Generating samples from generalized hyperbolic distributions and non-central chi-square distributions by inversion has become an important task for the simulation of recent models in finance in the framework of (quasi-) Monte Carlo. However, their distribution functions are quite expensive to evaluate and thus numerical methods like root finding algorithms are extremely slow. In this paper we demonstrate how our new method based on Newton interpolation and Gauss-Lobatto quadrature can be utilized for financial applications. Its fast marginal generation times make it competitive, even for situations where the parameters are not always constant. / Series: Research Report Series / Department of Statistics and Mathematics

We demonstrate that for the fast numerical inversion of the (generalized) inverse Gaussian distribution two algorithms based on polynomial interpolation are well-suited. Their precision is close to machine precision and they are much faster than the bisection method recently proposed by Y. Lai. / Series: Research Report Series / Department of Statistics and Mathematics

The present thrust in the electronics industry towards integrating multiple functions on a single chip while operating at very high frequencies has highlighted the need for efficient Electronic Design Automation (EDA) tools to shorten the design cycle and capture market windows. However, the increasing complexity in modern circuit design has made simulation a computationally cumbersome task. The notion of model order reduction has emerged as an effective tool to address this difficulty. Typically, there are numerous approaches and several issues involved in the implementation of model-order reduction techniques. Among the important ones of those issues is the problem of determining a suitable order (or size) for the reduced system. An optimal order would be the minimal order that enables the reduced system to capture the behavior of the original (more complex and larger) system up to a user-defined frequency. The contribution presented in this thesis describes a new approach aimed at determining the order of the reduced system. The proposed approach is based on approximating the impulse response of the original system in the time-domain. The core methodology in obtaining that approximation is based on numerically inverting the Laplace-domain of the representation of the impulse response from the complex-domain (s-domain) into the time-domain. The main advantage of the proposed approach is that it allows the order selection algorithm to operate directly on the time-domain form of the impulse response. It is well-known that numerically generating the impulse response in the time-domain is very difficult and its not impossible, since it requires driving the original network with the Dirac-delta function, which is a mathematical abstraction rather than a concrete waveform that can be implemented on a digital computer. However, such a difficulty is avoided in the proposed approach since it uses the Laplace-domain image of the impulse response to obtain its time-domain representation. The numerical simulations presented in the thesis demonstrate that using the time-domain waveform of the impulse response, computed using the proposed approach and properly filtered with a Butterworth filter, guides the order selection algorithm to select a smaller order, i.e., the reduced system becomes more compact in size. The phrase "smaller or more compact" in this context refers to the comparison with existing techniques currently in use, which seek to generate some form of time-domain approximations for the impulse response through driving the original network with pulse-shaped function (e.g., Gaussian pulse).

Nous proposons une nouvelle procédure de mesure permettant de tracer la courbe de résistivité en fonction de l'écartement électrode de courant-électrode de potentiel relié à la profondeur d'investigation et au gradient de teneur en eau. Les mesures de résistivité au droit des armatures sont réalisées et une procédure d'inversion permettant de remonter au gradient de résistivité est mise en place. Deux types de mesures ont été effectués expérimentalement, sur des dalles en béton armé et non armée. Deux types de gradient sont étudiés : le séchage et l'humidification. La présence de l'armature diminue significativement la résistivité à cause d'un effet de court-circuit. Les mesures montrent l'intérêt de la procédure pour l'étude du gradient de teneur en eau. Pour la partie numérique la simulation a montré que la mesure de résistivité permet effectivement de déterminer le gradient de résistivité et de définir les paramètres électrochimiques de l'acier via le modèle de Butler-Volmer. / We propose a new measurement procedure that allows drawing a resistivity curve as a function of the spacing between current electrode and potential electrode which is linked to the investigation depth and the water content gradient. The resistivity measurements above the rebars are performed and an inversion procedure to assess the resistivity gradient is carried out. For the experimental part two kinds of measurement are performed on concrete slabs (reinforced and unreinforced). Two gradient types are studied: drying and wetting. The presence of the reinforcement significantly decreases the measurement of the resistivity. The measurements show the interest of the proposed procedure for the study of the water content gradient. Two measurement results were considered in numerical analyses. The simulation shows that it is possible to assess the resistivity gradient and the electrochemical parameters of the rebar via the Butler-Volmer model.

En théorie de la commande, un modèle du système est généralement utilisé pour construire la loi de commande et assurer ses performances. Les équations mathématiques qui représentent le système à contrôler sont utilisées pour assurer que le contrôleur associé va stabiliser la boucle fermée. Mais, en pratique, le système réel s’écarte du comportement théorique modélisé. Des non-linéarités ou des dynamiques rapides peuvent être négligées, les paramètres sont parfois difficiles à estimer, des perturbations non maitrisables restent non modélisées. L’approche proposée dans ce travail repose en partie sur la connaissance du système à piloter par l’utilisation d’un modèle analytique mais aussi sur l’utilisation de données expérimentales hors ligne ou en ligne. A chaque pas de temps la valeur de la commande qui amène au mieux le système vers un objectif choisi a priori, est le résultat d’un algorithme qui minimise une fonction de coût ou maximise une récompense. Au centre de la technique développée, il y a l’utilisation d’un modèle numérique de comportement du système qui se présente sous la forme d’une fonction de prédiction tabulée ayant en entrée un n-uplet de l’espace joint entrées/état ou entrées/sorties du système. Cette base de connaissance permet l’extraction d’une sous-partie de l’ensemble des possibilités des valeurs prédites à partir d’une sous-partie du vecteur d’entrée de la table. Par exemple, pour une valeur de l’état, on pourra obtenir toutes les possibilités d’états futurs à un pas de temps, fonction des valeurs applicables de commande. Basé sur des travaux antérieurs ayant montré la viabilité du concept en entrées/état, de nouveaux développements ont été proposés. Le modèle de prédiction est initialisé en utilisant au mieux la connaissance a priori du système. Il est ensuite amélioré par un algorithme d’apprentissage simple basé sur l’erreur entre données mesurées et données prédites. Deux approches sont utilisées : la première est basée sur le modèle d’état (comme dans les travaux antérieurs mais appliquée à des systèmes plus complexes), la deuxième est basée sur un modèle entrée-sortie. La valeur de commande qui permet de rapprocher au mieux la sortie prédite dans l’ensemble des possibilités atteignables de la sortie ou de l’état désiré, est trouvée par un algorithme d’optimisation. Afin de valider les différents éléments proposés, cette commande a été mise en œuvre sur différentes applications. Une expérimentation réelle sur un quadricoptère et des essais réels de suivi de trajectoire sur un véhicule électrique du laboratoire montrent sacapacité et son efficacité sur des systèmes complexes et rapides. D’autres résultats en simulation permettent d’élargir l’étude de ses performances. Dans le cadre d’un projet partenarial, l’algorithme a également montré sa capacité à servir d’estimateur d’état dans la reconstruction de la vitesse mécanique d’une machine asynchrone à partir des signaux électriques. Pour cela, la vitesse mécanique a été considérée comme l’entrée du système. / In classical control theory, the control law is generally built, based on the theoretical model of the system. That means that the mathematical equations representing the system dynamics are used to stabilize the closed loop. But in practice, the actual system differs from the theory, for example, the nonlinearity, the varied parameters and the unknown disturbances of the system. The proposed approach in this work is based on the knowledge of the plant system by using not only the analytical model but also the experimental data. The input values stabilizing the system on open loop, that minimize a cost function, for example, the distance between the desired output and the predicted output, or maximize a reward function are calculated by an optimal algorithm. The key idea of this approach is to use a numerical behavior model of the system as a prediction function on the joint state and input spaces or input-output spaces to find the controller’s output. To do this, a new non-linear control concept is proposed, based on an existing controller that uses a prediction map built on the state-space. The prediction model is initialized by using the best knowledge a priori of the system. It is then improved by using a learning algorithm based on the sensors’ data. Two types of prediction map are employed: the first one is based on the state-space model; the second one is represented by an input-output model. The output of the controller, that minimizes the error between the predicted output from the prediction model and the desired output, will be found using optimal algorithm. The application of the proposed controller has been made on various systems. Some real experiments for quadricopter, some actual tests for the electrical vehicle Zoé show its ability and efficiency to complex and fast systems. Other the results in simulation are tested in order to investigate and study the performance of the proposed controller. This approach is also used to estimate the rotor speed of the induction machine by considering the rotor speed as the input of the system.

Cette thèse vise à améliorer la résolution en trois dimensions grâce à une technique récente d’imagerie : la microscopie tomographique diffractive (MTD). Son principe est d’éclairer l’objet successivement sous différents angles en lumière cohérente, de détecter le champ diffracté en phase et en amplitude, et de reconstruire la carte 3D de permittivité de l’objet par un algorithme d’inversion. La MTD s’est avérée capable de combiner plusieurs modalités utiles pour la microscopie sans marquage, telles que plein champ, champ sombre, à contraste de phase, confocale, ou encore la microscopie à synthèse d’ouverture 2D ou 3D. Toutes sont basées sur des approximations scalaires et linéaires, ce qui restreint leur domaine d’application pour restituer l’objet de manière quantitative. A l’aide d’une inversion numérique rigoureuse prenant en compte la polarisation du champ et le phénomène de diffusion multiple, nous sommes parvenus à reconstruire la carte 3D de permittivité d’objets avec une résolution de λ/4. Une amélioration supplémentaire la portant à λ/10 a été rendue possible par l’insertion d’information a priori sur l’objet dans l’algorithme d’inversion. Enfin, la résolution axiale est moins bonne du fait de l’asymétrie des schémas d’illumination et de détection dans les microscopes. Pour s’affranchir de cette limitation, une configuration de tomographie assistée par miroir a été implémentée et a mis en évidence un pouvoir de séparation axial meilleur que λ/2. Au final, la MTD s’est illustrée comme un outil de caractérisation puissant pour reconstruire en 3D les objets ainsi que leurs indices optiques, à des résolutions bien supérieures à celles des microscopes conventionnels. / This PhD thesis is devoted to the three-dimensional isotropic resolution improvement using optical tomographic diffraction microscopy (TDM), an emerging optical microscope technique. The principle is to illuminate the sample successively with various angles of coherent light, collect the complex (amplitude and phase) diffracted field and reconstruct the sample 3D permittivity map through an inversion algorithm. A single TDM measurement was shown to combine several popular microscopy techniques such as bright-field microscope, dark-field microscope, phase-contrast microscope, confocal microscope, 2D and 3D synthetic aperture microscopes. All rely on scalar and linear approximations that assume a linear link between the object and the field diffracted by it, which limit their applicability to retrieve the object quantitatively. Thanks to a rigorous numerical inversion of the TDM diffracted field data which takes into account the polarization of the field and the multiple scattering process, we were able to reconstruct the 3D permittivity map of the object with a λ/4 transverse resolution. A further improvement to λ/10 transverse resolution was achieved by providing a priori information about the sample to the non-linear inversion algorithm. Lastly, the poor axial resolution in microscopes is due to the fundamental asymmetry of illumination and detection. To overcome this, a mirror-assisted tomography configuration was implemented, and has demonstrated a sub-λ/2 axial resolution capability. As a result, TDM can be seen as a powerful tool to reconstruct objects in three-dimensions with their optical material properties at resolution far superior to conventional microscopes.

This thesis considers new methods that exploit recent developments in computer technology to address three extant problems in the area of Finance and Econometrics. The problem of Asian option pricing has endured for the last two decades in spite of many attempts to find a robust solution across all parameter values. All recently proposed methods are shown to fail when computations are conducted using standard machine precision because as more and more accuracy is forced upon the problem, round-off error begins to propagate. Using recent methods from numerical analysis based on multi-precision arithmetic, we show using the Mathematica platform that all extant methods have efficacy when computations use sufficient arithmetic precision. This creates the proper framework to compare and contrast the methods based on criteria such as computational speed for a given accuracy. Numerical methods based on a deformation of the Bromwich contour in the Geman-Yor Laplace transform are found to perform best provided the normalized strike price is above a given threshold; otherwise methods based on Euler approximation are preferred. The same methods are applied in two other contexts: the simulation of stable distributions and the computation of unit root densities in Econometrics. The stable densities are all nested in a general function called a Fox H function. The same computational difficulties as above apply when using only double-precision arithmetic but are again solved using higher arithmetic precision. We also consider simulating the densities of infinitely divisible distributions associated with hyperbolic functions. Finally, our methods are applied to unit root densities. Focusing on the two fundamental densities, we show our methods perform favorably against the extant methods of Monte Carlo simulation, the Imhof algorithm and some analytical expressions derived principally by Abadir. Using Mathematica, the main two-dimensional Laplace transform in this context is reduced to a one-dimensional problem.

為了改進Black-Scholes模式的實證現象，許多其他的模型被建議有leptokurtic特性以及波動度聚集的現象。然而對於其他的模型分析的處理依然是一個問題。在本論文中，我們建議使用馬可夫跳躍擴散過程，不僅能整合leptokurtic與波動度微笑特性，而且能產生波動度聚集的與長記憶的現象。然後，我們應用Lucas的一般均衡架構計算選擇權價格，提供均衡下當跳躍的大小服從一些特別的分配時則選擇權價格的解析解。特別地，考慮當跳躍的大小服從兩個情況，破產與lognormal分配。當馬可夫跳躍擴散模型的馬可夫鏈有兩個狀態時，稱為轉換跳躍擴散模型，當跳躍的大小服從lognormal分配我們得到選擇權公式。使用轉換跳躍擴散模型選擇權公式，我們給定一些參數下研究公式的數值極限分析以及敏感度分析。 / To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address the leptokurtic feature of the asset return distribution, and the effects of volatility clustering phenomenon. However, analytical tractability remains a problem for most of the alternative models. In this dissertation, we propose a Markov jump diffusion model, that can not only incorporate both the leptokurtic feature and volatility smile, but also present the economic features of volatility clustering and long memory. Next, we apply Lucas's general equilibrium framework to evaluate option price, and to provide analytical solutions of the equilibrium price for European call options when the jump size follows some specific distributions. In particular, two cases are considered, the defaultable one and the lognormal distribution. When the underlying Markov chain of the Markov jump diffusion model has two states, the so-called switch jump diffusion model, we write an explicit analytic formula under the jump size has a lognormal distribution. Numerical approximations of the option prices as well as sensitivity analysis are also given.