La dualité et la robustesse sont des outils essentiels dans les processus d'aide à la décision. Cette thèse concerne trois thèmes: dualité en optimisation convexe conique à données incertaines, dualité et régularisation en convexité généralisée, et la maximisation du rayon de robustesse en optimisation continue. Dans la première partie de ce travail, on considère les notions de pire valeur et de valeur robuste d'un problème d'optimisation conique à données incertaines. On donne une condition nécessaire et suffisante pour l'égalité entre la pire valeur et la valeur robuste de ce problème avec exactitude de la pire valeur. On déduit une condition suffisante permettant d'obtenir la propriété de dualité robuste forte pour ce problème. La deuxième partie est consacrée à la dualité et à la régularisation de fonctions numériques à valeurs réelles étendues. Deux points de vue sont considérés: l'approche par les niveaux et l'approche par les épigraphes. On étend ainsi à la convexité généralisée des résultats récents concernant le passage de la dualité quasi-convexe à la dualité convexe. On applique cette théorie à un problème d'optimisation pour déduire un résultat de dualité forte. La troisième partie de ce travail porte une étude du problème de maximisation du rayon de stabilité. On définit le rayon de robustesse pour un problème de décision en milieu incertain, et on étudie certaines de ces propriétés analytiques (concavité et semi-continuité). La contrepartie robuste d'un problème d'optimisation à données incertaines au sens du rayon de robustesse est introduite. On étudie le lien en termes d'ensemble de solutions optimales entre la contrepartie robuste au sens du rayon de robustesse et celle au sens de l'optimisation robuste d'un problème incertain d'optimisation continue. Un modèle générique du problème de maximisation du rayon de robustesse regroupant une large classe de cas pratique est proposé. On examine ce modèle dans un cas polyédral, dans le cas de la régression linéaire puis dans un cas quadratique. Notre stratégie dans ces différents cas, consiste à expliciter le rayon de robustesse et/ou à transformer le problème de maximisation du rayon de stabilité en un programme tractable. Une application à un problème de conception d'antenne circulaire est proposée dans le cas de la régression et une application au calcul d'un estimateur robuste est proposée dans le cas quadratique. / Duality and robustness are two important tools in decision making process. This thesis deals with tree topics : duality for an uncertain convex conical optimization problem, duality and regularity in generalized convexity, and the maximization of the stability radius. In the first part of this work, we consider the notions of worst value and robust value of an uncertain convex conical optimization problem. We give a necessary and sufficient condition to obtain the equality between the robust value and the worst value with exactness for the worst value. We derive a sufficient condition to obtain a robust strong duality property for this problem. The second part of this work is devoted to duality and regularity of the extended real-valued functions. Two points of view are considered: the sub-level set approach and the epigraphical approach. We then extend some recent results concerning the passage from the quasi-convex duality to convex duality to the generalized convexity. We apply this theory to an optimization problem to derive a strong duality property for this problem. The third part of this work is devoted to the study of the problem of maximization of the stability radius. We define the stability radius for a decision problem under data uncertainty, and study some of its analytical properties (e.g concavity and upper semi-continuity). The robust counterpart of an uncertain optimization problem according to the stability radius is introduced. We study the relation between the solution set of this counterpart and the solution set of the robust counterpart according to the robust optimization approach. We propose a generic model of the maximization of stability radius which covers a large class of applications. We study this problem in a polyhedral case, in the case of regression and in quadratic case. In each case, we compute the stability radius and/ or transform the problem of maximization of the stability radius to a tractable problem. An application to a circular antenna design problem is given in the regression case, and an application to compute a robust estimator is provided in the quadratic case.

Since the inverse problem in Diffusive Optical Tomography (DOT) is nonlinear and severely ill-posed, only low resolution reconstructions are feasible when noise is added to the data nowadays. The purpose of this thesis is to improve image reconstruction in DOT of the main optical properties of tissues with some novel mathematical methods. We have used the Landweber (L) method, the Landweber-Kaczmarz (LK) method and its improved Loping-Landweber-Kaczmarz (L-LK) method combined with sparsity or with total variation regularizations for single and simultaneous image reconstructions of the absorption and scattering coefficients. The sparsity method assumes the existence of a sparse solution which has a simple description and is superposed onto a known background. The sparsity method is solved using a smooth gradient and a soft thresholding operator. Moreover, we have proposed an improved sparsity method. For the total variation reconstruction imaging, we have used the split Bregman method and the lagged diffusivity method. For the total variation method, we also have implemented a memory-efficient method to minimise the storage of large Hessian matrices. In addition, an individual and simultaneous contrast value reconstructions are presented using the level set (LS) method. Besides, the shape derivative of DOT based on the RTE is derived using shape sensitivity analysis, and some reconstructions for the absorption coefficient are presented using this shape derivative via the LS method.\\Whereas most of the approaches for solving the nonlinear problem of DOT make use of the diffusion approximation (DA) to the radiative transfer equation (RTE) to model the propagation of the light in tissue, the accuracy of the DA is not satisfactory in situations where the medium is not scattering dominant, in particular close to the light sources and to the boundary, as well as inside low-scattering or non-scattering regions. Therefore, we have solved the inverse problem in DOT by the more accurate time-dependant RTE in two dimensions.

A ocupação irregular em áreas ambientalmente protegidas é um problema inerente à urbanização e está cada vez mais presente no espaço urbano brasileiro. Esse tipo de cenário se forma, de maneira geral, porque a população de baixa renda, sem condições financeiras de se inserir formalmente no mercado imobiliário, acaba ocupando áreas impróprias e que apresentam restrições de uso, seja por definições legais ou então por possuírem algum grau de risco à saúde ou à segurança da população. Tal situação gera uma série de impactos negativos ao meio ambiente e à população residente nesses locais. Com o objetivo de minimizar esses impactos o poder público desenvolve e executa processos de intervenção, como a regularização fundiária, por meio de ações de implantação de melhorias urbanísticas e habitacionais que acabam por gerar, também, melhorias ambientais. Esse processo, porém, é bastante complexo, pois envolve uma série de atores com interesses sociais, econômicos e políticos bastante diferentes. Diante desse contexto, o presente trabalho tem por objetivo problematizar a possibilidade de efetivação de um processo de regularização fundiária em uma área ambientalmente protegida, conciliando a inserção da população na cidade formal com a proteção ambiental. O objeto empírico é a Ilha Grande dos Marinheiros, Porto Alegre/RS, que foi escolhida por se configurar como um mosaico de ocupações de alta e baixa renda em áreas de proteção ambiental, com distintos graus de restrição de uso e ocupação. Os resultados apontam que a população de baixa renda residente na ilha é consciente e favorável a um processo de regularização fundiária, desde que possam permanecer em seus locais de moradia. A partir das análises e dos resultados obtidos nessa pesquisa espera-se auxiliar na elaboração de ações de regularização fundiária que garantam a manutenção dos moradores de baixa renda nas áreas ambientalmente protegidas, buscando melhorias nas condições de vida e de bem-estar da população e a proteção do ambiente natural. / The irregular settlements in environmentally protected areas are an inherent problem to the urbanization and it is increasingly present in the Brazilian urban space. This type of scenario is formed, generally, due to low-income population that have no financial position to formally achieve the property market terminate to occupy unsuitable areas which presents restrictions on their use, either for legally definitions or else by having some degree of risk to the health or safety of the population. Such situation generates a lot of negative impacts to the environment and the local population residing in these spots. Aiming to minimize these impacts the government develops and implements intervention processes such as land tenure regularization, through actions of deployment of urban and housing improvements that eventually also generate environmental enhancements. This process, however, is quite complex because it involves a number of actors with social, economic and political rather different interests. Given this context, this paper aims to discuss the possibility of effecting a process of regularization in an environmentally protected area, accommodating the insertion of the population in the formal city with environmental protection. The empirical object is the Ilha Grande dos Marinheiros (Big Island of Sailors), Porto Alegre / RS, which was chosen for configuring a mosaic of occupations of high and low income population in areas of environmental protection, with varying degrees of restriction on use and occupancy. The results show that the low-income residents on the island is conscious and favor a process of regularization, provided that they can remain in their places of residence. From the analysis and the results obtained in this research are expected it to assists in developing action for land tenure regularization, which ensure the maintenance of low-income residents in environmentally protected areas and seeking improvements in living conditions, well-being and protection of the natural environment.

This research investigates the use of an alternate means of modelling the performance of a train of feed water heaters in a steam cycle power plant, using machine learning. The goal of this study was to use a simple artificial neural network (ANN) to predict the behaviour of the plant system, specifically the inlet bled steam (BS) mass flow rate and the outlet water temperature of each feedwater heater. The output of the model was validated through the use of a thermofluid engineering model built for the same plant. Another goal was to assess the ability of both the thermofluid model and ANN model to predict plant behaviour under out of normal operating circumstances. The thermofluid engineering model was built on FLOWNEX® SE using existing custom components for the various heat exchangers. The model was then tuned to current plant conditions by catering for plant degradation and maintenance effects. The artificial neural network was of a multi-layer perceptron (MLP) type, using the rectified linear unit (ReLU) activation function, mean squared error (MSE) loss function and adaptive moments (Adam) optimiser. It was constructed using Python programming language. The ANN model was trained using the same data as the FLOWNEX® SE model. Multiple architectures were tested resulting in the optimum model having two layers, 200 nodes or neurons in each layer with a batch size of 500, running over 100 epochs. This configuration attained a training accuracy of 0.9975 and validation accuracy of 0.9975. When used on a test set and to predict plant performance, it achieved a MSE of 0.23 and 0.45 respectively. Under normal operating conditions (six cases tested) the ANN model performed better than the FLOWNEX® SE model when compared to actual plant behaviour. Under out of normal conditions (four cases tested), the FLOWNEX SE® model performed better than the ANN. It is evident that the ANN model was unable to capture the “physics” of a heat exchanger or the feed heating process as a result of its poor performance in the out of normal scenarios. Further tuning by way of alternate activation functions and regularisation techniques had little effect on the ANN model performance. The ANN model was able to accurately predict an out of normal case only when it was trained to do so. This was achieved by augmenting the original training data with the inputs and results from the FLOWNEX SE® model for the same case. The conclusion drawn from this study is that this type of simple ANN model is able to predict plant performance so long as it is trained for it. The validity of the prediction is highly dependent on the integrity of the training data. Operating outside the range which the model was trained for will result in inaccurate predictions. It is recommended that out of normal scenarios commonly experienced by the plant be synthesised by engineering modelling tools like FLOWNEX® SE to augment the historic plant data. This provides a wider spectrum of training data enabling more generalised and accurate predictions from the ANN model.

V posledních letech se metody slepé dekonvoluce rozšířily do celé řady technických a vědních oborů zejména, když nejsou již limitovány výpočetně. Techniky zpracování signálu založené na slepé dekonvoluci slibují možnosti zlepšení kvality výsledků dosažených zobrazením pomocí elektronového mikroskopu. Hlavním úkolem této práce je formulování problému slepé dekonvoluce obrazů z elektronového mikroskopu a hledání vhodného řešení s jeho následnou implementací a porovnáním s dostupnou funkcí Matlab Image Processing Toolboxu. Úplným cílem je tedy vytvoření algoritmu korigujícícho vady vzniklé v procesu zobrazení v programovém prostředí Matlabu. Navržený přístup je založen na regularizačních technikách slepé dekonvoluce.

La sélection d’un modèle approprié est l’une des tâches essentielles de l’apprentissage statistique. En général, pour une tâche d’apprentissage donnée, on considère plusieurs classes de modèles ordonnées selon un certain ordre de « complexité». Dans ce cadre, le processus de sélection de modèle revient `a trouver la « complexité » optimale, permettant d’estimer un modèle assurant une bonne généralisation. Ce problème de sélection de modèle se résume à l’estimation d’un ou plusieurs hyper-paramètres définissant la complexité du modèle, par opposition aux paramètres qui permettent de spécifier le modèle dans la classe de complexité choisie. L’approche habituelle pour déterminer ces hyper-paramètres consiste à utiliser une « grille ». On se donne un ensemble de valeurs possibles et on estime, pour chacune de ces valeurs, l’erreur de généralisation du meilleur modèle. On s’intéresse, dans cette thèse, à une approche alternative consistant à calculer l’ensemble des solutions possibles pour toutes les valeurs des hyper-paramètres. C’est ce qu’on appelle le chemin de régularisation. Il se trouve que pour les problèmes d’apprentissage qui nous intéressent, des programmes quadratiques paramétriques, on montre que le chemin de régularisation associé à certains hyper-paramètres est linéaire par morceaux et que son calcul a une complexité numérique de l’ordre d’un multiple entier de la complexité de calcul d’un modèle avec un seul jeu hyper-paramètres. La thèse est organisée en trois parties. La première donne le cadre général des problèmes d’apprentissage de type SVM (Séparateurs à Vaste Marge ou Support Vector Machines) ainsi que les outils théoriques et algorithmiques permettant d’appréhender ce problème. La deuxième partie traite du problème d’apprentissage supervisé pour la classification et l’ordonnancement dans le cadre des SVM. On montre que le chemin de régularisation de ces problèmes est linéaire par morceaux. Ce résultat nous permet de développer des algorithmes originaux de discrimination et d’ordonnancement. La troisième partie aborde successivement les problèmes d’apprentissage semi supervisé et non supervisé. Pour l’apprentissage semi supervisé, nous introduisons un critère de parcimonie et proposons l’algorithme de chemin de régularisation associé. En ce qui concerne l’apprentissage non supervisé nous utilisons une approche de type « réduction de dimension ». Contrairement aux méthodes à base de graphes de similarité qui utilisent un nombre fixe de voisins, nous introduisons une nouvelle méthode permettant un choix adaptatif et approprié du nombre de voisins. / The selection of a proper model is an essential task in statistical learning. In general, for a given learning task, a set of parameters has to be chosen, each parameter corresponds to a different degree of “complexity”. In this situation, the model selection procedure becomes a search for the optimal “complexity”, allowing us to estimate a model that assures a good generalization. This model selection problem can be summarized as the calculation of one or more hyperparameters defining the model complexity in contrast to the parameters that allow to specify a model in the chosen complexity class. The usual approach to determine these parameters is to use a “grid search”. Given a set of possible values, the generalization error for the best model is estimated for each of these values. This thesis is focused in an alternative approach consisting in calculating the complete set of possible solution for all hyperparameter values. This is what is called the regularization path. It can be shown that for the problems we are interested in, parametric quadratic programming (PQP), the corresponding regularization path is piece wise linear. Moreover, its calculation is no more complex than calculating a single PQP solution. This thesis is organized in three chapters, the first one introduces the general setting of a learning problem under the Support Vector Machines’ (SVM) framework together with the theory and algorithms that allow us to find a solution. The second part deals with supervised learning problems for classification and ranking using the SVM framework. It is shown that the regularization path of these problems is piecewise linear and alternative proofs to the one of Rosset [Ross 07b] are given via the subdifferential. These results lead to the corresponding algorithms to solve the mentioned supervised problems. The third part deals with semi-supervised learning problems followed by unsupervised learning problems. For the semi-supervised learning a sparsity constraint is introduced along with the corresponding regularization path algorithm. Graph-based dimensionality reduction methods are used for unsupervised learning problems. Our main contribution is a novel algorithm that allows to choose the number of nearest neighbors in an adaptive and appropriate way contrary to classical approaches based on a fix number of neighbors.

The Convolutional Neural Network (CNN) is a mathematical model designed to distill input information into a more useful representation. This distillation process removes information over time through a series of dimensionality reductions, which ultimately, grant the model the ability to resist noise, and generalize effectively. However, CNNs often contain elements that are ineffective at contributing towards useful representations. This Thesis aims at providing a remedy for this problem by introducing Absolute Cosine Value Regularization (ACVR). This is a regularization technique hypothesized to increase the representational power of CNNs by using a Gradient Descent Orthogonalization algorithm to force the vectors that constitute their filters at any given convolutional layer to occupy unique positions in Rⁿ. This method should in theory, lead to a more effective balance between information loss and representational power, ultimately, increasing network performance. The following Thesis proposes and examines the mathematics and intuition behind ACVR, and goes on to propose Dynamic-ACVR (D-ACVR). This Thesis also proposes and examines the effects of ACVR on the filters of a low-dimensional CNN, as well as the effects of ACVR and D-ACVR on traditional Convolutional filters in VGG-19. Finally, this Thesis proposes and examines regularization of the Pointwise filters in MobileNetv1.

Ces travaux s’attachent à décrire et quantifier les mécanismes de ruptures des compositestissés 3D sous chargement de traction quasi-statique et à mettre en place une méthode de simulationnumérique adaptée et robuste, pouvant à terme être appliquée en bureau d’études.Dans cette optique, une étude expérimentale a été menée afin de quantifier la propagation defissures dans ces matériaux. Celle-ci a permis de mettre en place un scenario de rupture, entirant parti de la multi-instrumentation des essais. L’étude a également été effectuée sur deséprouvettes de géométries et de tailles variées et a mis en évidence d’importantes variations dutaux de restitution d’énergie avec les conditions d’essai. Un formalisme d’analyse et de modélisationintroduisant des longueurs internes a ensuite été présenté et adapté aux mécanismes derupture des composites tissés 3D. Ce formalisme est étayé par la recherche des mécanismes àl’aide de l’analyse des faciès de rupture. Les longueurs introduites ont ainsi été mises en relationavec les paramètres du tissage. Une méthode d’identification des paramètres a été proposée etles conséquences de ce comportement sur le dimensionnement de pièces composites discutées.Enfin, le transfert de ces résultats a été effectué vers des simulations numériques robustes. Desméthodes de régularisation des modèles d’endommagement continu ont été présentées et évaluéesà l’aune de leur capacité à assurer, d’une part, la robustesse des résultats et, d’autre part,la bonne retranscription des effets d’échelle expérimentaux. La prise en compte de ces considérationsnumériques et physiques nous a amené à proposé un modèle d’endommagement Non-Local.Une méthode d’identification des paramètres et de la longueur interne à partir des données expérimentalesa été proposée. / This work aims to describe and quantify the failure mechanisms of 3D woven composites underquasi-static tensile loading and to implement an adapted and robust numerical simulationmethod, that can be applied in industry. To this end, an experimental study was carried out toquantify the propagation of cracks in these materials. Thus, a crack propagation scenario wasestablished, thanks to the multi-instrumentation used during the tests. The experimental campaignwas carried out on specimens of various geometries and sizes and highlighted significantvariations in the fracture toughness with the test conditions. A modelisation framework introducinginternal lengths was then presented and adapted to 3D woven composites. This frameworkis supported by the identification of the failure mechanisms subsequent to the analysis of thecrack profile. The introduced lengths were thus related to the weaving parameters. A method foridentifying the parameters was proposed and the consequences of this behaviour on the designof the composite parts discussed. Finally, these results were transferred to robust numerical simulations.Regularisation methods of continuous damage models were presented and evaluatedin terms of their ability to ensure, on the one hand, the robustness of the results and, on theother hand, the correct transcription of experimental size effects. Taking into account these numericaland physical considerations led us to propose a Non-Local damage model. A method foridentifying the parameters and the internal length on experimental data was proposed.

<p>&nbsp;U okviru ove doktorske disertacije posmatrani su zakoni održanja sa funkcijom fluksa koja ima prekid u x = 0, pri čemu delovi fluksa levo i desno od x = 0 imaju smo po jedan ekstrem. U prvoj glavi se može naći pregled osnovnih pojmova, definicija i teorema. U drugoj&nbsp; glavi su opisani hiperbolični sistemi zakona održanja, slaba re&scaron;enja, kao&nbsp;<br />i numerički postupci za njihovo re&scaron;avanje. U trećoj glavi su predstavljeni&nbsp; diskretni profili darnih talasa. U četvrtoj glavi su opisani zakoni održanja&nbsp; sa prekidnom funkcijom fluksa i ukratko su predstvaljeni rezultati drugih autora iz ove oblasti. U petoj glavi je najpre analizirana tzv. jednačina sa dva fluksa u slučaju kada oba dela fluksa levo i desno od&nbsp; x = 0 imaju minimum, a pri tome se seku u najvi&scaron;e jednoj tačci unutar intervala. Primenom regularizacije na intervalu [&minus;<em>&epsilon;, &epsilon;</em>], za<em> &epsilon;</em> &gt; 0 dovoljno malo, dokazano je postojanje diskretnih udarnih profila za postupak Godunova za zakone održanja sa promenljivom funkcijom fluksa. Definisan je i odgovarajući diskretan uslov entropije, a postojanje entropijskog diskretnog&nbsp; udarnog profila je postavljen kao kriterijum za dopustivost udarnih talasa. Potom je analizirana ista jednačina u slucaju kada deo fluksa levo&nbsp; od x = 0 ima maksimum, a deo fluksa desno od x = 0 minimum, dok se oba dela fluksa seku na&nbsp; krajevima posmatranog intervala. U ovom slučaju, uop&scaron;ten je uslov entropije. U okviru ove glave je prikazano nekoliko numeričkih primera za oba opisana slučaja. Numerički rezultati&nbsp; su dobijeni kori&scaron;cenjem softvera razvijenog za potrebe ove teze u pro<br />gramskom paketu <em>Mathematica</em>.</p> / <p>We consider conservation laws with a flux discontinuity at x = 0, where the flux parts from both left and right hand side of x = 0 have at most one extreme on the&nbsp; observed &nbsp;domain. The first chapter provides elementary definitions and theorems..The second chapter refers to hyperbolic systems of conservation laws, their solutions, and&nbsp; numerical procedures. The third chapter is devoted to discrete&nbsp; shock profiles. The fourth chapter describes conservation laws with discontinuous flux functions and provides basic information upon known results in this field. In the&nbsp; fifth chapter, we first &nbsp;analyse the two-flux equation when both flux parts have a minimum and cross at most&nbsp;&nbsp;&nbsp; at one point in the interior of the domain. Using a flux regularization on the interval [&minus;&epsilon;,&nbsp;&nbsp; &epsilon;], for &epsilon; &gt; 0 small enough, we show the existence of discrete shock profiles for Godunov&rsquo;s scheme for conservation laws with discontinuous flux functions. We also define a discrete entropy condition accordingly, and use the existence of an entropy discrete shock profile as an entropy criterion for shocks. Then we analyse the same problem in the case when the flux part on the left of x = 0 has a maximum and the part on the right of x = 0 has a minimum, whereas the fluxes cross at the edges of the interval. We derive a more general discrete entropy condition in this case. We provide several numerical examples in both of the above mentioned flux cases. All the&nbsp; presented numerical results are obtained using a program written in Mathematica. Finally, in chapter six, we prove the existence of&nbsp; singular shock waves in the case when the graph of one of the flux parts is above the graph of the other one on the entire domain. For that purpose, we use the shadow wave technique. At the end of this chapter, we provide a numerical verification of the obtained singular solution.</p>

We study the regularization problem for linear, constant coefficient descriptor systems $E x^. = AX + Bu, y_1 = Cx, y_2=\Gamma x^.$ by proportional and derivative mixed output feedback. Necessary and sufficient conditions are given, which guarantee that there exist output feedbacks such that the closed-loop system is regular, has index at most one and $E +BG\Gamma$ has a desired rank, i.e. there is a desired number of differential and algebraic equations. To resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of E is decreased. Numerical procedures are derived to construct the desired feedbacks gains. These numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way.