L'objectif de notre thèse de <p>doctorat est d’étudier et de décrire les propriétés chimiques et mé- <p>caniques du moteur moléculaire F1 -ATPase. Le moteur F1 -ATPase <p>est un moteur rotatif, d’aspect sphérique et d’environ 10 nanomètre <p>de rayon, qui utilise l’énergie de l’hydrolyse de l’ATP comme car- <p>burant moléculaire. <p>Des questions fondamentales se posent sur le fonctionnement de <p>ce moteurs et sur la quantité de travail qu’il peut fournir. Il s’agit <p>de questions qui concernent principalement la thermodynamique <p>des processus irréversibles. De plus, comme ce moteur est de <p>taille nanométrique, il est fortement influencé par les fluctuations <p>moléculaires, ce qui nécessite une approche stochastique. <p>C’est en créant deux modéles stochastiques complémentaires de <p>ce moteur que nous avons contribué à répondre à ces questions <p>fondamentales. <p>Le premier modèle discuté au chapitre 5 de la thèse, est un mod- <p>èle continu dans le temps et l’espace, décrit par des équations de <p>Fokker-Planck, est construit sur des résultats expérimentaux. <p>Ce modèle tient compte d’une description explicite des fluctua- <p>tions affectant le degré de liberté mécanique et décrit les tran- <p>sitions entre les différents états chimiques discrets du moteur, <p>par un processus de sauts aléatoires entre premiers voisins. Nous <p>avons obtenus des résultats précis concernant la chimie d’hydrolyse <p>et de synthèse de l’ATP, et pour les dépendences du moteur en les <p>différentes variables mécaniques, à savoir, la friction et le couple <p>de force extérieur, ainsi que la dépendence en la température. <p>Les résultats que nous avons obtenus avec ce modèle sont en ex- <p>cellent accord avec les observations expérimentales. <p>Le second modèle est discret dans l’espace et continu dans le <p>temps et est décrit dans le chapitre 6. L’analyse des résultats <p>obtenus par simulations numériques montre que le modèle est <p>en accord avec les observations expérimentales et il permet en <p>outre de dériver des grandeurs thermodynamiques analytique- <p>ment, décrites au chapitre 4, ce que le modèle continu ne permet <p>pas. <p>La comparaison des deux modèles révele la nature du fonction- <p>nement du moteur, ainsi que son régime de fonctionnement loin <p>de l’équilibre. Le second modèle a éte soumis récemment pour <p>publication. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

L’objectif central de la thèse est d’étudier diverses mesures du risque de modèle, exprimées en terme monétaire, qui puissent être appliquées de façon cohérente à une collection hétérogène de produits financiers. Les deux premiers chapitres traitent cette problématique, premièrement d’un point de vue théorique, ensuite en menant un étude empirique centrée sur le marché du gaz naturel. Le troisième chapitre se concentre sur une étude théorique du risque dit de base (en anglais basis risk). Dans le premier chapitre, nous nous sommes intéressés à l’évaluation de produits financiers complexes, qui prend en compte le risque de modèle et la disponibilité dans le marché de produits dérivés basiques, appelés aussi vanille. Nous avons en particulier poursuivi l’approche du transport optimal (connue dans la littérature) pour le calcul des bornes de prix et des stratégies de sur (sous)-couverture robustes au risque de modèle. Nous reprenons en particulier une construction de probabilités martingales sous lesquelles le prix d’une option exotique atteint les dites bornes de prix, en se concentrant sur le cas des martingales positives. Nous mettons aussi en évidence des propriétés significatives de symétrie dans l’étude de ce problème. Dans le deuxième chapitre, nous approchons le problème du risque de modèle d’un point de vue empirique, en étudiant la gestion optimale d’une unité de gaz naturel et en quantifiant l’effet de ce risque sur sa valeur optimale. Lors de cette étude, l’évaluation de l’unité de stockage est basée sur le prix spot, alors que sa couverture est réalisée avec des contrats à termes. Comme mentionné auparavant, le troisième chapitre met l’accent sur le risque de base, qui intervient lorsque l’on veut couvrir un actif conditionnel basé sur un actif non traité (par exemple la température) en se servant d’un portefeuille constitué d’actifs traités sur le marché. Un critère de couverture dans ce contexte est celui de la minimisation de la variance qui est étroitement lié à la décomposition dite de Föllmer-Schweizer. Cette décomposition peut être déduite de la résolution d’une certaine équation différentielle stochastique rétrograde (EDSR) dirigée par une martingale éventuellement à sauts. Lorsque cette martingale est un mouvement brownien standard, les EDSR sont fortement associées aux EDP paraboliques semi linéaires. Dans le cas général nous formulons un problème déterministe qui étend les EDPs mentionnées. Nous appliquons cette démarche à l’important cas particulier de la décomposition de Föllmer-Schweizer, dont nous donnons des expressions explicites de la décomposition du payoff d’une option lorsque les sous-jacents sont exponentielles de processus additifs. / The main objective of this thesis is the study of the model risk and its quantification through monetary measures. On the other hand we expect it to fit a large set of complex (exotic) financial products. The first two chapters treat the model risk problem both from the empirical and the theoretical point of view, while the third chapter concentrates on a theoretical study of another financial risk called basis risk. In the first chapter of this thesis, we are interested in the model-independent pricing and hedging of complex financial products, when a set of standard (vanilla) products are available in the market. We follow the optimal transport approach for the computation of the option bounds and the super (sub)-hedging strategies. We characterize the optimal martingale probability measures, under which the exotic option price attains the model-free bounds; we devote special interest to the case when the martingales are positive. We stress in particular on the symmetry relations that arise when studying the option bounds. In the second chapter, we approach the model risk problem from an empirical point of view. We study the optimal management of a natural gas storage and we quantify the impact of that risk on the gas storage value. As already mentioned, the last chapter concentrates on the basis risk, which is the risk that arises when one hedges a contingent claim written on a non-tradable but observable asset (e.g. the temperature) using a portfolio of correlated tradable assets. One hedging criterion is the mean-variance minimization, which is closely related to the celebrated Föllmer-Schweizer decomposition. That decomposition can be deduced from the resolution of a special Backward Stochastic Differential Equations (BSDEs) driven by a càdlàg martingale. When this martingale is a standard Brownian motion, the related BSDEs are strongly related to semi-linear parabolic PDEs. In that chapter, we formulate a deterministic problem generalizing those PDEs to the general context of martingales and we apply this methodology to discuss some properties of the Föllmer-Schweizer decomposition. We also give an explicit expression of such decomposition of the option payoff when the underlying prices are exponential of additives processes.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

Cette thèse s'intéresse à une étude variée des EDSRs. Une grande partie des résultats sont obtenus sous l'hypothèse d'une croissance de type quadratique du générateur en sa dernière variable. Un premier lien entre EDSRs quadratiques unidimensionnelles et théorie des jeux nous amène à développer des résultats avec générateurs convexes. La théorie du contrôle optimal nécessite quant à elle de traiter du cas multidimensionnel, dans lequel existence et unicité globales ne sont obtenues que pour des générateurs diagonalement quadratiques. Les résultats majeurs sur les EDSRs réfléchies (dont la solution est contrainte à rester dans un domaine) concernent des générateurs Lipschitziens. C'est dans ce cadre que nous développons un résultat de propagation du chaos, avec une contrainte portant sur la loi de la solution plutôt que sur sa trajectoire. Nous dressons enfin un pont entre EDSRs quadratiques et EDSRs réfléchies grâce aux EDSRs quadratiques de type champ moyen. Nous donnons plusieurs nouveaux résultats sur la possibilité de résoudre une équation quadratique dont le générateur dépend également de la moyenne des deux variables. / In this thesis, we are interested in studying variously Backward Stochastic Differential Equations. A large proportion of the results are obtained under the assumption that the driver is of quadratic growth in its last variable. A first link between one-dimensional quadratic BSDEs and game theory leads us to develop results with convex drivers. Optimal control theory requires as for it to deal with the multidimensional case, in which global existence and uniqueness are obtained only for diagonaly quadratic drivers. Major achievements in reflected BSDEs (whose solution is constrained to remain in a domain) are reached for Lipschitz drivers. We develop a result of chaos propagation in this setting, with a constraint on the law of the solution rather than on its path. We finaly build bridge between quadratic BSDEs and reflected BSDEs thanks to mean field quadratic BSDEs. We give several new results on solvability of a quadratic BSDE whose driver depends also on the mean of both variables.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

M. Tech. (Department of Industrial Engineering and Operations Management, Faculty of Engineering), Vaal University of Technology. / The current business environments in many companies are characterized by markets facing tough competitions, from which customer requirements and expectations are becoming increasingly high in terms of quality, cost and delivery dates, etc. These emerging expectations are even getting stronger due to rapid development of new information and communication technologies that provide direct connections between companies and their clients. As a result, companies should have powerful control mechanisms at their disposal. To achieve this, companies rely on a number of functions including production scheduling. This function has always been present within companies, but today, it is facing increasing complexities because of the large number of jobs that must be executed simultaneously. Amongst many factors, it is time driven. This study demonstrates that several disciplines can be married into one model (i.e. a unified model) to solve scheduling problems after disruptions, and clears the way for future multi-disciplinary research efforts. Scheduling problem is modeled as follows: Ito’s stochastic differential rule is used to analyse the time evolution of random or stochastic processes. Multifactor productivity is used to unify various disruption factors. Theory of line balancing is also employed to determine the required number of resources to minimize bottleneck. Reliability: disruptions are considered to be equivalent to system failure. The failure rate of the system is translated to the reliability of the system mathematically. The probabilities of failure are used as indicators of disruptions, and the theory of reliability is then applied. Bernoulli’s principle is also employed to relate pressure to production flow and aid in managing bottleneck situations. Results indicate that the amount of resources needed after disruption depends on the nature of disruption, and that the scheduler should plan to increase number of facilities following a trend that is only predicted by the nature of disruptions. It is also shown that disruption of one type may not greatly affect productivity of a certain company layout, whilst similar disruptions can have devastating effect on another type. It is further concluded that impacts of disruption are dependent on the type of company layouts.

[ES] La naturaleza aleatoria que caracteriza algunos fenómenos en sistemas físicos reales (e.g., ingeniería, biología, economía, finanzas, epidemiología y otros) nos ha planteado el desafío de un cambio de paradigma del modelado matemático y el análisis de sistemas dinámicos, y a tratar los fenómenos aleatorios como variables aleatorias o procesos estocásticos. Este enfoque novedoso ha traído como consecuencia nuevas especificidades que la teoría clásica del modelado y análisis de sistemas dinámicos deterministas no ha podido cubrir. Afortunadamente, maravillosas contribuciones, realizadas sobre todo en el último siglo, desde el campo de las matemáticas por científicos como Kolmogorov, Langevin, Lévy, Itô, Stratonovich, sólo por nombrar algunos; han abierto las puertas para un estudio bien fundamentado de la dinámica de sistemas físicos perturbados por ruido. En la presente tesis se discute el uso de ecuaciones diferenciales algebraicas estocásticas (EDAEs) para el modelado de sistemas multifísicos en red afectados por perturbaciones estocásticas, así como la evaluación de su estabilidad asintótica a través de exponentes de Lyapunov (ELs). El estudio está enfocado en EDAEs d-index-1 y su reformulación como ecuaciones diferenciales estocásticas ordinarias (EDEs). Fundamentados en la teoría ergódica, es factible analizar los ELs a través de sistemas dinámicos aleatorios (SDAs) generados por EDEs subyacentes. Una vez garantizada la existencia de ELs bien definidas, hemos procedido al uso de técnicas de simulación numérica para determinar los ELs numéricamente. Hemos implementado métodos numéricos basados en descomposición QR discreta y continua para el cómputo de la matriz de solución fundamental y su uso en el cálculo de los ELs. Las características numéricas y computacionales más relevantes de ambos métodos se ilustran mediante pruebas numéricas. Toda esta investigación sobre el modelado de sistemas con EDAEs y evaluación de su estabilidad a través de ELs calculados numéricamente, tiene una interesante aplicación en ingeniería. Esta es la evaluación de la estabilidad dinámica de sistemas eléctricos de potencia. En el presente trabajo de investigación, implementamos nuestros métodos numéricos basados en descomposición QR para el test de estabilidad dinámica en dos modelos de sistemas eléctricos de potencia de una-máquina bus-infinito (OMBI) afectados por diferentes perturbaciones ruidosas. El análisis en pequeña-señal evidencia el potencial de las técnicas propuestas en aplicaciones de ingeniería. / [CA] La naturalesa aleatòria que caracteritza alguns fenòmens en sistemes físics reals (e.g., enginyeria, biologia, economia, finances, epidemiologia i uns altres) ens ha plantejat el desafiament d'un canvi de paradigma del modelatge matemàtic i l'anàlisi de sistemes dinàmics, i a tractar els fenòmens aleatoris com a variables aleatòries o processos estocàstics. Aquest enfocament nou ha portat com a conseqüència noves especificitats que la teoria clàssica del modelatge i anàlisi de sistemes dinàmics deterministes no ha pogut cobrir. Afortunadament, meravelloses contribucions, realitzades sobretot en l'últim segle, des del camp de les matemàtiques per científics com Kolmogorov, Langevin, Lévy, Itô, Stratonovich, només per nomenar alguns; han obert les portes per a un estudi ben fonamentat de la dinàmica de sistemes físics pertorbats per soroll. En la present tesi es discuteix l'ús d'equacions diferencials algebraiques estocàstiques (EDAEs) per al modelatge de sistemes multifísicos en xarxa afectats per pertorbacions estocàstiques, així com l'avaluació de la seua estabilitat asimptòtica a través d'exponents de Lyapunov (ELs). L'estudi està enfocat en EDAEs d-index-1 i la seua reformulació com a equacions diferencials estocàstiques ordinàries (EDEs). Fonamentats en la teoria ergòdica, és factible analitzar els ELs a través de sistemes dinàmics aleatoris (SDAs) generats per EDEs subjacents. Una vegada garantida l'existència d'ELs ben definides, hem procedit a l'ús de tècniques de simulació numèrica per a determinar els ELs numèricament. Hem implementat mètodes numèrics basats en descomposició QR discreta i contínua per al còmput de la matriu de solució fonamental i el seu ús en el càlcul dels ELs. Les característiques numèriques i computacionals més rellevants de tots dos mètodes s'illustren mitjançant proves numèriques. Tota aquesta investigació sobre el modelatge de sistemes amb EDAEs i avaluació de la seua estabilitat a través d'ELs calculats numèricament, té una interessant aplicació en enginyeria. Aquesta és l'avaluació de l'estabilitat dinàmica de sistemes elèctrics de potència. En el present treball de recerca, implementem els nostres mètodes numèrics basats en descomposició QR per al test d'estabilitat dinàmica en dos models de sistemes elèctrics de potència d'una-màquina bus-infinit (OMBI) afectats per diferents pertorbacions sorolloses. L'anàlisi en xicotet-senyal evidencia el potencial de les tècniques proposades en aplicacions d'enginyeria. / [EN] The random nature that characterizes some phenomena in the real-world physical systems (e.g., engineering, biology, economics, finance, epidemiology, and others) has posed the challenge of changing the modeling and analysis paradigm and treat these phenomena as random variables or stochastic processes. Consequently, this novel approach has brought new specificities that the classical theory of modeling and analysis for deterministic dynamical systems cannot cover. Fortunately, stunning contributions made overall in the last century from the mathematics field by scientists such as Kolmogorov, Langevin, Lévy, Itô, Stratonovich, to name a few; have opened avenues for a well-founded study of the dynamics in physical systems perturbed by noise. In the present thesis, we discuss stochastic differential-algebraic equations (SDAEs) for modeling multi-physical network systems under stochastic disturbances, and their asymptotic stability assessment via Lyapunov exponents (LEs). We focus on d-index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Supported by the ergodic theory, it is feasible to analyze the LEs via the random dynamical system (RDSs) generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important numerical and computational features of both methods are illustrated through numerical tests. All this investigation concerning systems modeling through SDAEs and their stability assessment via computed LEs finds an appealing engineering application in the dynamic stability assessment of power systems. In this research work, we implement our QR-based numerical methods for testing the dynamic stability in two types of single-machine infinite-bus (SMIB) power system models perturbed by different noisy disturbances. The analysis in small-signal evidences the potential of the proposed techniques in engineering applications. / Mi agradecimiento al estado ecuatoriano que, a través del Programa de Becas para el Fortalecimiento y Desarrollo del Talento Humano en Ciencia y Tecnología 2012 de la Secretaría Nacional de Educación Superior, Ciencia y Tecnología (SENESCYT), han financiado mis estudios de doctorado. / González Zumba, JA. (2020). Dynamic Modeling and Stability Analysis of Stochastic Multi-Physical Systems Applied to Electric Power Systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/158558

In mathematical finance, the sensitivities of option prices to various market parameters, also known as the “Greeks”, reflect the exposure to different sources of risk. Computing these is essential to predict the impact of market moves on portfolios and to hedge them adequately. This is commonly done using Monte Carlo simulations. However, obtaining accurate estimates of the Greeks can be computationally costly. Multilevel Monte Carlo offers complexity improvements over standard Monte Carlo techniques. However the idea has never been used for the computation of Greeks. In this work we answer the following questions: can multilevel Monte Carlo be useful in this setting? If so, how can we construct efficient estimators? Finally, what computational savings can we expect from these new estimators? We develop multilevel Monte Carlo estimators for the Greeks of a range of options: European options with Lipschitz payoffs (e.g. call options), European options with discontinuous payoffs (e.g. digital options), Asian options, barrier options and lookback options. Special care is taken to construct efficient estimators for non-smooth and exotic payoffs. We obtain numerical results that demonstrate the computational benefits of our algorithms. We discuss the issues of convergence of pathwise sensitivities estimators. We show rigorously that the differentiation of common discretisation schemes for Ito processes does result in satisfactory estimators of the the exact solutions’ sensitivities. We also prove that pathwise sensitivities estimators can be used under some regularity conditions to compute the Greeks of options whose underlying asset’s price is modelled as an Ito process. We present several important results on the moments of the solutions of stochastic differential equations and their discretisations as well as the principles of the so-called “extreme path analysis”. We use these to develop a rigorous analysis of the complexity of the multilevel Monte Carlo Greeks estimators constructed earlier. The resulting complexity bounds appear to be sharp and prove that our multilevel algorithms are more efficient than those derived from standard Monte Carlo.

Ce travail présente quelques résultats concernant des méthodes numériques déterministes et probabilistes pour des équations aux dérivées partielles à coefficients aléatoires, avec des applications à l'hydrogéologie. On s'intéresse tout d'abord à l'équation d'écoulement dans un milieu poreux en régime stationnaire avec un coefficient de perméabilité lognormal homogène, incluant le cas d'une fonction de covariance peu régulière. On établit des estimations aux sens fort et faible de l'erreur commise sur la solution en tronquant le développement de Karhunen-Loève du coefficient. Puis on établit des estimations d'erreurs éléments finis dont on déduit une extension de l'estimation d'erreur existante pour la méthode de collocation stochastique, ainsi qu'une estimation d'erreur pour une méthode de Monte-Carlo multi-niveaux. On s'intéresse enfin au couplage de l'équation d'écoulement considérée précédemment avec une équation d'advection-diffusion, dans le cas d'incertitudes importantes et d'une faible longueur de corrélation. On propose l'analyse numérique d'une méthode numérique pour calculer la vitesse moyenne à laquelle la zone contaminée par un polluant s'étend. Il s'agit d'une méthode de Monte-Carlo combinant une méthode d'élements finis pour l'équation d'écoulement et un schéma d'Euler pour l'équation différentielle stochastique associée à l'équation d'advection-diffusion, vue comme une équation de Fokker-Planck. / This work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.

Ziel dieser Arbeit ist es eine Beziehung zwischen den biophysikalischen Eigenschaften der Nervenmembran, und den ausgeführten Berechnungen und Filtereigenschaften eines tonisch feuernden Neurons, unter Einbeziehen intrinsischer Fluktuationen, herzustellen. Zu diesem Zweck werden zu erst die mikroskopischen Fluktuationen, die durch das stochastische Öffnen und Schließen der Ionenkanäle verursacht werden, zu makroskopischer Varibilität in den Zeitpunkten des Auftretens der Aktionspotentiale übersetzt, denn es sind diese Spikezeiten die in vielen sensorischen Systemen informationstragenden sind. Die Methode erlaubt es das stochastischer Verhalten komplizierter Ionenkanalstrukturen mit einer großen Zahl an Untereinheiten, in Spikezeitenvariabilität zu übersetzen. Als weiteres werden die Filtereigenschaften der Nervenzellen in der überschwelligen Dynamik, also bei Existenz eines stabilen Grenzzyklus, aus ihren Phasenantwortkurven (PAK), einer Eigenschaft des linearisierten adjungierten Flusses auf dem Grenzzyklus, in einem stöhrungstheoretischen Ansatz berechnet. Es ergibt sich, dass Charakteristika des Filter, wie beispielsweise die DC Komponente und die Eigenschaften des Filters um die Fundamentalfrequenz und ihrer Harmonien, von den Fourierkomponenten der PAK abhängen. Unter Verwendung der hergeleiteten Filter und weiterer Annahmen ist es möglich das frequenzabhängige Signal-zu-Rauschen Verhältnis zu berechnen, und damit eine untere Schranke für die Informationstransferrate eines Leitfähigkeitsmodells zu berechnen. Unter Zuhilfenahme der numerischen Kontinuierungsmethode ist es möglich die Veränderungen in der Spikevariabilität und den Filtern für jeden biophysikalischen Parameter des System zu verfolgen. Weiterhin wurde die verwendete Phasenreduktion durch eine Korrektur ergänzt, die die Radialdynamik einbezieht. Es zeigt sich, dass die Krümmung der Isochronen einen Einfluss darauf hat ob das Rauschen einen positiven oder negativen Frequenzschift hervorruft. / The goal of the thesis is to establish quantitative, analytical relations between the biophysical properties of nerve membranes and the performed neuronal computations for neurons in a tonically spiking regime and in the presence of intrinsic noise. For this purpose, two major lines of investigation are followed. Firstly, microscopic noise caused by the stochastic opening and closing of ion channels is mapped to the macroscopic spike jitter that affects neural coding. The method is generic enough to allow one to treat Markov channel models with complicated, high-dimensional state spaces and calculate from them the noise in the coding variable, i.e., the spike time. Secondly, the suprathreshold filtering properties of neurons are derived, based on the phase response curves (PRCs) by perturbing the associated Fokker-Planck equations. It turns out that key characteristics of the filter, such as the DC component of the gain and the behaviour near the fundamental frequency and its harmonics are related to the particular Fourier components of the PRC and hence the bifurcation type of the neuron. With the help of the derived filter and further approximations one is able to calculate the frequency resolved signal-to-noise ration and finally the total information transmission rate of a conductance based model. Using the method of numerical continuation it is possible to calculate the change in spike time noise level as well as the filtering properties for arbitrary changes in biophysical parameter such as varying channel densities or mean input to the cell. We extend the phase reduction to include correction terms from the amplitude dynamics that are related to the curvature of the isochrons and provide a method to identify the required amplitude sensitivities numerically. It can be shown that the curvature of the isochron has a direct consequence for the noise induced frequency shift.