur la méthode de la frontière immergée généralisée et la théorie des poutres de Reissner-Simo. Cette dernière, comme les équations des fluides polaires, est dérivée de la mécanique des milieux continus puis les équations obtenues sont discrétisées afin de les amener à une résolution numérique. Pour la première fois, la théorie des schémas de Runge-Kutta additifs est combinée à celle des schémas de Runge-Kutta-Munthe-Kaas pour engendrer une méthode d’ordre de convergence formel arbitraire. De plus, les opérations d’interpolation et d’étalement sont traitées d’un nouveau point de vue qui suggère l’usage des splines interpolatoires nodales en lieu et place des fonctions d’étalement traditionnelles. Enfin, de nombreuses vérifications numériques sont faites avant de considérer les simulations de la nage. / This paper first discusses various attempts at modeling and simulating anguilliform swimming, then we develop a new technique, based on a method of generalized immersed boundary and the beam theory of Reissner-Simo. Subsequent to the derivation of the equations of polar fluids, the beam theory is derived from continuum mechanics and the resulting equations are then discretized, allowing a numerical solution. For the first time, the theory of additive Runge-Kutta schemes are combined with the Runge-Kutta-Munthe-Kaas method to generate schemes of arbitrarily high formal order of convergence. Moreover, the interpolation and spreading operations are handled from a new point of view that suggests the use of interpolatory nodal splines instead of spreading traditional functions. Finally, many numerical verifications are done before considering simulations of swimming.

Understanding and characterising the diffusive transport of capillary oxygen and nutrients in striated muscles is key to assessing angiogenesis and investigating the efficacy of experimental and therapeutic interventions for numerous pathological conditions, such as chronic ischaemia. In articular, the influence of both muscle tissue and microvascular heterogeneities on capillary oxygen supply is poorly understood. The objective of this thesis is to develop mathematical and computational modelling frameworks for the purpose of extending and generalising the current use of histology in estimating the regions of tissue supplied by individual capillaries to facilitate the exploration of functional capillary oxygen supply in striated muscles. In particular, we aim to investigate the balance between local capillary supply of oxygen and oxygen demand in the presence of various anatomical and functional heterogeneities, by capturing tissue details from histological imaging and estimating or predicting regions of capillary supply. Our computational method throughout is based on a finite element framework that captures the anatomical details of tissue cross sections. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe oxygen transport from capillaries to uniform muscle tissues (e.g. cardiac muscle). Transport is then explored in terms of oxygen levels and capillary supply regions. In Chapter 3 we extend this modelling framework to explore the influence of the surrounding tissue by accounting for the spatial anisotropies of fibre oxygen demand and diffusivity and the heterogeneity in fibre size and shape, as exemplified by mixed muscle tissues (e.g. skeletal muscle). We additionally explore the effects of diffusion through the interstitium, facilitated--diffusion by myoglobin, and Michaelis--Menten kinetics of tissue oxygen consumption. In Chapter 4, a further extension is pursued to account for intracellular heterogeneities in mitochondrial distribution and diffusive parameters. As a demonstration of the potential of the models derived in Chapters 2--4, in Chapter 5 we simulate oxygen transport in myocardial tissue biopsies from rats with either impaired angiogenesis or impaired arteriolar perfusion. Quantitative predictions are made to help explain and support experimental measurements of cardiac performance and metabolism. In the final chapter we summarize the main results and indicate directions for further work.

We investigate the biological constraints determined by the complex relationships between ecological and immunological processes of host-pathogen interactions, with emphasis on influenza viruses in human, which are responsible for a number of pandemics in the last 150 years. We begin by discussing prolegomenous reviews of historical perspectives on the use of theoretical modelling as a complementary tool in public health and epidemiology, current biological background motivating the objective of the thesis, and derivations of mathematical models of multi-locus-allele systems for infectious diseases with co-circulating serotypes. We provide detailed analysis of the multi-locus-allele model and its age-specific extension. In particular, we establish the necessary conditions for the local asymptotic stability of the steady states and the existence of oscillatory behaviours. For the age-structured model, results on the existence of a mild solution and stability conditions are presented. Numerical studies of various strain spaces show that the dynamic features are preserved. Specifically, we demonstrate that discrete antigenic forms of pathogens can exhibit three distinct dynamic features, where antigenic variants (i) fully self-organize and co-exist with no strain structure (NSS), (ii) sort themselves into discrete strain structure (DSS) with non-overlapping or minimally overlapping clusters under the principle of competitive exclusion, or (iii) exhibit cyclical strain structure (CSS) where dominant antigenic types are cyclically replaced with sharp epidemics dominated by (1) a single strain dominance with irregular emergence and re-emergence of certain pathogenic forms, (2) ordered alternating appearance of a single antigenic type in periodic or quasi-periodic form similar to periodic travelling waves, (3) erratic appearance and disappearance of synchrony between discrete antigenic types, and (4) phase-synchronization with uncorrelated amplitudes. These analyses allow us to gain insight into the age-specific immunological profile in order to untangle the effects of strain structures as captured by the clustering behaviours, and to provide public health implications. The age-structured model can be used to investigate the effect of age-specific targeting for public health purposes.

This dissertation focused on development and utilization of numerical and experimental approaches to improve the CFD modeling of fluidization flow of cohesive micron size particles. The specific objectives of this research were: (1) Developing a cluster prediction mechanism applicable to Two-Fluid Modeling (TFM) of gas-solid systems (2) Developing more accurate drag models for Two-Fluid Modeling (TFM) of gas-solid fluidization flow with the presence of cohesive interparticle forces (3) using the developed model to explore the improvement of accuracy of TFM in simulation of fluidization flow of cohesive powders (4) Understanding the causes and influential factor which led to improvements and quantification of improvements (5) Gathering data from a fast fluidization flow and use these data for benchmark validations. Simulation results with two developed cluster-aware drag models showed that cluster prediction could effectively influence the results in both the first and second cluster-aware models. It was proven that improvement of accuracy of TFM modeling using three versions of the first hybrid model was significant and the best improvements were obtained by using the smallest values of the switch parameter which led to capturing the smallest chances of cluster prediction. In the case of the second hybrid model, dependence of critical model parameter on only Reynolds number led to the fact that improvement of accuracy was significant only in dense section of the fluidized bed. This finding may suggest that a more sophisticated particle resolved DNS model, which can span wide range of solid volume fraction, can be used in the formulation of the cluster-aware drag model. The results of experiment suing high speed imaging indicated the presence of particle clusters in the fluidization flow of FCC inside the riser of FIU-CFB facility. In addition, pressure data was successfully captured along the fluidization column of the facility and used as benchmark validation data for the second hybrid model developed in the present dissertation. It was shown the second hybrid model could predict the pressure data in the dense section of the fluidization column with better accuracy.

Microcirculatory vessels are lined by endothelial cells (ECs) which are surrounded by a single or multiple layer of smooth muscle cells (SMCs). Spontaneous and agonist induced spatiotemporal calcium (Ca2+) events are generated in ECs and SMCs, and regulated by complex bi-directional signaling between the two layers which ultimately determines the vessel tone. The contractile state of microcirculatory vessels is an important factor in the determination of vascular resistance, blood flow and blood pressure. This dissertation presents theoretical insights into some of the important and currently unresolved phenomena in microvascular tone regulation. Compartmental and continuum models of isolated EC and SMC, coupled EC-SMC and a multi-cellular vessel segment with deterministic and stochastic descriptions of the cellular components were developed, and the intra- and inter-cellular spatiotemporal Ca2+ mobilization was examined. Coupled EC-SMC model simulations captured the experimentally observed localized subcellular EC Ca2+ events arising from the opening of EC transient receptor vanilloid 4 (TRPV4) channels and inositol triphosphate receptors (IP3Rs). These localized EC Ca2+ events result in endothelium-derived hyperpolarization (EDH) and Nitric Oxide (NO) production which transmit to the adjacent SMCs to ultimately result in vasodilation. The model examined the effect of heterogeneous distribution of cellular components and channel gating kinetics in determination of the amplitude and spread of the Ca2+ events. The simulations suggested the necessity of co-localization of certain cellular components for modulation of EDH and NO responses. Isolated EC and SMC models captured intracellular Ca2+ wave like activity and predicted the necessity of non-uniform distribution of cellular components for the generation of Ca2+ waves. The simulations also suggested the role of membrane potential dynamics in regulating Ca2+ wave velocity. The multi-cellular vessel segment model examined the underlying mechanisms for the intercellular synchronization of spontaneous oscillatory Ca2+ waves in individual SMC. From local subcellular events to integrated macro-scale behavior at the vessel level, the developed multi-scale models captured basic features of vascular Ca2+ signaling and provide insights for their physiological relevance. The models provide a theoretical framework for assisting investigations on the regulation of vascular tone in health and disease.

Le but de cette thèse CIFRE est de construire un portefeuille de stratégies de trading algorithmique intraday. Au lieu de considérer les prix comme une fonction du temps et d'un aléa généralement modélisé par un mouvement brownien, notre approche consiste à identifier les principaux signaux auxquels sont sensibles les donneurs d'ordres dans leurs prises de décision puis alors de proposer un modèle de prix afin de construire des stratégies dynamiques d'allocation de portefeuille. Dans une seconde partie plus académique, nous présentons des travaux de pricing d'options européennes et asiatiques. / The aim of this CIFRE thesis is to build a portfolio of intraday algorithmic trading strategies. Instead of considering stock prices as a function of time and a brownian motion, our approach is to identify the main signals affecting market participants when they operate on the market so we can set up a prices model and then build dynamical strategies for portfolio allocation. In a second part, we introduce several works dealing with asian and european option pricing.

Les travaux de cette thèse s'inscrivent dans le domaine des équations aux dérivées partielles et sont liés à la problématique des changements d'échelle dans le contexte de la cinétique des gaz. En effet, sachant qu'il existe plusieurs niveaux de description pour un gaz, on cherche à relier les différentes échelles associées dans un cadre où une part d'aléa intervient. Dans une première partie, on établit la dérivation rigoureuse de l'équation de Boltzmann linéaire sans cut-off en partant d'un système de particules interagissant via un potentiel à portée infinie en partant d'un équilibre perturbé.La deuxième partie traite du passage d'un modèle BGK stochastique avec champ fort à une loi de conservation scalaire avec forçage stochastique. D'abord, on établit l'existence d'une solution au modèle BGK considéré. Sous une hypothèse additionnelle, on prouve alors la convergence vers une formulation cinétique associée à la loi de conservation avec forçage stochastique.Au cours de la troisième partie, on quantifie dans le cas à vitesses discrètes le défaut de régularité dans les lemmes de moyenne et on établit un lemme de moyenne stochastique dans ce même cas. On applique alors le résultat au cadre de l'approximation de Rosseland pour établir la limite diffusive associée à ce modèle.Enfin, on s'intéresse à l'étude numérique du modèle de Uchiyama de particules carrées à quatre vitesses en dimension deux. Après avoir adapté les méthodes de simulation développées dans le cas des sphères dures, on effectue une étude statistique des limites à différentes échelles de ce modèle. On rejette alors l'hypothèse d'un mouvement Brownien fractionnaire comme limite diffusive / The work of this thesis belongs to the field of partial differential equations and is linked to the problematic of scale changes in the context of kinetic of gas. Indeed, knowing that there exists different scales of description for a gas, we want to link these different associated scales in a context where some randomness acts, in initial data and/or distributed on all the time interval. In a first part, we establish the rigorous derivation of the linear Boltzmann equation without cut-off starting from a particle system interacting via a potential of infinite range starting from a perturbed equilibrium. The second part deals with the passage from a stochastic BGK model with high-field scaling to a scalar conservation law with stochastic forcing. First, we establish the existence of a solution to the considered BGK model. Under an additional assumption, we prove then the convergence to a kinetic formulation associated to the conservation law with stochastic forcing. In the third part, first we quantify in the case of discrete velocities the defect of regularity in the averaging lemmas. Then, we establish a stochastic averaging lemma in that same case. We apply then the result to the context of Rosseland approximation to establish the diffusive limit associated to this model.Finally, we are interested into the numerical study of Uchiyama's model of square particles with four velocities in dimension two. After adapting the methods of simulation which were developed in the case of hard spheres, we carry out a statistical study of the limits at different scales of this model. We reject the hypothesis of a fractional Brownian motion as diffusive limit