Recent research has shown that compressible signals can be recovered from a very limited number of measurements by minimizing nonconvex functions that closely resemble the L0-norm function. These functions have sparse minimizers and, therefore, are called sparsity-promoting functions (SPFs). Recovery is achieved by solving a nonconvex optimization problem when using these SPFs. Contemporary methods for the solution of such difficult problems are inefficient and not supported by robust convergence theorems. New signal-recovery methods for compressive sensing that can be used to solve nonconvex problems efficiently are proposed. Two categories of methods are considered, namely, sequential convex formulation (SCF) and proximal-point (PP) based methods. In SCF methods, quadratic or piecewise-linear approximations of the SPF are employed. Recovery is achieved by solving a sequence of convex optimization problems efficiently with state-of-the-art solvers. Convex problems are formulated as regularized least-squares, second-order cone programming, and weighted L1-norm minimization problems. In PP based methods, SPFs that entail rich optimization properties are employed. Recovery is achieved by iteratively performing two fundamental operations, namely, computation of the PP of the SPF and projection of the PP onto a convex set. The first operation is performed analytically or numerically by using a fast iterative method. The second operation is performed efficiently by computing a sequence of closed-form projectors. The proposed methods have been compared with the leading state-of-the-art signal-recovery methods, namely, the gradient-projection method of Figueiredo, Nowak, and Wright, the L1-LS method of Kim, Koh, Lustig, Boyd, and Gorinevsky, the L1-Magic method of Candes and Romberg, the spectral projected-gradient L1-norm method of Berg and Friedlander, the iteratively reweighted least squares method of Chartrand and Yin, the difference-of-two-convex-functions method of Gasso, Rakotomamonjy, and Canu, and the NESTA method of Becker, Bobin, and Candes. The comparisons concerned the capability of the proposed and competing algorithms in recovering signals in a wide range of test problems and also the computational efficiency of the various algorithms. Simulation results demonstrate that improved reconstruction performance, measurement consistency, and comparable computational cost are achieved with the proposed methods relative to the competing methods. The proposed methods are robust, are supported by known convergence theorems, and lead to fast convergence. They are, as a consequence, particularly suitable for the solution of hard recovery problems of large size that entail large dynamic range and, are, in effect, strong candidates for use in many real-world applications. / Graduate / 0544 / eng.flavio.teixeira@gmail.com

Dry powder inhalers (DPIs) are a common therapeutic modality for lung diseases such as asthma, but they are also used to treat systemic diseases such as diabetes. Advantages of DPIs include their portable design and low manufacturing costs. Another advantage of DPIs is their breath activation, which makes them popular among patients. In a passive DPI drug is only released when the patient inhales. When the patient inhales, air flows through the device. The flow of air entrains a dry powder formulation inside the device and carries it to the lung. Currently, no DPI exists which can deliver drug independent of the patient to the desired target site in the lung. This is because drug release depends on the patient’s inhalation manoeuvre. To maximize the effect of the treatment it is necessary to optimize DPIs to achieve drug delivery that (A) is independent of the inhalation manoeuvre and (B) is targeted to the correct site in the lung. Therefore, this thesis aims to apply numerical and experimental methods to optimize DPIs systematically. First, two clinically justifiable cost functions have been developed corresponding to the DPI design objectives (A) and (B). An Eulerian-Eulerian (EE) computational fluid dynamics (CFD) approach has then been used to optimize a DPI entrainment geometry. Three different optimized entrainment geometries have been found corresponding to three different therapeutic applications. Second, the CFD approach has been validated experimentally. This is the first experimental study to validate an EE CFD approach for DPI modelling. Third, a personalized medicine approach to DPI design has been proposed. The development of this approach makes it possible to achieve the design objectives for patients with highly different lung functions. Finally, an adaptive DPI with a variable bypass element has been developed. This DPI achieves design objectives (A) and (B) for patients with highly different lung functions with a single device. In contrast to the personalized medicine approach, there is no need to select the optimal amount of bypass, since the device adapts automatically.

A new, novel numerical optimization algorithm is developed, tested, and used to solve difficult numerical problems from the field of astrodynamics. First, a brief review of optimization theory is presented and common numerical optimization techniques are discussed. Then, the new method, called the Learning Approach to Sampling Optimization (LA) is presented. Simple, illustrative examples are given to further emphasize the simplicity and accuracy of the LA method. Benchmark functions in lower dimensions are studied and the LA is compared, in terms of performance, to widely used methods. Three classes of problems from astrodynamics are then solved. First, the N - impulse orbit transfer and rendezvous problems are solved by using the LA optimization technique along with derived bounds that make the problem computationally feasible. This marriage between analytical and numerical methods allows an answer to be found for an order of magnitude greater number of impulses than are currently published. Next, the N -impulse work is applied to design periodic close encounters (PCE) in space. The encounters are defined as an open rendezvous, meaning that two spacecraft must be at the same position at the same time, but their velocities are not necessarily equal. The PCE work is extended to include N -impulses and other constraints, and new examples are given. Finally, a trajectory optimization problem is solved using the LA algorithm and comparing performance with other methods based on two models-with varying complexity-of the Cassini-Huygens mission to Saturn. The results show that the LA consistently outperforms commonly used numerical optimization algorithms.

O FBST (Full Bayesian Significance Test) é um procedimento para testar hipóteses precisas, apresentado por Pereira e Stern (1999), e baseado no cálculo da probabilidade posterior do conjunto tangente ao conjunto que define a hipótese nula. Este procedimento é uma alternativa Bayesiana aos testes de significância usuais. Neste trabalho, estudamos a relação entre os resultados do FBST e de um teste freqüentista, o TRVG (Teste da Razão de Verossimilhanças Generalizado), através de alguns problemas clássicos de testes de hipóteses. Apresentamos, também, todos os procedimentos computacionais utilizados para a resolução automática dos dois testes para grandes amostras, necessária ao estudo da relação entre os testes. / FBST (Full Bayesian Significance Test) is a procedure to test precise hypotheses, presented by Pereira and Stern (1999), which is based on the calculus of the posterior probability of the set tangent to the set that defines the null hypothesis. This procedure is a Bayesian alternative to the usual significance tests. In the present work we study the relation between the FBST\'s results and those of a frequentist test, GLRT (Generalised Likelihood Ratio Test) through some classical problems in hypotesis testing. We also present all computer procedures that compose the automatic solutions for applying FBST and GLRT on big samples what was necessary for studying the relation between both tests.

O FBST (Full Bayesian Significance Test) é um procedimento para testar hipóteses precisas, apresentado por Pereira e Stern (1999), e baseado no cálculo da probabilidade posterior do conjunto tangente ao conjunto que define a hipótese nula. Este procedimento é uma alternativa Bayesiana aos testes de significância usuais. Neste trabalho, estudamos a relação entre os resultados do FBST e de um teste freqüentista, o TRVG (Teste da Razão de Verossimilhanças Generalizado), através de alguns problemas clássicos de testes de hipóteses. Apresentamos, também, todos os procedimentos computacionais utilizados para a resolução automática dos dois testes para grandes amostras, necessária ao estudo da relação entre os testes. / FBST (Full Bayesian Significance Test) is a procedure to test precise hypotheses, presented by Pereira and Stern (1999), which is based on the calculus of the posterior probability of the set tangent to the set that defines the null hypothesis. This procedure is a Bayesian alternative to the usual significance tests. In the present work we study the relation between the FBST\'s results and those of a frequentist test, GLRT (Generalised Likelihood Ratio Test) through some classical problems in hypotesis testing. We also present all computer procedures that compose the automatic solutions for applying FBST and GLRT on big samples what was necessary for studying the relation between both tests.

One of the key open problems in the field of theoretical fluid mechanics concerns the possibility of the singularity formation in solutions of the 3D Navier-Stokes system in finite time. This phenomenon is associated with the behaviour of the enstrophy, which is an L2 norm of the vorticity and must become unbounded if such a singularity occurs. Although there is no blow-up in the 2D Navier-Stokes equation, we would like to investigate how much enstrophy can a planar incompressible flow in a bounded domain produce given certain initial enstrophy. We address this issue by formulating an optimization problem in which the time derivative of the enstrophy serves as the objective functional and solve it using tools of the optimization theory and calculus of variations. We propose an efficient computational approach which is based on the iterative steepest-ascent procedure. In addition, we introduce an easy-to-implement method of computing the gradient of the objective functional. Finally, we present computational results addressing the key question of this project and provide numerical evidence that the maximum enstrophy growth exhibits the scaling dE/dt ~ C*E*E for C>0 and very small E. All computations are performed using the Chebyshev spectral method. / Thesis / Master of Science (MSc) / For many decades, scientists have been investigating fundamental aspects of the Navier-Stokes equation, a central mathematical model arising in fluid mechanics. Although the equation is widely used by engineers to describe numerous flow phenomena, it is still an open question whether the Navier-Stokes system always admits physically meaningful solutions. To address this issue, we want to explore its mathematical aspects deeper by analyzing the behaviour of the enstrophy, which is a quantity associated with the vorticity of the flow and a convenient measure of the regularity of the solution. In this study, we consider a planar and incompressible flow bounded by solid walls. Using basic tools of mathematical analysis and optimization theory, we propose a computational method enabling us to find out how much enstrophy can such a flow produce instantaneously. We present numerical evidence that this instantaneous growth of enstrophy has a well-defined asymptotic behavior, which is consistent with physical assumptions.

Le travail présenté dans cette thèse est divisé en deux parties. Dans la première partie, un modèle pour la représentation unifiée de problèmes d'optimisation numérique est proposé. Ce modèle permet de définir un problème d'optimisation indépendamment de la stratégie utilisée pour le résoudre. Cette représentation unifiée est particulièrement appréciable en robotique où une solution analytique des problèmes est rarement possible. La seconde partie traite de l'exécution de mouvements complexes asservis sur un robot humanoïde. Lors de la locomotion d'un tel système, les glissements des points de contact entraînent une dérive qu'il est nécessaire de corriger. Nous proposons ici un modèle permettant d'asservir une tâche de locomotion sur un capteur externe afin de compenser les erreurs d'exécution des mouvements. Un modèle est également proposé permettant de représenter des séquences de tâches de locomotion et de manipulation asservies. Enfin, une méthodologie pour le développement d'applications robotiques complexes est établie. Les stratégies proposées dans le cadre de cette thèse ont été validées sur la plate-forme expérimentale HRP-2. / The presented work is divided into two parts. In the first one, an unified computer representation for numerical optimization problems is proposed. This model allows to define problems independently from the algorithm used to solve it. This unified model is particularly interesting in robotics where exact solutions are difficult to find. The second part is dealing with complex trajectory execution on humanoid robots with sensor feedback. When a biped robots walks, contact points often slip producing a drift which is necessary to compensate. We propose here a closed-loop control scheme allowing the use of sensor feedback to cancel execution errors. To finish, a method for the the development of complex robotics application is detailed. This thesis contributions have been implemented on the HRP-2 humanoid robot.

The results section of my thesis includes three chapters. The first two chapters are on theoretical game theory. In both chapters, by mathematical modelling and game theoretical tools, I am predicting the behaviour of the players in some real world issues. Hoteling-Downs model plays an important role in the modern political interpretations. The first chapter of this study investigates an extension of Hoteling-Downs model to have multi-dimensional strategy space and asymmetric candidates. Chapter 3 looks into the inspection game where the inspections are not the same in the series of sequential inspections. By modelling the game as a series of recursive zero-sum games I find the optimal strategy of the players in the equilibrium. The forth chapter investigates direct optimization methods for large scale problems. Using Matlab implementations of Genetic and Nelder-Mead algorithms, I compare the efficiency and accuracy of the most famous direct optimization methods for unconstraint optimization problems based on differing number of variables.

Dans le contexte de la vision par ordinateur cette thèse étudie le problème d’appariement d’images dans le cadre de la télédétection pour la géologie. Plus précisément, nous disposons dans ce travail de deux images de la même scène géographique, mais acquises à partir de deux points de vue différents et éventuellement à un autre moment. La tâche d’appariement est d'associer à chaque pixel de la première image un pixel de la seconde image.Bien que ce problème soit relativement facile pour les êtres humains, il reste difficile à résoudre par un ordinateur. De nombreuses approches pour traiter cette tâche ont été proposées. Les techniques les plus prometteuses formulent la tâche comme un problème d'optimisation numérique. Malheureusement, le nombre d'inconnues ainsi que la nature de la fonction à optimiser rendent ce problème extrêmement difficile à résoudre. Cette thèse étudie deux approches avec un schéma multi-échelle pour résoudre le problème numérique sous-jacent / This thesis studies the computer vision problem of image registration in the context of geological remote sensing surveys. More precisely we dispose in this work of two images picturing the same geographical scene but acquired from two different view points and possibly at a different time. The task of registration is to associate to each pixel of the first image its counterpart in the second image.While this problem is relatively easy for human-beings, it remains an open problem to solve it with a computer. Numerous approaches to address this task have been proposed. The most promising techniques formulate the task as a numerical optimization problem. Unfortunately, the number of unknowns along with the nature of the objective function make the optimization problem extremely difficult to solve. This thesis investigates two approaches along with a coarsening scheme to solve the underlying numerical problem

The design of optimal strictly positive real (SPR) compensators using numerical optimization is considered. The plants to be controlled are linear and nonlinear flexible manipulators. For the design of SISO and MIMO linear SPR controllers, the optimization objective function is defined by reformulating the H2-optimal control problem subject to the constraint that the controllers must be SPR. Various controller parameterizations using transfer functions/matrices and state-space equations are considered. Depending on the controller form, constraints are enforced (i) using simple inequalities guaranteeing SPRness, (ii) in the frequency domain, or (iii) by implementing the Kalman-Yakubovich- Popov lemma. The design of a gain-scheduled SPR controller using numerical optimization is also considered. Using a family of linear SPR controllers, the time dependent scheduling signals are parameterized, and the objective function of the optimizer seeks to find the form of the scheduling signals which minimizes the manipulator tip tracking error while minimizing the control effort.