Error estimation must be used to find the accuracy of a designed classifier, an issue that is critical in biomarker discovery for disease diagnosis and prognosis in genomics and proteomics. This dissertation is concerned with the analytical formulation of the joint distribution of the true error of misclassification and two of its commonly used estimators, resubstitution and leave-one-out, as well as their marginal and mixed moments, in the context of the Linear Discriminant Analysis (LDA) classification rule. In the first part of this dissertation, we obtain the joint sampling distribution of the actual and estimated errors under a general parametric Gaussian assumption. Exact results are provided in the univariate case and an accurate approximation is obtained in the multivariate case. We show how these results can be applied in the computation of conditional bounds and the regression of the actual error, given the observed error estimate. In practice the unknown parameters of the Gaussian distributions, which figure in the expressions, are not known and need to be estimated. Using the usual maximum-likelihood estimates for such parameters and plugging them into the theoretical exact expressions provides a sample-based approximation to the joint distribution, and also sample-based methods to estimate upper conditional bounds. In the second part of this dissertation, exact analytical expressions for the bias, variance, and Root Mean Square (RMS) for the resubstitution and leave-one-out error estimators in the univariate Gaussian model are derived. All probabilistic characteristics of an error estimator are given by the knowledge of its joint distribution with the true error. Partial information is contained in their mixed moments, in particular, their second mixed moment. Marginal information regarding an error estimator is contained in its marginal moments, in particular, its mean and variance. Since we are interested in estimator accuracy and wish to use the RMS to measure that accuracy, we desire knowledge of the second-order moments, marginal and mixed, with the true error. In the multivariate case, using the double asymptotic approach with the assumption of knowing the common covariance matrix of the Gaussian model, analytical expressions for the first moments, second moments, and mixed moment with the actual error for the resubstitution and leave-one-out error estimators are derived. The results provide accurate small sample approximations and this is demonstrated in the present situation via numerical comparisons. Application of the results is discussed in the context of genomics.

Dans cette thèse, nous étudions les propriétés spectrales du Laplacien avec la condition de bord de Robin attractive sur des domaines du plan à coins. Notre but est de comprendre l’influence des coins convexes sur l’asymptotique des valeurs propres de cet opérateur lorsque le paramètre de Robin est grand. Nous montrons en particulier que l’asymptotique des premières valeurs propres de Robin sur des polygones curvilignes est déterminée par des opérateurs modèles : les Laplaciens agissant sur les secteurs tangents au domaine. Pour une certaine classe de polygones droits, nous montrons l’existence d’un opérateur effectif sur le bord du domaine qui détermine l’asymptotique des valeurs propres suivantes. Enfin, des asymptotiques de Weyl pour différents seuils dépendant du paramètre de Robin sont obtenues. / In this thesis, we are interested in the spectral properties of the Laplacian with the attractive Robin boundary condition on planar domains with corners. The aim is to understand the influence of the convex corners on the spectral properties of this operator when the Robin parameter is large. In particular, we show that the asymptotics of the first Robin eigenvalues on curvilinear polygons is determined by model operators: the Robin Laplacians acting on infinite sectors. For a particular class of polygons with straight edges, we prove the existence of an effective operator acting on the boundary of the domain and determining the asymptotics of the further eigenvalues. Finally, some Weyl-type asymptotics for different thresholds depending on the Robin parameter are obtained.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: There are many examples of cooperation in action in society and in nature. In some cases cooperation leads to the increase of the overall welfare of those involved, and in other cases cooperation may be to the detriment of the larger society. The presence of cooperation seems natural if there is a direct bene t to individuals who choose to cooperate. However, in examples of cooperation this bene t is not always immediately obvious. The so called prisoner's dilemma is often used as an analogy to study cooperation and tease out the factors that lead to cooperation. In classical game theory, each player is assumed to be rational and hence typically seeks to select his strategy in such a way as to maximise his own expected pay-o . In the case of the classical prisoner's dilemma, this causes both players to defect. In evolutionary game theory, on the other hand, it is assumed that players have limited knowledge of the game and only bounded rationality. Games in evolutionary game theory are repeated in rounds and players are a orded the opportunity to adapt and learn as this repetition occurs. Past studies have revealed that cooperation may be a viable strategy if the prisoner's dilemma is placed in an evolutionary context, where the evolutionary tness of a strategy is directly related to the pay-o achieved by the player adopting the strategy. One of the mechanisms that promote the persistence of cooperation in the evolutionary prisoner's dilemma is structured interaction between players. A mathematical framework for representing the evolutionary prisoner's dilemma (ESPD) is developed in this thesis. The mathematical framework is used to undertake an analytical approach (i.e. avoiding the use of simulation) towards investigating the dynamics of the ESPD with a path, cycle, plane grid or toroidal grid as underlying graph. The objective of this investigation is to determine the likelihood of the emergence of persistent cooperation between players. The ESPD on a path or a cycle admits two fundamentally di erent parameter regions; large values of the temptation-to-defect parameter are not capable of inducing persistent cooperation, while small values of this parameter allow for the possibility of persistent cooperation. It is found that the likelihood of cooperation increases towards certainty as the order of the underlying graph increases if the underlying graph is a path or cycle. The state space of the ESPD with a plane or toroidal grid graph as underlying graph grows very quickly as a function of the graph order. The automorphism classes of game states are enumerated to determine exactly how fast the size of the state space of the game grows as a function of the order of the underlying graph. Finally, the dynamics of the ESPD is investigated for a grid graph as underlying graph (in cases where the state space is small enough) by means of constructing the corresponding state graphs of the ESPD. / AFRIKAANSE OPSOMMING: Daar is baie voorbeelde van samewerking in the gemeenskap en in die natuur. In sommige gevalle lei samewerking tot 'n toename in die algehele welvaart van die betrokkenes, terwyl samewerking in ander gevalle tot nadeel van die bre er gemeenskap mag wees. Die voorkoms van samewerking blyk natuurlik te wees indien daar 'n direkte voordeel vir die individue is wat kies om saam te werk. In voorbeelde van samewerking is s o 'n voordeel egter nie altyd voor-diehand- liggend nie. Die sogenaamde prisoniersdilemma word dikwels as voorbeeld in die studie van samewerking gebruik om die faktore wat na samewerking lei, te ontbloot. In klassieke speleteorie word daar aangeneem dat elke speler rasioneel is en dus poog om sy spelstrategie op s o 'n manier te kies dat sy eie verwagte uitbetaling gemaksimeer word. In die geval van die klassieke prisoniersdilemma veroorsaak dit dat beide spelers mekaar verraai. In evolusion^ere speleteorie, daarenteen, word daar slegs aangeneem dat elke speler oor beperkte kennis van die spel en begrensde rasionaliteit beskik. Spele in evolusion^ere speleteorie word in rondtes herhaal en spelers word die geleentheid gebied om gedurende hierdie herhalingsproses aan te pas en te leer. Vorige studies het getoon dat samewerking 'n lewensvatbare strategie is indien die prisoniersdilemma in 'n evolusion^ere konteks gespeel word, waar die evolusion^ere ksheid van 'n strategie direk afhang van die uitbetaling van 'n speler wat die strategie volg. Een van die meganismes wat volhoubare samewerking in die evolusion^ere prisoniersdilemma voortbring, is gestruktureerde interaksie tussen spelers. 'n Wiskundige raamwerk word vir die voorstelling van die evolusion^ere prisoniersdilemma in hierdie tesis ontwikkel. Hierdie wiskundige raamwerk word gebruik om 'n analitiese studie (met ander woorde sonder die gebruik van simulasie) van die dinamika van die prisoniersdilemma op 'n pad, siklus, rooster in die vlak, of rooster op die torus as onderliggende gra ek van stapel te stuur. Die doel van hierdie studie is om die waarskynlikheid vir die ontstaan van volhoubare samewerking tussen spelers te bepaal. Die prisoniersdilemma op 'n pad of siklus as onderliggende gra ek het twee fundamenteel verskillende parametergebiede tot gevolg; groot waardes van die versoeking-om-te-verraai parameter lei nie tot volhoubare samewerking nie, terwyl volhoubare samewerking wel vir klein waardes van hierdie parameter moontlik is. Daar word gevind dat die kans vir volhoubare samewerking toeneem tot sekerheid namate die orde van die onderliggende gra ek groei. Die toestandsruimte van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek groei baie vinnig as 'n funksie van die orde van die gra ek. Die outomor smeklasse van die speltoestande word getel met die doel om te bepaal presies hoe vinnig die toestandsruimte van die spel as 'n funksie van die orde van die onderliggende gra ek groei. Die dinamika van die prisoniersdilemma met 'n rooster in die vlak of 'n rooster op die torus as onderliggende gra ek word laastens deur middel van konstruksies van die ooreenstemmende toestandsgra eke ondersoek (in gevalle waar die toestandsruimte klein genoeg is).

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

La synchronisation temporelle des systèmes MIMO a été abondamment étudiée dans les quinze dernières années, mais la plupart des techniques existantes supposent que le bruit est blanc temporellement et spatialement, ce qui ne permet pas de modéliser la présence d'interférence. Nous considérons donc le cas de bruits blancs temporellement mais pas spatialement, dont la matrice de covariance spatiale est inconnue. En formulant le problème de l'estimation de l'instant de synchronisation comme un test d'hypothèses, nous aboutissons au test du rapport de vraisemblance généralisé (GLRT) qui donne lieu à la comparaison avec un seuil d'une statistique de test eta_GLRT. Cependant, pour des raisons de complexité, l'utilisation de cette statistique n'est pas toujours considérée comme réaliste. La première partie de ce travail a donc été consacrée à mettre en évidence des tests alternatifs moins complexes à mettre en œuvre, tout en ayant des performances similaires. Une analyse comparative exhaustive, prenant en considération le bruit et l'interférence, le type de canal, le nombre d'antennes en émission et en réception, et l'orthogonalité de la séquence de synchronisation est réalisée. Enfin, nous étudions le problème de l'optimisation du nombre d'antennes en émission K pour la synchronisation temporelle, montrant que pour un RSB élevé, les performances augmentent avec K dès que le produit de K avec le nombre d'antennes de réception M n'est pas supérieur à 8.Le deuxième aspect de ce travail est une analyse statistique de eta_GLRT dans le cas où la taille de la séquence d'apprentissage N est du même ordre de grandeur que M, ce qui conduit naturellement à étudier le comportement de eta_GLRT dans le régime asymptotique des grands systèmes M tend vers l'infini, N tend l'infini de telle sorte que M/N tende vers une constante non nulle. Nous considérons le cadre applicatif d'un système muni d'une unique antenne d'émission et d'un canal à trajets multiples, qui est formellement identique à celui d'un système MIMO dont le nombre d'antennes d'émissions correspondrait au nombre de trajets. Lorsque le nombre de trajets L est beaucoup plus faible que N et M, nous établissons que eta_GLRT a un comportement gaussien avec l'espérance asymptotique L log (1 / (1-M/N)) et la variance (L/N)*(M/N)/(1-M/N). Ceci est en contraste avec le régime asymptotique standard quand N tend vers l'infini et M et L fixe où eta_GLRT a un comportement chi2. Sous l'hypothèse H_1, eta_GLRT a aussi un comportement gaussien. Nous considérons également le cas où le nombre de trajets L tend vers l'infini à la même vitesse que M et N. Nous utilisons des résultats connus concernant le comportement des statistiques linéaires des valeurs propres des grandes F matrices, et déduisons que dans le régime où L,M,N tendent vers l'infini à la même vitesse, eta_GLRT a encore un comportement gaussien sous H_0, mais avec une espérance et variance différentes. L'analyse de eta_GLRT sous H_1 lorsque L,M,L convergent vers l'infini nécessite l'établissement d'un théorème central limite pour les statistiques linéaires des valeurs propres de matrices F de moyennes non-nulles, une tâche difficile. Motivé par les résultats obtenus dans le cas où L reste fini, nous proposons d'approximer la distribution asymptotique par une distribution gaussienne dont l'espérance et la variance sont la somme de l'espérance et la variance asymptotique sous H_0quand L tend vers l'infini avec l'espérance et la variance asymptotique sous H_1 dans le régime classique N tend vers l'infini et M fixé. Des simulations numériques permettent de comparer les courbes ROC des différents approximant avec des courbes ROC empiriques. Les résultats montrent que nos approximant de grandes dimensions fournissent de meilleurs résultats quand M/N augmente, tout en permettant de capturer la performance réelle pour les petites valeurs de M/N / Time synchronization of MIMO systems have been strongly studied in the last fifteen years, but most of the existing techniques assume a spatially and temporally white noise, which does not allow modeling the presence of interference. We consider thus a temporally white but spatially colored noise, with an unknown covariance matrix. Formulating the estimation problem as a hypothesis testing problem, we obtain a Generalized likelihood ratio test (GLRT), which gives us a synchronization statistics eta_GLRT. However, for complexity reasons, it is not always considered realistic for practical situations. A part of this work has thus been devoted to showing that there exist non-GLRT statistics that are less complex to implement than theet a_GLRT, while having similar performance. Furthermore, we perform a comparative parameter analysis, taking into consideration the noise type, channel type, the number of transmit and receive antennas, and the orthogonality of the synchronization sequence. Lastly, the problem of optimization of the number of transmit antennas K for time synchronization has been investigated. showing, for high SNR, increasing performance with K as long as the product KM is not larger than 8, where M is the number of receive antennas. The second aspect of MIMO synchronization studied in thesis is asymptotic analysis of the same GLRT, but for large M. In this context, the synchronization sequence length N is the same order of magnitude as M, and this leads us naturally to the study of the the behavior of eta_GLRT in the asymptotic regime where M,N go towards infinity such that M/N go towards a non-zero constant. We consider the case of a single transmit antenna in a multi-path channel, which formally is equivalent to the MIMO system where the transmit antennas correspond to the number of paths. We address the case When the number of paths L does not scale with M and N, we establish that eta_GLRT has a Gaussian behavior with asymptotic mean L log (1/ (1 - M/N))and variance (L/N)*(M/N)/(1-M/N).This is in contrast with the standard asymptotic regime N goes to infinity and M fixed where eta_GLRT has a chi^2 behaviour. Under hypothesis H_1, eta_GLRT still has a Gaussian behaviour. The corresponding asymptotic mean and variance are obtained as the sum of the asymptotic mean and variance in the standard regime N goes to infinity and M fixed, and L log(1/(1-/M/N))L log (1 / (1-M/N)) and (L/N)*(M/N)/(1-M/N)respectively, i.e. the asymptotic mean and variance under H_0.We also consider the case where the number of paths L converges towards infinity at the same rate as M and N. Using known results of concerning the behaviour of linear statistics of the eigenvalues of large F-matrices, we deduce that in the regime where L,M,N converge to infinity at the same rate, eta_GLRT still has a Gaussian behaviour under H_0, but with a different mean and variance. The analysis of eta_GLRT under H_1 whenL,M,N converge to infinity needs to establish a central limit theorem for linear statistics of the eigenvalues of large non zero-mean F-matrices, a difficult ask. Motivated by the results obtained in the case where L remains finite, we propose to approximate the asymptotic distribution of eta_GLRT by a Gaussian distribution whose mean and variance are the sum of the asymptotic mean and variance under H_0when L goes to infinity with the asymptotic mean and variance under H_1 in the standard regime N goes to infinity and M fixed. Numerical simulations allow to compare the ROC curves obtained with the different approximations with the empirical ROC curves. The results show that the large-system approximations provide better results when M/N increases, while also allowing to capture the actual performance for small values of M/N

Made available in DSpace on 2015-03-04T18:50:29Z (GMT). No. of bitstreams: 1 Apresentacao.pdf: 103220 bytes, checksum: c76acce6b0debd619e9db9533aa20f11 (MD5) Previous issue date: 2003-02-13 / Conselho Nacional de Desenvolvimento Cientifico e Tecnologico / The Topological Sensitivity Analysis results in a scalar function, denoted as Topological Derivative, that supplies for each point of the domain of definition of the problem the sensitivity of a given cost function when a small hole is created. However, when a hole is introduced, it is no longer possible to stablish a homeomorphism between the domains. Due to this mathematical difficulty the Topological Derivative may become restrictive, nevertheless be extremely general. Thus, in the present work it is proposed a new method to calculte the Topological Derivative via Shape Sensitivity Analysis. This result, formally proved through a theorem, leads to a simpler and more general methodology than the others found in the literature. The Topological Sensitivity Analysis is performed for several Engineering problems, and the obtained results are used to improve the design of mechanical devices by introducing holes. The same theory developed to calculate the Topological Derivative is used to determine the sensitivity of the cost function when a small incrustation is introduced in each position of the domain, resulting in a novel concept denoted as Configurational Sensitivity Analysis, being discussed some possible applications in the context of Inverse Problems and modelling of phenomena that experiment changes in the physical properties of the medium. Thus, the methodology developed in the present work results in a framework with potential applications in Topology Optimization, Inverse Problems and Mechanical Modelling, which may be seen, from now on, not only as a method to calculate the Topological Derivative, but as a promising research area in Computational Modelling. / A análise de Sensibilidade Topológica resulta em uma função escalar, denominada Derivada Topológica, que fornece para cada ponto do domínio de definição do problema a sensibilidade de uma dada função custo quando um pequeno furo é criado. No entanto, ao introduzir um furo, não é mais possível estabelecer um homeomorfismo entre os domínios envolvidos. Devido a essa dificuldade matemática a Derivada Topológica pode se tornar restritiva, não obstante seja extremamente geral. No presente trabalho, portanto, é proposto um novo método de cálculo da Derivada Topológica via Análise de Sensibilidade à Mudança de Forma. Este resultado, formalmente demonstrado através de um teorema, conduz a uma metodologia mais simples e geral do que as demais encontradas na literatura. A Análise de Sensibilidade Topológica é então realizada em diversos problemas da Engenharia e os resultados obtidos são empregados para melhorar o projeto de componentes mecânicos mediante a introdução de furos. A mesma teoria desenvolvida para calcular a Derivada Topológica é utilizada para determinar a sensibilidade da função custo ao introduzir uma pequena incrustação numa dada posição do domínio, resultando em um novo conceito denominado Análise de Sensibilidade Configuracional, sendo discutidas suas possíveis aplicações no contexto de Problemas Inversos e de modelagem de fenômenos que experimentam mudanças nas propriedades físicas do meio. Assim, a metodologia aqui desenvolvida é uma ferramenta em potencial tanto de Otimização Topológica quanto de Problemas Inversos e de Modelagem Mecânica, podendo ser vista, a partir de agora, não somente como um método de cálculo da Derivada Topológica, mas como uma promissora área de pesquisa em Modelagem Computacional.

Submitted by Maria Cristina (library@lncc.br) on 2015-11-25T12:00:01Z No. of bitstreams: 1 Tese - Análise de Sensibilidade Topológica.pdf: 447139 bytes, checksum: d7d9c80ad59acb3e3cf12ae2d457887f (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2015-11-25T12:00:22Z (GMT) No. of bitstreams: 1 Tese - Análise de Sensibilidade Topológica.pdf: 447139 bytes, checksum: d7d9c80ad59acb3e3cf12ae2d457887f (MD5) / Made available in DSpace on 2015-11-25T12:00:35Z (GMT). No. of bitstreams: 1 Tese - Análise de Sensibilidade Topológica.pdf: 447139 bytes, checksum: d7d9c80ad59acb3e3cf12ae2d457887f (MD5) Previous issue date: 2015-11-03 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / The topological derivative concept has been proved to be useful in many relevant applications such as topology optimization, inverse problems, image processing, multi-scale constitutive modeling, fracture mechanics and damage evolution modeling. The topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial di erential equations. On the other hand, the topological derivatives associated with coupled problems have been derived only in their abstract forms. In this paper, therefore, we deal with the Reissner-Mindlin plate bending model, which is written in the form of a coupled system of partial di erential equations. In particular, the topological asymptotic analysis of the associated total potential energy is developed and the topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form.Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justi cation for the derived formulas. / O conceito de derivada topológica tem se mostrado útil em muitas aplicações, tais como otimização topológica, problemas inversos, processamento de imagens, modelagem constitutiva multi-escala, mecânica da fratura e modelagem da evolução de dano. A análise assintótica topológica foi amplamente desenvolvida para uma grande variedade de problemas modelados por equações diferenciais parciais. Por outro lado, a derivada topológica associada a problemas acoplados é conhecida apenas em sua forma abstrata. Neste trabalho, portanto, considera-se o modelo de flexão de placa de Reissner-Mindlin, que é escrito na forma de um sistema acoplado de equações diferenciais parciais. Em particular, a análise assintótica topológica da energia potencial total associada é desenvolvida e a derivada topológica com relação a nucleação de uma inclusão circular é obtida na sua forma fechada. Finalmente, os resíduos da expansão assintótica topológica são estimados e uma justificativa matemática completa para a derivada topológica é apresentada.

Submitted by Maria Cristina (library@lncc.br) on 2016-07-27T18:52:15Z No. of bitstreams: 1 tese_Suelen.pdf: 922374 bytes, checksum: f324427616027a422decc0eaf56c7ae2 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2016-07-27T18:52:32Z (GMT) No. of bitstreams: 1 tese_Suelen.pdf: 922374 bytes, checksum: f324427616027a422decc0eaf56c7ae2 (MD5) / Made available in DSpace on 2016-07-27T18:52:42Z (GMT). No. of bitstreams: 1 tese_Suelen.pdf: 922374 bytes, checksum: f324427616027a422decc0eaf56c7ae2 (MD5) Previous issue date: 2016-04-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / In this work a new method for obstacles reconstruction from partial boundary measurements is proposed. For a given boundary excitation, we want to determine the quantity, locations and sizes of a number of obstacles embedding whiting a geometrical domain, from partial boundary measurements related to such an excitation. This problem is written in the form of ill-posed and over-determinated partial differential equation. The idea therefore is to rewrite it as an optimization problem where a shape functional measuring the misfit between the boundary measurement and the solution to an auxiliary boundary value problem is minimized with respect to a set of ball-shaped holes. The topological derivative concept is used for solving the resulting topology optimization problem, leading to a second-order reconstruction algorithm free of initial guess. The resulting algorithm is non-iterative and thus very robust with respect to noisy data. Finally, some numerical results are presented in order to demonstrate the effectiveness of proposed reconstruction algorithm. / O objetivo deste trabalho é apresentar um novo método de reconstrução de obstáculos. Mais precisamente, dada uma excitação deseja-se obter a solução de um problema inverso de reconstrução consistindo na determinação da quantidade, localização e tamanho de obstáculos no interior de um dado domínio geométrico a partir de leituras parciais da resposta à referida excitação. Este problema é escrito na forma de uma equação diferencial parcial sobredeterminada. Essa dificuldade é contornada reescrevendo o problema inverso na forma de um problema de otimização. A ideia básica consiste em minimizar um funcional de forma que mede a diferença entre o dado lido e o calculado numericamente em relação ao próprio domínio geométrico. Em particular o conceito de derivada topológica é utilizado, o que conduz a um algoritmo de reconstrução de segunda ordem e independente de qualquer chute inicial. Como o algoritmo resultante é não-iterativo, o processo de reconstrução torna-se extremamente robusto à presença de ruído. Vários exemplos numéricos de reconstrução são apresentados donde se verifica a validade dos resultados obtidos.

Este projeto de doutorado apresenta uma nova abordagem MAP-MRF para a classificação contextual de imagens multiespectrais utilizando combinação de modelos de Campos Aleatórios Markovianos definidos em sistemas de ordens superiores. A modelagem estatística para o problema de classificação segue o paradigma Bayesiano, com a definição de um modelo Markoviano para os dados observados (Gaussian Markov Random Field multiespectral) e outro modelo para representar o conhecimento a priori (Potts). Nesse cenário, o parâmetro β do modelo de Potts atua como um parâmetro de regularização, tendo papel fundamental no compromisso entre as observações e o conhecimento a priori, de modo que seu correto ajuste é necessário para a obtenção de bons resultados. A introdução de sistemas de vizinhança de ordens superiores requer a definição de novos métodos para a estimação dos parâmetros dos modelos Markovianos. Uma das contribuições desse trabalho é justamente propor novas equações de pseudo-verossimilhança para a estimação desses parâmetros no modelo de Potts em sistemas de segunda e terceira ordens. Apesar da abordagem por máxima pseudo-verossimilhança ser amplamente utilizada e conhecida na literatura de campos aleatórios, pouco se conhece acerca da acurácia dessa estimação. Foram derivadas aproximações para a variância assintótica dos estimadores propostos, caracterizando-os completamente no caso limite, com o intuito de realizar inferências e análises quantitativas sobre os parâmetros dos modelos Markovianos. A partir da definição dos modelos e do conhecimento dos parâmetros, o próximo estágio é a classificação das imagens multiespectrais. A solução para esse problema de inferência Bayesiana é dada pelo critério de estimação MAP, onde a solução ótima é determinada maximizando a probabilidade a posteriori, o que define um problema de otimização. Como não há solução analítica para esse problema no caso de prioris Markovianas, algoritmos iterativos de otimização combinatória foram empregados para aproximar a solução ótima. Nesse trabalho, adotam-se três métodos sub-ótimos: Iterated Conditional Modes, Maximizer of the Posterior Marginals e Game Strategy Approach. Porém, é demonstrado na literatura que tais métodos convergem para máximos locais e não globais, pois são altamente dependentes de sua condição inicial. Isto motivou o desenvolvimento de uma nova abordagem para combinação de classificadores contextuais, que utiliza múltiplas inicializações simultâneas providas por diferentes classificadores estatísticos pontuais. A metodologia proposta define um framework MAP-MRF bastante robusto para solução de problemas inversos, pois permite a utilização e a integração de diferentes condições iniciais em aplicações como classificação, filtragem e restauração de imagens. Como medidas quantitativas de desempenho, são adotados o coeficiente Kappa de Cohen e o coeficiente Tau de Kendall para verificar a concordância entre as saídas dos classificadores e a verdade terrestre (amostras pré-rotuladas). Resultados obtidos mostram que a inclusão de sistemas de vizinhança de ordens superiores é de fato capaz de melhorar significativamente não apenas o desempenho da classificação como também a estimação dos parâmetros dos modelos Markovianos, reduzindo tanto o erro de estimação quanto a variância assintótica. Além disso, a combinação de classificadores contextuais através da utilização de múltiplas inicializações simultâneas melhora significativamente o desempenho da classificação se comparada com a abordagem tradicional com apenas uma inicialização. / This work presents a novel MAP-MRF approach for multispectral image contextual classification by combining higher-order Markov Random Field models. The statistical modeling follows the Bayesian paradigm, with the definition of a multispectral Gaussian Markov Random Field model for the observations and a Potts MRF model to represent the a priori knowledge. In this scenario, the Potts MRF model parameter (β) plays the role of a regularization parameter by controlling the tradeoff between the likelihood and the prior knowledge, in a way that a suitable tunning for this parameter is required for a good performance in contextual classification. The introduction of higher-order MRF models requires the specification of novel parameter estimation methods. One of the contributions of this work is the definition of novel pseudo-likelihood equations for the estimation of these MRF parameters in second and third order neighborhood systems. Despite its widely usage in practical MRF applications, little is known about the accuracy of maximum pseudo-likelihood approach. Approximations for the asymptotic variance of the proposed MPL estimators were derived, completely characterizing their behavior in the limiting case, allowing statistical inference and quantitative analysis. From the statistical modeling and having the model parameters estimated, the next step is the multispectral image classification. The solution for this Bayesian inference problem is given by the MAP criterion, where the optimal solution is obtained by maximizing the a posteriori distribution, defining an optimization problem. As there is no analytical solution for this problem in case of Markovian priors, combinatorial optimization algorithms are required to approximate the optimal solution. In this work, we use three suboptimal methods: Iterated Conditional Modes, Maximizer of the Posterior Marginals and Game Strategy Approach, a variant approach based on non-cooperative game theory. However, it has been shown that these methods converge to local maxima solutions, since they are extremelly dependent on the initial condition. This fact motivated the development of a novel approach for combination of contextual classifiers, by making use of multiple initializations at the same time, where each one of these initial conditions is provided by different pointwise pattern classifiers. The proposed methodology defines a robust MAP-MRF framework for the solution of general inverse problems since it allows the use and integration of several initial conditions in a variety of applications as image classification, denoising and restoration. To evaluate the performance of the classification results, two statistical measures are used to verify the agreement between the classifiers output and the ground truth: Cohens Kappa and Kendalls Tau coefficient. The obtained results show that the use of higher-order neighborhood systems is capable of significantly improve not only the classification performance, but also the MRF parameter estimation by reducing both the estimation error and the asymptotic variance. Additionally, the combination of contextual classifiers through the use of multiple initializations also improves the classificatoin performance, when compared to the traditional single initialization approach.

Cette thèse est dédiée à l'étude mathématique et numérique d'une classe d'équations cinétiques collisionnelles, de type équation de Boltzmann. Nous avons porté un intérêt tout particulier à l'équation des milieux (ou gaz) granulaires, initialement introduite dans la littérature physique pour décrire le comportement hors équilibre de matériaux composés d'un grand nombre de grains, ou particules, non nécessairement microscopiques, et interagissant par des collisions dissipant l'énergie cinétique. Ces modèles se sont révélés avoir une structure mathématique très riche. Cette thèse se structure en trois partie pouvant être lues de manière indépendante, mais néanmoins en rapport avec des équations cinétiques collisionnelles en général, et l'équation des milieux granulaires en particulier. La première partie est dédiée à l'étude mathématique du comportement asymptotique de certaines équations cinétiques collisionnelles dans un cadre homogène en espace. Nous y montrons des résultats de type explosion et convergence vers la solution autosimilaire avec calcul explicite des taux, pour des opérateurs de type Boltzmann, grâce à l'utilisation (entre autre) d'une nouvelle méthode de changement de variables dépendant directement de la solution de l'équation considérée. En particulier, nous démontrons que pour un modèle de gaz granulaire - dit anormal - il est possible d'observer une explosion en temps fini. Dans la deuxième partie, orientée analyse numérique et calcul scientifique, nous nous intéressons développement et à l'étude de méthodes spectrales pour la résolution de problèmes multi-échelles, issus de la théorie des équations cinétiques collisionnelles. Les méthodes de changement de variables tiennent aussi une place importante dans cette partie, et permettent d'observer numériquement des phénomènes non triviaux qui apparaissent lors de l'étude de gaz granulaires, comme la création d'amas de matière ou la caractérisation précise du retour vers l'équilibre. La troisième et dernière partie est dédiée à l'étude spectrale de l'opérateur des milieux granulaires avec bain thermique, linéarisé au voisinage d'un équilibre homogène en espace, afin d'établir des résultats de type stabilité et convergence vers une limite hydrodynamique. Ce travail est en fait la généralisation d'un résultat célèbre dans la théorie de l'équation de Boltzmann, dû à R. Ellis et M. Pinsky, et établissant rigoureusement la première limite hydrodynamique vers les équations d'Euler compressibles linéaires puis Navier-Stokes de cette équation / This dissertation is dedicated to the mathematical and numerical study of a class of collisional kinetic equations, such as the Boltzmann equation of perfect gases. We took a particular interest in the granular media (or gases) equation, which has been first introduced in the physical literature to describe the nonnequilibrium behavior of materials composed of a large number of grains (the particles) of macroscopic size, interacting through energy dissipative collisions. These models have a very rich mathematical structure. This dissertation is divided in three independent part, all related to the theory of collisional kinetic equation, with a strong emphasis on granular media. The first part concerns the mathematical study of the asymptotic behavior of space homogeneous Boltzmann-like kinetic equations. We prove some blow up results, as well as convergence towards self-similarity, with explicit rates for two different models. One of the key tools of our proofs is the use of a new scaling method, where the scaling function depends on the solution itself. We especially prove that for a particular model of granular gases (also know as anomalous), finite time blow up occurs. The second part is dedicated to the development and study of spectral methods for the resolution of multi-scale problems, coming from the theory of collisional kinetic equations. Some rescaling methods take a very important place in this part, allowing to observe numerically some nontrivial phenomena such as the clustering in space which occurs in the time evolution of a space inhomogeneous granular gas, or to investigate numerically the trend to equilibrium for this equation. The whole third (and last) part is dedicated to the spectral study of the granular gases operator with a thermal bath, linearized near a space homogeneous self-similar profile. The goal of this work is to prove some stability results for the complete space inhomogeneous equation, and to investigate the hydrodynamic limit of the model. This work is based and extend the famous result of R. Ellis and M. Pinsky on the spectrum of the linearized Boltzmann equation, intended to establish rigorously the hydrodynamic limit of this equation towards the linearized Euler and Navier-Stokes equations