Dans cette thèse nous considérons l’équation des ondes amorties vectorielle sur une variété riemannienne compacte, lisse et sans bord. L’amortisseur est ici une fonction lisse allant de la variété dans l’espace des matrices hermitiennes de taille n. Les solutions de cette équation sont donc à valeurs vectorielles. Nous commençons dans un premier temps par calculer le meilleur taux de décroissance exponentiel de l’énergie en fonction du terme d’amortissement. Ceci nous permet d’obtenir une condition nécessaire et suffisante la stabilisation forte de l’équation des ondes amorties vectorielle. Nous mettons aussi en évidence l’apparition d’un phénomène de sur-amortissement haute fréquence qui n’existait pas dans le cas scalaire. Dans un second temps nous nous intéressons à la répartition asymptotique des fréquences propres de l’équation des ondes amorties vectorielle. Nous démontrons que, à un sous ensemble de densité nulle près, l’ensemble des fréquences propres est contenu dans une bande parallèle à l’axe imaginaire. La largeur de cette bande est déterminée par les exposants de Lyapunov d’un système dynamique défini à partir du coefficient d’amortissement. / In this thesis we are considering the vectorial damped wave equation on a compact and smooth Riemannian manifold without boundary. The damping term is a smooth function from the manifold to the space of Hermitian matrices of size n. The solutions of this équation are thus vectorial. We start by computing the best exponential energy decay rate of the solutions in terms of the damping term. This allows us to deduce a sufficient and necessary condition for strong stabilization of the vectorial damped wave equation. We also show the appearance of a new phenomenon of high-frequency overdamping that did not exists in the scalar case. In the second half of the thesis we look at the asymptotic distribution of eigenfrequencies of the vectorial damped wave equation. Were show that, up to a null density subset, all the eigenfrequencies are in a strip parallel to the imaginary axis. The width of this strip is determined by the Lyapunov exponents of a dynamical system defined from the damping term.

La connaissance du microclimat et de l’évapotranspiration ou flux de chaleur latente qui représente la consommation réelle en eau de la culture à l’échelle des parcelles agricoles est une donnée importante pour comprendre le développement des cultures. La plupart des modèles permettant d’estimer l’évapotranspiration sont utilisés sur des surfaces homogènes sans tenir compte des interactions surface - atmosphère et de la variabilité spatiale du domaine agricole. Nous avons utilisé un modèle de couche limite atmosphérique afin de prendre en compte ces interactions. Une approche dite « patchée » permet d’introduire la variabilité spatiale des surfaces dans le modèle à partir des diverses proportions et des caractéristiques des principaux couverts végétaux qui composent le paysage. Une méthode d’assimilation variationnelle a été implémentée afin d’estimer certains paramètres du modèle difficile à connaître précisément. La méthode est basée sur le calcul de l’adjoint du modèle et utilise une température de surface observée par télédétection. L’approche développée est comparée à des approches plus simples considérant chaque type de surface indépendamment, mettant en évidence le rôle de la prise en compte de la variabilité spatiale de la surface sur la simulation du microclimat et des flux de surface / Knowledge of climate at regional scale and evapotranspiration (or latent heat flux which represents the actual water consumption of culture) is a key to understand the development of crops. Most of the methods aiming at estimating evapotranspiration assume homogeneous or decoupled atmospheric variables over the modelling domain without accounting for the feedback between surface and atmosphere. In order to analyse such dependencies and to predict microclimate and land surface fluxes we have developed a coupled atmospheric boundary layer - land surface model which accounts for the landscape heterogeneity using a tiled approach. We have implemented appropriate procedures (variational data assimilation) for assimilating remote sensing data into the model allowing to retrieve some input parameters difficult to estimate spatially (soil moisture and aerodynamic roughness). The developed method is compared to classical approaches considering each type of surface independently. Results are discussed in this paper

Dans cette thèse, nous présentons une étude rigoureuse des barres d'erreurs et des sensibilités de paramètres neutroniques (tels le keff) aux données nucléaires de base utilisées pour les calculer. Notre étude commence au niveau fondamental, i.e. les fichiers de données ENDF et leurs incertitudes, fournies sous la forme de matrices de variance/covariance, et leur traitement. Lorsqu'un calcul méthodique et consistant des sensibilités est consenti, nous montrons qu'une approche déterministe utilisant des formalismes bien connus est suffisante pour propager les incertitudes des bases de données avec un niveau de précision équivalent à celui des meilleurs outils disponibles sur le marché, comme les codes Monte-Carlo de référence. En appliquant notre méthodologie à trois exercices proposés par l'OCDE, dans le cadre des Benchmarks UACSA, nous donnons des informations, que nous espérons utiles, sur les processus physiques et les hypothèses sous-jacents aux formalismes déterministes utilisés dans cette étude. / This dissertation presents a comprehensive study of sensitivity/uncertainty analysis for reactor performance parameters (e.g. the k-effective) to the base nuclear data from which they are computed. The analysis starts at the fundamental step, the Evaluated Nuclear Data File and the uncertainties inherently associated with the data they contain, available in the form of variance/covariance matrices. We show that when a methodical and consistent computation of sensitivity is performed, conventional deterministic formalisms can be sufficient to propagate nuclear data uncertainties with the level of accuracy obtained by the most advanced tools, such as state-of-the-art Monte Carlo codes. By applying our developed methodology to three exercises proposed by the OECD (UACSA Benchmarks), we provide insights of the underlying physical phenomena associated with the used formalisms.

La caractérisation de la neige est un enjeu important pour la gestion des ressources en eau et pour la prévision des risques d'avalanche. L'avènement des nouveaux satellites Radar de Synthèse d'Ouverture (RSO) bande X à haute résolution permet d'acquérir des données de résolution métrique avec une répétitivité journalière. Dans ce travail, un modèle de rétrodiffusion des ondes électromagnétiques de la neige sèche est adapté à la bande X et aux fréquences plus élevées. L'algorithme d'assimilation de données 3D-VAR est ensuite implémenté pour contraindre le modèle d'évolution de la neige SURFEX/Crocus à l'aide des observations satellitaires. Enfin, l'ensemble de ces traitements sont évalué à partir de données du satellite TerraSAR-X acquises sur le glacier d'Argentière dans la vallée de Chamonix. Cette première comparaison montre le fort potentiel de l'assimilation des données RSO bande X pour la caractérisation du manteau neigeux. / Characterization of snowpack structure is an important issue for the management of water resources and the prediction of avalanche risks. New Synthetic Aperture Radar (SAR) satellites in X-band at high-resolution allow us to acquire image data with metric resolution and daily observations. In this work, an electromagnetic backscattering model applicable for dry snow is adapted for X-band and higher frequencies. The 3D-VAR data assimilation algorithm is then implemented to constrain the evolution of the snow metamorphisme model SURFEX/Crocus using satellite observations. Finally, the algorithm is evaluated using image data acquired from TerraSAR-X satellite on the Argentiere glacier in the Chamonix Valley of the French Alps. This first comparison shows the high potential of the data assimilation assimilation method using X-band SAR data for characterization of the snowpack.

When modelling risk in the banking book, a simple capital level approach can fail to capture the interactions between different risk measures or risk classes since they are modelled separately. In this thesis we propose a model for projecting the book value of a run-off balance sheet portfolio of fixed and variable rate loans, while also calculating net interest income, economic value of equity, capital requirement and capital cost within the same model. Using adjoint algorithmic differentiation, we also retrieve the sensitivities of each measure and the balance sheet towards a term structure of zero rates, for the lifetime of the portfolio. The model is an attempt at a holistic approach to modelling interest rate risk in the banking book, and its design allows for extensions to other financial risk classes such as credit risk and liquidity risk. / Vid modellering av risker i bankboken kan en enkel kapitalmodell misslyckas med att fånga interaktionerna mellan olika riskmått eller riskklasser eftersom de modelleras separat. I den här masteruppsatsen föreslår vi en modell för att projicera balansräkningen framåt i tiden för en portfölj av lån med fast och rörlig ränta, och beräknar samtidigt Räntenetto, Ekonomiskt Värde av Eget Kapital, kapitalkrav och kapitalkostnad inom samma modell. Med hjälp av adjoint algoritmisk derivering beräknar vi även känsligheterna för varje mått samt bokfört värde av portföljen mot en terminstruktur av nollräntor under portföljens livstid. Modellen är ett försök att holistisk modellera ränterisk i bankboken, och dess utformning möjliggör utvidgning till andra finansiella riskklasser som kreditrisk och likviditetsrisk.

Data assimilation arises in a vast array of different topics: traditionally in meteorological and oceanographic modelling, wind tunnel or water tunnel experiments and recently from biomedical engineering. Data assimilation is a process for combine measured or observed data with a mathematical model, to obtain estimates of the expected data. The measured data usually contains inaccuracies and is given with low spatial and/or temporal resolution. In this thesis data assimilation for time dependent fluid flow is considered. The flow is assumed to satisfy a given partial differential equation, representing the mathematical model. The problem is to determine the initial state which leads to a flow field which satisfies the flow equation and is close to the given data. In the first part we consider one-dimensional flow governed by Burgers’ equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L2-norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L2-minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods. In the second part three-dimensional inviscid compressible flow represented by the Euler equations is considered. Adjoint technique is used to obtain an explicit formula for the gradient to the optimization problem. The gradient is used in combination with a quasi-Newton method to obtain a solution. The main focus regards the derivation of the adjoint equations with boundary conditions. An existing flow solver EDGE has been modified to solve the adjoint Euler equations and the gradient computations are validated numerically. The proposed iteration method are applied to a test problem where the initial pressure state is reconstructed, for exact data as well as when disturbances in data are present. The numerical convergence and the result are satisfying.

<p>The stability and the receptivity of three-dimensional flat plate boundary layers is studied employing parabolised stability equations. These allow for computationally efficient parametric studies. Two different sets of equations are used. The stability of modal disturbances in the form of crossflow vortices is studied by means of the well-known classical parabolised stability equations (PSE). A new method is developed which is applicable to more general vortical-type disturbances. It is based on a modified version of the classical PSE and describes both modal and non-modal growth in three-dimensional boundary layers. This modified PSE approach is used in conjunction with a Lagrange multiplier technique to compute spatial optimal disturbances in three-dimensional boundary layers. These take the form of streamwise oriented tilted vortices initially and develop into streaks further downstream. When entering the domain where modal disturbances become unstable optimal disturbances smoothly evolve into crossflow modes. It is found that non-modal growth is of significant magnitude in three-dimensional boundary layers. Both the lift-up and the Orr mechanism are identified as the physical mechanisms behind non-modal growth. Furthermore, the modified PSE are used to determine the response of three-dimensional boundary layers to vortical free-stream disturbances. By comparing to results from direct numerical simulations it is shown that the response, including initial transient behaviour, is described very accurately. Extensive parametric studies are performed where effects of free-stream turbulence are modelled by filtering with an energy spectrum characteristic for homogeneous isotropic turbulence. It is found that a quantitative prediction of the boundary layer response to free-stream turbulence requires detailed information about the incoming turbulent flow field. Finally, the adjoint of the classical PSE is used to determine the receptivity of modal disturbances with respect to localised surface roughness. It is shown that the adjoint approach yields perfect agreement with results from Finite-Reynold-Number Theory (FRNT) if the boundary layer is assumed to be locally parallel.  Receptivity is attenuated if nonlocal and non-parallel effects are accounted for. Comparisons to direct numerical simulations and extended parametric studies are presented.</p>

This dissertation describes the application of two principled variance reduction strategies to increase the efficiency for two applications within medical physics. The first, called correlated Monte Carlo (CMC) applies to patient-specific, permanent-seed brachytherapy (PSB) dose calculations. The second, called adjoint-biased forward Monte Carlo (ABFMC), is used to compute cone-beam computed tomography (CBCT) scatter projections. CMC was applied for two PSB cases: a clinical post-implant prostate, and a breast with a simulated lumpectomy cavity. CMC computes the dose difference between the highly correlated dose computing homogeneous and heterogeneous geometries. The particle transport in the heterogeneous geometry assumed a purely homogeneous environment, and altered particle weights accounted for bias. Average gains of 37 to 60 are reported from using CMC, relative to un-correlated Monte Carlo (UMC) calculations, for the prostate and breast CTV’s, respectively. To further increase the efficiency up to 1500 fold above UMC, an approximation called interpolated correlated Monte Carlo (ICMC) was applied. ICMC computes using CMC on a low-resolution (LR) spatial grid followed by interpolation to a high-resolution (HR) voxel grid followed. The interpolated, HR is then summed with a HR, pre-computed, homogeneous dose map. ICMC computes an approximate, but accurate, HR heterogeneous dose distribution from LR MC calculations achieving an average 2% standard deviation within the prostate and breast CTV’s in 1.1 sec and 0.39 sec, respectively. Accuracy for 80% of the voxels using ICMC is within 3% for anatomically realistic geometries. Second, for CBCT scatter projections, ABFMC was implemented via weight windowing using a solution to the adjoint Boltzmann transport equation computed either via the discrete ordinates method (DOM), or a MC implemented forward-adjoint importance generator (FAIG). ABFMC, implemented via DOM or FAIG, was tested for a single elliptical water cylinder using a primary point source (PPS) and a phase-space source (PSS). The best gains were found by using the PSS yielding average efficiency gains of 250 relative to non-weight windowed MC utilizing the PPS. Furthermore, computing 360 projections on a 40 by 30 pixel grid requires only 48 min on a single CPU core allowing clinical use via parallel processing techniques.

Ao longo da última década o método adjunto tem sido consolidado como uma das mais versáteis e bem sucedidas ferramentas de otimização aerodinâmica e projeto inverso na Dinâmica dos Fluidos Computacional. Ele se tornou uma área de pesquisa por si só, criando uma grande variedade de aplicações e uma literatura prolífica. Entretanto, alguns aspectos relevantes do método permanecem ainda relativamente pouco explorados na literatura. Como é o caso das condições de contorno adjuntas e, mais especificamente, com respeito a fronteiras permeáveis. Esta dissertação discute detalhadamente uma nova forma de tratar o problema de contorno, que tem como objetivo assegurar que as equações adjuntas sejam bem-postas. O principal objetivo da otimização aerodinâmica consiste na tentativa de minimizar (ou maximizar) uma determinada medida de mérito. As aplicações de projeto inverso são desenvolvidas para escoamentos Euler 2-D ao redor de aerofólios, representados com a parametrização CST (Class-Shape function Transformation) proposta por Kulfan e Bussoletti (2006), em regime de vôo transônico e com domínio discretizado por malhas não-estruturadas de triângulos através de um ciclo de projeto, que utiliza o método steepest descent como algoritmo de busca da direção que minimiza (ou maximiza) a função de mérito. As equações adjuntas são derivadas na sua formulação contínua e suas condições de contorno são determinadas por equações diferenciais características adjuntas e relações de compatibilidade compatíveis com as variações realizáveis da física do escoamento. As variáveis adjuntas são, então, vistas como forças de vínculo generalizadas, que asseguram a realizabilidade de variações do escoamento. / Over the last decade the adjoint method has been consolidated as one of the most versatile and successful tools of aerodynamic optimization and inverse design in Computational Fluid Dynamics. It has become a research area of its own, spawning a large variety of applications and a prolific literature. Yet, some relevant aspects of the method remain relatively less explored in the literature. Such is the case with the adjoint boundary conditions and, more specifically, with regard to permeable boundaries. This dissertation discusses at length a novel approach to the boundary problem, which aims at ensuring the well-posedness of the adjoint equations. The main goal of aerodynamic optimization consists in attempting to minimize (or maximize) a certain mesure of merit. The inverse design applications are developed for 2-D Euler flows around airfoils, represented with the CST (Class-Shape function Transformation) parameterization proposed by Kulfan and Bussoletti (2006), in the transonic flight regime and domain discretized by triangle unstructured meshes in a design loop which makes use of the steepest descent method as search direction that minimizes (or maximizes) the mesure of merit. Adjoint equations are derived in the continuous formulation and their boundary conditions are determined by adjoint characteristic differential equations and compatibility relations. The latter are derived so as to be compatible with the realizable variations of physical quantities. The adjoint variables are seen as generalized constraint forces, which ensure the realizability of flow variations.

Um desafio constante no projeto aerodinâmico de uma superfície é obter a forma geométrica que permite, baseado em uma determinada medida de mérito, o melhor desempenho possível. No contexto de projeto de aeronaves de transporte, o desempenho ótimo em cruzeiro é a principal meta do projetista. Nesse cenário, o uso da Dinâmica do Fluidos Computacional como não só uma ferramenta de análise mas também de síntese torna-se uma forma atrativa para melhorar o projeto de aeronaves que é uma atividade dispendiosa em termos de tempo e recursos financeiros. O método adotado para projeto aerodinâmico é baseado na teoria de controle ótimo. Essa abordagem para o problema de otimização aerodinâmica foi inicialmente proposta por Jameson (1997) e é chamada de método adjunto. Esse método apresenta uma grande diminuição de custo computacional se comparado com a abordagem de diferenças finitas para a otimização baseada em gradiente. Essa dissertação apresenta o método adjunto contínuo aplicado às equações de Euler. Tal método está inserido no contexto de um ciclo de projeto inverso aerodinâmico. Nesse ciclo, tanto o código computacional de solução das equações do escoamento quanto o código de solução das equações adjuntas foram desenvolvidos ao longo desse trabalho. Além disso, foi adotada uma metodologia de redução do gradiente da função de mérito em relação às variáveis de projeto. O algorítmo utilizado para a busca do mínimo da função de mérito é o steepest descent. Os binômios de Bernstein foram escolhidos para representar a geometria do aerofólio de acordo com a parametrização proposta por Kulfan e Bussoletti (2006). Apresenta-se um estudo dessa parametrização mostrando suas características relevantes para a otimização aerodinâmica. Os resultados apresentados estão divididos em dois grupos: validação do ciclo de projeto inverso e aplicações práticas. O primeiro grupo consiste em exercícios de projeto inverso nos quais são estabelecidas distribuições de pressão desejadas obtidas a partir de geometrias conhecidas, desta forma garante-se que tais distribuições são realizáveis. No segundo grupo, porém, as distribuições desejadas são propostas pelo projetista baseado em sua experiência e, portanto, não sendo garantida a realizabilidade dessas distribuições. Em ambos os grupos, incluem-se resultados nos regimes de escoamento transônico e subsônico incompressível. / A constant endeavor in aerodynamic design is to find the shape that yields optimum performance, according to some context-dependent measure of merit. In particular for transport aircrafts, an optimum cruise performance is usually the designers main goal. In this scenario the use of the Computational Fluid Dynamics (CFD) technique as not only an analysis tool but as a design tool becomes an attractive aid to the time and financial resource consuming activity that is aircraft design. The method adopted for aerodynamic design is based on optimal control theory. This approach to the design problem was first proposed by Jameson (1997) and it is called adjoint method. It shows a great computational cost advantage over the finite difference approach to gradient-based optimization. This dissertation presents an Euler adjoint method implemented in context of an inverse aerodynamic design loop. In this loop both the flow solver and the adjoint solver were developed during the course of this work and their formulation are presented. Further on, a gradient reduction methodology is used to obtain the gradient of the cost function with respect to the design variables. The method chosen to drive the cost function to its minimum is the steepest descent. Bernstein binomials were chosen to represent the airfoil geometry as proposed by Kulfan and Bussoletti (2006). A study of such geometric representation method is carried on showing its relevant properties for aerodynamic optimization. Results are presented in two groups: inverse design loop validation and practical application. The first group consists of inverse design exercises in which the target pressure distribution is from a known geometry, this way such distribution is guaranteed to be realizable. On the second group however, the target distribution is proposed based on the designers knowledge and its not necessarily realizable. In both groups the results include transonic and subsonic incompressible conditions.