Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

Sampling theory is an active field of research that spans a variety of disciplines from communication engineering to pure mathematics. Sampling theory provides the crucial connection between continuous and discrete representations of information that enables one store continuous signals as discrete, digital data with minimal error. It is this connection that allows communication engineers to realize many of our modern digital technologies including cell phones and compact disc players. This thesis focuses on certain non-Fourier generalizations of sampling theory and their applications. In particular, non-Fourier analogues of bandlimited functions and extensions of sampling theory to functions on curved manifolds are studied. New results in bandlimited function theory, sampling theory on curved manifolds, and the theory of self-adjoint extensions of symmetric operators are presented. Besides being of mathematical interest in itself, the research contained in this thesis has applications to quantum physics on curved space and could potentially lead to more efficient information storage methods in communication engineering.

In this thesis, I apply the adjoint for the Community Multiscale Air Quality model (hereafter CMAQ-ADJ) in a high spatial resolution study of the sensitivity of ozone to several of its precursors in the regions surrounding the Great Lakes. CMAQ-ADJ was originally developed for low spatial resolution applications. In order to use it in high spatial resolution (12 km) studies, it was necessary to resolve a conflict between the pre-set fixed output time step interval in CMAQ-ADJ and the CMAQ-calculated irregular synchronization time-step and also to modify the meteorological interface for the backward model integrations. To increase computation efficiency, the chemistry time-step in the modified CMAQ-ADJ is checkpointed instead of being re-calculated in the backward part of the model as before. I used the modified model to analyze the sensitivity of ozone to precursor species for cases of assumed high ozone episode in two target locations in southwestern and east-central Ontario. The studies examined the influence of pre-existing ozone, NO, CO, anthropogenic volatile organic compounds (VOCs) and isoprene on ozone level changes for the 69 hours immediately preceding the assumed high ozone event. The results are dominated by the long-distance advection, local meteorology (lake breezes), air temperature, the underlying surface features, and emissions in the pollutant pathway. Both production and titration of ozone by NOx is evident at different times and locations in the simulations. The industrial Midwest U.S. and Ohio Valley have been shown to be an important source of anthropogenic emission of NO and most VOCs that contribute to high ozone events in southwestern and east-central Ontario. Isoprene from the northern forest suppresses ozone in both target regions, with a greater magnitude in east-central Ontario. The response of ozone level in the two selected receptor regions in Ontario to different VOCs depends on the type of VOC, the time and location they are emitted, and the air temperature. Increasing VOC emissions in urban areas such as Toronto and Ottawa in the morning can enhance the ozone level by late afternoon. Increasing VOCs except ethylene and formaldehyde in regions with large VOC/NOx ratio in the morning tends to suppress the ozone level by late afternoon. Among all the species examined, NO has the largest impact on the target ozone level changes. CO is very unlikely to significantly influence the ozone level changes in southwestern or east-central Ontario.

Bei der Berechnung von Adjungierten, zum Debuggen und für ähnliche Anwendungen kann man die Umkehr der entsprechenden Programmauswertung verwenden. Der einfachste Ansatz, nämlich das Mitschreiben einer kompletten Mitschrift der Vorwärtsrechnung, welche anschließend rückwärts gelesen wird, verursacht einen enormen Speicherplatzbedarf. Als Alternative dazu kann man die Mitschrift auch stückweise erzeugen, indem die Programmauswertung von passend gewählten Checkpoints wiederholt gestartet wird. Das Ziel der Arbeit ist die Minimierung des von der Programmumkehr verursachten Zeit- und Speicherplatzbedarfs. Dieser wird gemessen in Auswertungswiederholungen bzw. verwendeten Checkpoints. Optimale Umkehrschemata werden für Ein- und Mehr-Schritt-Verfahren entwickelt, welche zum Beispiel bei der Diskretisierung einer gewöhnlichen Differentialgleichung Verwendung finden. Desweiteren erfolgte die Entwicklung von parallelen Umkehrschemata, d. h. mehrere Prozessoren werden für die Umkehrung der Programmauswertung eingesetzt. Diese zusätzlichen Prozessoren dienen dazu, die wiederholten Berechnungen des Programms zu parallelisieren, so daß ein Prozessor die Rückwartsrechnung ohne Unterbrechung durchführen kann. Sowohl für die seriellen als auch für die parallelen Umkehrschemata wurde gezeigt, daß die Länge der umzukehrenden Programmauswertung exponentiell in Abhängigkeit von der Zahl der verwendeten Checkpoints und der Zahl der wiederholten Auswertungen bzw. verwendeten Prozessoren wächst. / For adjoint calculations, parameter estimation, and similar purposes one may need to reverse the execution of a computer program. The simplest option is to record a complete execution log and then to read it backwards. This requires massive amounts of storage. Instead one may generate the execution log piecewise by restarting the ``forward'' calculation repeatedly from suitably placed checkpoints. The basic structure of the resulting reversal schedules is illustrated. Various strategies are analysed with respect to the resulting temporal and spatial complexity on serial and parallel machines. For serial machines known optimal compromises between operations count and memory requirement are explained, and they are extended to more general situations. For program execution reversal on multi-processors the new challenges and demands on an optimal reversal schedule are described. We present parallel reversal schedules that are provably optimal with regards to the number of concurrent processes and the total amount of memory required.

This thesis deals with the design of a robust and safe control algorithm to aim at an artificial pancreas. More precisely we will be interested in controlling the stabilizing part of a classical cure. To meet this objective, the design of a robust nonlinear model predictive controller based on the solution of a saddle point optimization problem is considered. Also, to test the controller performances in a realistic case, numerical simulations on a FDA validated testing platform are envisaged.In a first part, we present an extension of the usual nonlinear model predictive controller designed to robustly control, in a sampled-data framework, systems described by nonlinear ordinary differential equations. This controller, which computes the best control input by considering the solution of a constrained saddle point optimization problem, is called saddle point model predictive controller (SPMPC). Using this controller, it is proved that the closed-loop is Ultimately Bounded and, with some assumptions on the problem structure, Input-to State practically Stable. Then, we are interested in numerically solving the corresponding control problem. To do so, we propose an algorithm inspired from the augmented Lagrangian technique and which makes use of adjoint model.In a second part, we consider the application of this controller to the problem of artificial blood glucose control. After a modeling phase, two models are retained. A simple one will be used to design the controller and a complex one will be used to simulate realistic virtual patients. This latter is needed to validate our control approach. In order to compute a good control input, the SPMPC controller needs the full state value. However, the sensors can only provide the value of blood glucose. That is why the design of an adequate observer is envisaged. Then, numerical simulations are performed. The results show the interest of the approach. For all virtual patients, no hypoglycemia event occurs and the time spent in hyperglycemia is too short to induce damageable consequences. Finally, the interest of extending the SPMPC approach to consider the control of time delay systems in a sampled-data framework is numerically explored.

L'assimilation variationnelle de données 4D-Var est une technique très utilisée en géophysique, notamment en météorologie et océanographie. Elle consiste à estimer des paramètres d'un modèle numérique direct, en minimisant une fonction de coût mesurant l'écart entre les sorties du modèle et les mesures observées. La minimisation, qui est basée sur une méthode de gradient, nécessite le calcul du modèle adjoint (produit de la transposée de la matrice jacobienne avec le vecteur dérivé de la fonction de coût aux points d'observation). Lors de la mise en œuvre de l'AD 4D-Var, il faut faire face à des problèmes d'implémentation informatique complexes, notamment concernant le modèle adjoint, la parallélisation du code et la gestion efficace de la mémoire. Afin d'aider au développement d'applications d'AD 4D-Var, le logiciel YAO qui a été développé au LOCEAN, propose de modéliser le modèle direct sous la forme d'un graphe de flot de calcul appelé graphe modulaire. Les modules représentent des unités de calcul et les arcs décrivent les transferts des données entre ces modules. YAO est doté de directives de description qui permettent à un utilisateur de décrire son modèle direct, ce qui lui permet de générer ensuite le graphe modulaire associé à ce modèle. Deux algorithmes, le premier de type propagation sur le graphe et le second de type rétropropagation sur le graphe permettent, respectivement, de calculer les sorties du modèle direct ainsi que celles de son modèle adjoint. YAO génère alors le code du modèle direct et de son adjoint. En plus, il permet d'implémenter divers scénarios pour la mise en œuvre de sessions d'assimilation.Au cours de cette thèse, un travail de recherche en informatique a été entrepris dans le cadre du logiciel YAO. Nous avons d'abord formalisé d'une manière plus générale les spécifications deYAO. Par la suite, des algorithmes permettant l'automatisation de certaines tâches importantes ont été proposés tels que la génération automatique d'un parcours "optimal" de l'ordre des calculs et la parallélisation automatique en mémoire partagée du code généré en utilisant des directives OpenMP. L'objectif à moyen terme, des résultats de cette thèse, est d'établir les bases permettant de faire évoluer YAO vers une plateforme générale et opérationnelle pour l'assimilation de données 4D-Var, capable de traiter des applications réelles et de grandes tailles.

In this thesis, I apply the adjoint for the Community Multiscale Air Quality model (hereafter CMAQ-ADJ) in a high spatial resolution study of the sensitivity of ozone to several of its precursors in the regions surrounding the Great Lakes. CMAQ-ADJ was originally developed for low spatial resolution applications. In order to use it in high spatial resolution (12 km) studies, it was necessary to resolve a conflict between the pre-set fixed output time step interval in CMAQ-ADJ and the CMAQ-calculated irregular synchronization time-step and also to modify the meteorological interface for the backward model integrations. To increase computation efficiency, the chemistry time-step in the modified CMAQ-ADJ is checkpointed instead of being re-calculated in the backward part of the model as before. I used the modified model to analyze the sensitivity of ozone to precursor species for cases of assumed high ozone episode in two target locations in southwestern and east-central Ontario. The studies examined the influence of pre-existing ozone, NO, CO, anthropogenic volatile organic compounds (VOCs) and isoprene on ozone level changes for the 69 hours immediately preceding the assumed high ozone event. The results are dominated by the long-distance advection, local meteorology (lake breezes), air temperature, the underlying surface features, and emissions in the pollutant pathway. Both production and titration of ozone by NOx is evident at different times and locations in the simulations. The industrial Midwest U.S. and Ohio Valley have been shown to be an important source of anthropogenic emission of NO and most VOCs that contribute to high ozone events in southwestern and east-central Ontario. Isoprene from the northern forest suppresses ozone in both target regions, with a greater magnitude in east-central Ontario. The response of ozone level in the two selected receptor regions in Ontario to different VOCs depends on the type of VOC, the time and location they are emitted, and the air temperature. Increasing VOC emissions in urban areas such as Toronto and Ottawa in the morning can enhance the ozone level by late afternoon. Increasing VOCs except ethylene and formaldehyde in regions with large VOC/NOx ratio in the morning tends to suppress the ozone level by late afternoon. Among all the species examined, NO has the largest impact on the target ozone level changes. CO is very unlikely to significantly influence the ozone level changes in southwestern or east-central Ontario.

In this thesis, we investigate impulsive differential systems with delays of the form And more generally of the form The dissertation consists of five chapters. The first chapter serves as introduction, contains preliminary considerations and assertions that will be encountered in the sequel. In chapter 2, we construct the adjoint systems and obtain the variation of parameters formulas of the solutions in terms of fundamental matrices. The asymptotic behavior of solutions of systems satisfying the Perron condition is investigated in chapter 3. In chapter4, we give a result that characterizes the behavior of solutions in the case there is a bounded solution. Moreover, a necessary and sufficient condition for the existence of periodic solutions is obtained. In the last chapter, a series of consequences on the existence of periodic solutions of functionally equivlent impulsive systems with delays is established.

Numerical simulation is an important tool in understanding the electromagnetic field and how it interacts with the environment. Different topics for time-domain finite-difference (FDTD) and finite-element (FETD) methods for Maxwell's equations are treated in this thesis. Subcell models are of vital importance for the efficient modeling of small objects that are not resolved by the grid. A novel model for thin sheets using shell elements is proposed. This approach has the advantage of taking into account discontinuities in the normal component of the electric field, unlike previous models based on impedance boundary conditions (IBCs). Several results are presented to illustrate the capabilities of the shell element approach. Waveguides are of fundamental importance in many microwave applications, for example in antenna feeds. The key issues of excitation and truncation of waveguides are addressed. A complex frequency shifted form of the uniaxial perfectly matched layer (UPML) absorbing boundary condition (ABC) in FETD is developed. Prism elements are used to promote automatic grid generation and enhance the performance. Results are presented where reflection errors below -70dB are obtained for different types of waveguides, including inhomogeneous cases. Excitation and analysis via the scattering parameters are achieved using waveguide modes computed by a general frequency-domain mode solver for the vector Helmholtz equation. Huygens surfaces are used in both FDTD and FETD for excitation in waveguide ports. Inverse problems have received an increased interest due to the availability of powerful computers. An important application is non-destructive evaluation of material. A time-domain, minimization approach is presented where exact gradients are computed using the adjoint problem. The approach is applied to a general form of Maxwell's equations including dispersive media and UPML. Successful reconstruction examples are presented both using synthetic and experimental measurement data. Parameter reduction of complex geometries using simplified models is an interesting topic that leads to an inverse problem. Gradients for subcell parameters are derived and a successful reconstruction example is presented for a combined dielectric sheet and slot geometry.

Cette étude s’inscrit dans le cadre de l’inspection en service des soudures des réacteurs nucléaires de génération IV, en vue de contribuer à la démonstration de sûreté. La structure anisotrope et hétérogène des soudures multipasses en acier inoxydable austénitique rend leur contrôle ultrasonore difficile. Ainsi, afin d'interpréter correctement les signaux mesurés et de caractériser les défauts potentiels, une description de la soudure est utilisée. Elle constitue la connaissance a priori introduite dans la méthode de l'Energie Topologique. L’étude réalisée se décline en deux temps : le développement de la méthode en milieu borné et sa comparaison avec le Matched Field Processing, puis son application au cas de soudures réelles. L'extension de la méthode de l'Energie Topologique aux milieux bornés isotropes et homogènes vise à tirer parti des réflexions multiples. Plusieurs solutions du problème numérique de propagation, obtenues pour différentes conditions aux frontières, sont judicieusement associées afin de sélectionner les échos de diffraction porteurs d'information. Selon le type de défaut à imager des énergies topologiques spécifiques sont définies. La technique est introduite analytiquement avant d'être validée numériquement puis expérimentalement.Dans un second temps, la méthode est appliquée au milieu complexe de la soudure. La procédure est testée expérimentalement sur des soudures réelles afin d'évaluer les performances en localisation. Cependant, en raison de la variabilité de la structure, la qualité de l'image peut se dégrader selon les cas d'étude. La possibilité de générer des sources arbitraires permet de pallier en grande partie cette difficulté. / The present study has been done as part of the in-service inspection of weld structure belonging to generation IV nuclear reactors. It aims at checking both the safety and integrity of these components. The anisotropic and heterogeneous structure of austenitic stainless steel welds disturbs the ultrasonic non destructive testing. Thus, a weld description model is necessary to properly analyze the ultrasonic measured signals and to characterize potential flaws. The weld model makes a priori knowledge up in the Topological Energy method. The study is divided into two parts: development of the method in a bounded medium and comparison with the Matched Field Processing method, and then its application to real weld structures.The work firstly focuses on expanding the Topological Energy method to isotropic and homogeneous bounded medium to take advantage of multiple reflections between the flaw and edges. For that, different conditions are numerically applied to boundaries. By adding up these conditions it becomes possible to select the appropriate scattering signal. Modified topological energies are defined according to the type of analyzed flaws. The approach is analytically demonstrated before being validated firstly from synthetical data and then from experimental data.The second part deals with the application of the method to the complex weld structure. The process is experimentally tested on welds in order to evaluate efficiency of flaws localization. However, the image's quality can be deteriorated because of variability of the structure. By generating arbitrary ultrasonic source this difficulty is mostly overcame.