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Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.

L'optimisation multidisciplinaire (MDO) à base de gradients est efficace et très utilisée pour le dimensionnement structural d'ailes flexibles. Cependant, dans le contexte de simulations numériques haute-fidélité, le calcul efficace des gradients reste un défi majeur. L'objectif de ce travail est d'étudier les approches les mieux adaptées aux spécificités du calcul de sensibilité des efforts aéroélastiques par rapport à des paramètres structuraux.Deux techniques de calcul de gradient haute-fidélité adaptées aux systèmes aéroélastiques fortement couplés sont proposées. La technique la plus intrusive repose sur les formulations directe et adjointe qui nécessitent un effort d'implémentation logicielle substantiel. Alternativement, nous proposons une approche découplée et non-intrusive, moins lourde à implémenter et cependant capable de fournir une approximation précise des gradients. Ces deux techniques ont été intégrées dans le logiciel CFD elsA de l'Onera.La précision, l'efficience et l'applicabilité de ces méthodes sont démontrées sur le cas-test avion de transport civil Common Research Model (CRM). Nous résolvons un problème inverse dont l'objectif est de retrouver, en conditions de vol de croisière, une loi cible de vrillage voilure. Ces deux méthodes s'avèrent comparables en matière de précision et de coût. Elles offrent ainsi une souplesse supplémentaire de mise en œuvre en fonction du niveau d'intégration recherché dans le processus MDO. / To improve the structural design of flexible wings, gradient based Multidisciplinary Design Optimization (MDO) techniques are effective and widely used. However, gradients calculation is not trivial and can be costly when high-fidelity models are considered. Our objective is to study different suitable approaches to compute gradients of aeroelastic loads with respect to structural design parameters.To this end, two high-fidelity aero-structure gradient computation techniques for strongly coupled aeroelastic systems are proposed. The most intrusive technique includes the well-established direct and adjoint formulations that require substantial implementation effort. In contrast, we propose an alternative uncoupled non-intrusive approach easier to implement and yet capable of providing accurate gradients approximations. Both techniques have been implemented in the Onera elsA CFD software.Accuracy, efficiency and applicability of these methods are demonstrated on the civil transport aircraft Common Research Model (CRM) test-case. More specifically, an inverse design problem is set up with the objective of matching an in-flight target twist law distribution. These two methods prove to be comparable in terms of accuracy and cost. Thus they offer additional operational flexibility depending on the level of integration sought in the MDO process.

ix, 99 p. / We characterize the diagonals of four classes of self-adjoint operators on infinite dimensional Hilbert spaces. These results are motivated by the classical Schur-Horn theorem, which characterizes the diagonals of self-adjoint matrices on finite dimensional Hilbert spaces. In Chapters II and III we present some known results. First, we generalize the Schur-Horn theorem to finite rank operators. Next, we state Kadison's theorem, which gives a simple necessary and sufficient condition for a sequence to be the diagonal of a projection. We present a new constructive proof of the sufficiency direction of Kadison's theorem, which is referred to as the Carpenter's Theorem. Our first original Schur-Horn type theorem is presented in Chapter IV. We look at operators with three points in the spectrum and obtain a characterization of the diagonals analogous to Kadison's result. In the final two chapters we investigate a Schur-Horn type problem motivated by a problem in frame theory. In Chapter V we look at the connection between frames and diagonals of locally invertible operators. Finally, in Chapter VI we give a characterization of the diagonals of locally invertible operators, which in turn gives a characterization of the sequences which arise as the norms of frames with specified frame bounds. This dissertation includes previously published co-authored material. / Committee in charge: Marcin Bownik, Chair; N. Christopher Phillips, Member; Yuan Xu, Member; David Levin, Member; Dietrich Belitz, Outside Member

Nessa tese, mostramos que a Hamiltoniana H e o operador helicidade de uma partícula de Dirac que se movimenta em duas dimensões na presença de um tubo de fluxo magnético infinitamente fino na origem admitem, cada um, uma família de quatro parâmetros de extensões auto-adjuntas. Para cada extensão correspondem condições de contorno a serem satisfeitas pelas auto-fuções na origem. Apesar dos operadores H e formalmente comutarem antes da especificação das condições de contorno, para garantirmos a conservação da helicidade, não é suficiente obtermos as mesmas condições de contorno para ambos os operadores, ou seja, não é suficiente a determinação de um domínio comum a ambos. Mostramos que, para certas relações entre os parâmetros das extensões satisfeitas, é possível a determinação dos domínios mais gerais onde ambos os operadores H e são auto-adjuntos e onde a helicidade é conservada, simultaneamente com a preservação da simetria de Aharonov-Bohm ( + 1), onde é o fluxo magnético em unidades naturais. Nossos resultados implicam que, nem a conservação da helicidade nem a simetria de Aharonov-Bohn, resolvem o problema da escolha da condição de contorno fisicamente correta. / We show that both the Hamiltonian H and the helicity operator of a Dirac particle moving in two dimension in the presence of an infinitely thin magnetic flux tube admit each a four- parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BC\'s) to be satisfied by the eigenfunctions at the origin. Althou- gh the actions af these two operators commute before specification of boundary conditions, to ensure helicity conservation it is not sufficient to take the same BC\'s for both operators. We show that, given certain relations between the parameters of the extensions it is possible to write down the most general domain where both operators H and are self-adjoint with heli- city conservation and also Aharonov-Bohm symmetry ( + 1) preserved, where is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results im- ply that neither helicity conservation nor Aharonov-Bohm symmetry by themselves solves the problem of choosing the \"physical \"boundary conditions for this system.

O método adjunto tem sido extensivamente utilizado como ferramenta de síntese no projeto de aeronaves por permitir que se obtenham sensibilidades de distintas medidas de mérito com relação a parâmetros que controlam a geometria de superfícies aerodinâmicas. O presente trabalho visa uma ampliação das aplicações da formulação contínua do método, ao utilizar propriedades físicas do escoamento nas fronteiras permeáveis do domínio computacional como parâmetros de controle de uma particular medida de mérito. Desse modo é possível, entre muitas possibilidades, determinar a sensibilidade de integrais como sustentação ou arrasto de uma aeronave com relação às condições de cruzeiro, por exemplo. Mais do que isso, essa informação pode ser obtida com a mesma solução adjunta computada para realizar otimização de forma. Vale destacar, ainda, que para que se consiga obter essa informação a partir das equações adjuntas, é necessário que se implemente condições de contorno baseadas em equações diferenciais características, resolvendo o problema de Riemann completo nas fronteiras do domínio. A implementação das usuais condições de contorno homogêneas, vastamente difundidas na literatura, resultaria em gradientes nulos. Esta nova abordagem do método é então aplicada a escoamentos modelados pelas equações de Euler 2-D compressíveis em estado estacionário. Ambos os problemas, físico e adjunto, são resolvidos numericamente com um código computacional que utiliza o método dos volumes finitos com segunda ordem de precisão no espaço e discretização centrada com dissipação artificial. As soluções estacionárias são obtidas ao se postular um termo tempo-dependente e integra-lo com um esquema Runge-Kutta de 5 passos e 2a ordem de precisão. As simulações são realizadas em malhas não-estruturadas formadas por elementos triangulares em 4 geometrias distintas: um bocal divergente, um perfil diamante, um aerofólio simétrico (NACA 0012) e o outro assimétrico (RAE 2822). Os gradientes adjuntos são então validados por meio da comparação com os obtidos pelo método de diferenças finitas nos regimes de escoamento subsônico, supersônico e transônico. / The adjoint method has been extensively used as an aircraft design tool, since it enables one to obtain sensitivities of many different mesures of merit with respect to parameters that control the aerodynamic surface geometry. This works aims to open up the possibilities of the method\'s applications by using flow physical properties at the permeable boundaries of the computational domain as control parameters of a particular measure of merit. This way it is possible, among many possibilities, to compute lift or drag sensitivities of an aircraft with respect to cruise conditions, for instance. Moreover, this information can be obtained with the same adjoint solution used to perform shape optimization. It is also worth noting that in order to obtain this information from the adjoint equations it is necessary to implement characteristics-based boundary conditions, resolving the complete Riemann problem at the boundaries of the computational domain. The use of the traditional homogeneous boundary conditions, widely spread in the literature, would lead the gradient to vanish. This new approach of the method is, then, applied to flows modeled by the 2-D steady state compressible Euler equations. Both, physical and adjoint problems are numerically solved with a computational code that makes use of a 2nd order finite volume method and central differences with artifficial dissipation. The steady solutions are obtained by postulating a time-dependent term and integrating it with a 5-stage 2nd order Runge-Kutta scheme. The simulations are performed on unstructured triangular meshes to 4 different geometries: a divergent nozzle, a diamond profile, a symmetric airfoil (NACA 0012) and a assymmetric airfoil (RAE 2822). The adjoint gradients are then validated by comparison with those obtained by finite differences method in subsonic, supersonic and transonic flow regimes.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Nos dias atuais a Ãgua se constitui em um bem natural que limita o desenvolvimento socioeconÃmico e, atà mesmo, a subsistÃncia da populaÃÃo. Como tentativa de minimizar o problema da escassez de Ãgua tem-se utilizado a explotaÃÃo da Ãgua subterrÃnea. Entretanto, esse crescimento da utilizaÃÃo de Ãguas subterrÃneas foi feito de forma desordenada e com a construÃÃo inadequada de poÃos. Essa prÃtica acabou por colocar em risco a qualidade das Ãguas subterrÃneas. Assim, a gestÃo dos recursos hÃdricos subterrÃneos tem se tornado um grande desafio. Essa tese propÃe o desenvolvimento um modelo para a simulaÃÃo de fluxo hÃdrico e de transporte de massa para problemas transientes em aqÃÃferos costeiros sujeitos à intrusÃo salina, por meio do desenvolvimento de um modelo numÃrico. Em seguida à desenvolvida uma anÃlise de sensitividade com o objetivo de possibilitar, atravÃs do melhor conhecimento dos parÃmetros locais e suas influÃncias, uma melhor adequaÃÃo do modelo à realidade. / Today the water is a natural well which limits the socioeconomic development and even the subsistence of the population. An attempt to minimize the problem of water scarcity has used the farming of groundwater. However, this growth of the use of groundwater was done inappropriately and with inadequate wells construction. This practice was eventually put at risk the quality of groundwater. Thus, the management of groundwater resources has become a major challenge. This thesis proposes developing a model for the simulation of water flow and mass transport for transient problems in coastal aquifers subject to saline intrusion, through the development of a numerical model. Then we developed a sensitivity analysis with the goal of enabling through better knowledge of local parameters and their influences, a best fit of model to reality.

Problemas de otimização se fazem cada vez mais presentes nos mais diversos ramos da Engenharia. Encontrar configurações ótimas para um determinado problema significa, por exemplo, melhorar desempenho, reduzir custos entre outros ganhos. Existem hoje diversas maneiras de atacar um problema de otimização, cada qual com suas particularidades, vantagens e desvantagens. Dentre os métodos de otimização que utilizam gradientes de sensibilidade, o cálculo numérico dos mesmos consiste em uma importante etapa do projeto que, dependendo do problema, pode acarretar em custos computacionais muito elevados inviabilizando a abordagem escolhida. Este trabalho visa desenvolver e apresentar uma nova metodologia para o cálculo desses gradientes de sensibilidade, com base no Método Adjunto. O Método Adjunto é um método amplamente estudado e com diversas aplicações principalmente em Engenharia Aeronáutica. Nesse trabalho, todo o conhecimento prévio é utilizado para a derivação do método para aplicá-lo a escoamentos viscosos e incompressíveis. É desenvolvido também o cálculo do gradiente de sensibilidade com respeito a parâmetros geométricos e não geométricos. Para validar a metodologia proposta são feitas simulações numéricas das equações governantes do escoamento e adjuntas utilizando dois códigos computacionais distintos, SEMTEX e FreeFem++, o primeiro baseado no Método dos Elementos Espectrais e o segundo no Método dos Elementos Finitos, mostrando assim a independência do Método Adjunto na sua formulação contínua em relação a métodos computacionais. Para a validação são cujos gradientes possam ser calculados de outras formas permitindo comparações para calibrar e aperfeiçoar o cálculo do gradiente de sensibilidade. / Optimization problems are widely present in differents fields of Engineering. Finding optimal configurations in a problem means, for example, improving performance, reducing costs, among other achievements. There are several wellknown ways to tackle an optimization problem, each one has its own advantages and disadvantages. Considering the gradient-based optimization methods, the step of their numerical calculation is extremely important, as it may result in huge computational costs, thus making the chosen method impracticable. This work aims to develop and present a new methodology to compute these sensitivity gradients based on the Adjoint Method. The Adjoint Method is a widely studied method with several applications chiefly in A eronautical Engineering. In the present work, all the previous knowledge will be used to derive the equations of the method in order to apply them to viscous incompressible flows. The calculation of the sensitivity gradient, with respect to both geometric and non-geometric paramatersm will be developed as well. To validate the proposed methodology, numerical simulations of the governing and adjoint equations are carried out, using two computational codes called SEMTEX and FreeFem++, the former is based on the Spectral Element Method and the later, on the Finite Element Method, thus showing that the Adjoint Method, in its continuous formulation, is independent of the particular numerical method that is used. In order to validate the algorithm, simple problems are chosen, for which the gradients can be computed by other methods. This choice admits comparison between numerical values of gradients in order to calibrate and improve the methodology proposed.

O presente trabalho é o início de um estudo da aplicação do método de otimização conhecido como adjunto em escoamentos incompressíveis, viscosos e periódicos, envolvendo um problema de bastante interesse: a análise da aplicação de splitter plates em cilindros de seção circular. Conhecido por sua simplicidade, o splitter plate, que consiste em uma placa plana alinhada ao escoamento e colocada a jusante do corpo, é um dispositivo efetivo na mudança de comportamento da esteira de vórtices de Von Kárman. A partir da revisão bibliográfica foi possível entender a dinâmica do escoamento, proporcionando uma calibração dos modelos numéricos. Complementando esta etapa, foi efetuada uma análise da qualidade das malhas computacionais. Através de uma geração de diversas malhas computacionais, o espaço de soluções foi explorado buscando encontrar o mínimo arrasto para diversos comprimentos de splitter plate e diferentes números de Reynolds (Re). Foi observada a influência da placa na formação da esteira de vórtices, obtendo uma redução dos coeficientes de força do cilindro. Com esses dados, foi possível desenvolver o método de otimização voltado para análise do gradiente de sensibilidade conhecido como método adjunto baseado nas equações de Navier Stokes utilizando o problema descrito como base para validação dos resultados. A abordagem do método adjunto caracteriza-se pela busca dos extremos de funções conhecidas como medidas de mérito. Essas funções podem ser integrais de sustentação e arrasto por exemplo. Na literatura, o método adjunto é apresentado como possuindo duas grandes vantagens: a primeira é a imposição das equações do escoamento como restrições do problema, o que sempre confinará as variações da medida de mérito dentro do universo de soluções realizáveis; já a segunda é conseqüência da primeira, pois as restrições permitem uma simplificação no cálculo do gradiente de sensibilidade, reduzindo o custo computacional. Para o cálculo do gradiente de sensibilidade, o objetivo é otimizar o arrasto do cilindro sob efeito do splitter plate variando os parâmetros de controle (comprimento e posicionamento do splitter plate). A direção de busca e o cálculo do passo da geometria são obtidos a partir da relação entre a solução numérica do escoamento e as variáveis adjuntas calculadas. Nesta dissertação, será apresentada a pesquisa bibliográfica, os resultados do método tentativa e erro, a formulação do método adjunto baseado nas equações de Navier Stokes e um exemplo de sua solução numérica, demonstrando sua existência. / The report is the beginning of a research about the application of the so called adjoint method in the optimization of incompressive, viscous and periodic flows. The study involves a problem of great interest: an analysis of the implementation of splitter plates in the flow around cylinders with circular section. Widely recognized for its simplicity, the splitter plate consists of a flat plate, which is placed in the wake of a cylinder, in the stream wise direction, and it works by changing the way the shear layers interact with one another. Based on a literature survey, it is possible to understand the physics of this class of flows. As a better result, one learns what to expect from the numerical solutions and hence, one can calibrate its parameters. Moreover, we study the best configuration of the computational mesh, thus reducing the computational cost. After the generation of meshes, the universe of solutions was explored to find the minimum drag for various lengths of splitter plate and Reynolds number (Re). The influence of the plate in the interaction of the shear layers was observed in the reduction of drag coefficient. These results form a the basis for comparison, upon one can develop the optimization by the adjoint method. The adjoint method can be used to search the extreme of objective functionals. These functionals can be the lift and drag integrals for example. The theory presents two advantages to the method: first, the imposing the equations that govern the flow as variational constraints one limits the variations to the universe of realizable solutions; second, these constrains simplify the computation of the sensitivity gradient, by reducing its computational cost. To compute the sensitivity gradient, the objective functional can be defined as the average drag coefficient of the circular cylinder with a splitter plate. The control parameters are the length of the plate and the distance between it and the body, which known as gap. The search direction and the variation of geometry can be obtained by the relationship between the solutions to the flow and the adjoint equations. This final report shows the literature survey, the results of trial and error method and the formulation and one result of adjoint equations based on the incompressible NavierStokes equations.

In this dissertation we investigate methods of solving various optimization problems with PDE constraints, i.e. optimization problems that have a system of partial differential equations in the set of constraints, and develop frameworks for a number of practically inspired problems that were not considered in the literature before. Such problems arise in areas like fluid mechanics, chemical engineering, finance, and other areas where a physical system needs to be optimized. In most of the literature sources on PDE-constrained optimization only relatively simple systems of PDEs are considered, they are either linear, or the size of the system is small. On the contrary, in our case, we search for solution methods to problems constrained by large (8 to 10 equations) and non-linear systems of PDEs. More specifically, in the first part of the dissertation we consider a multiphysics phenomenon where electromagnetic and mechanical fields interact within an electrically conductive body, and develop the optimization framework to find an efficient way to control one field through another. We also apply the developed PDE-constrained optimization framework to a financial options portfolio optimization problem, and more specifically consider the case that to the best of our knowledge is not covered in the literature.