Many tasks in machine learning require learning some kind of input-output relation (function), for example, recognising handwritten digits (from image to number) or learning the motion behaviour of a dynamical system like a pendulum (from positions and velocities now to future positions and velocities). We consider this problem using the Bayesian framework, where we use probability distributions to represent the state of uncertainty that a learning agent is in. In particular, we will investigate methods which use Gaussian processes to represent distributions over functions. Gaussian process models require approximations in order to be practically useful. This thesis focuses on understanding existing approximations and investigating new ones tailored to specific applications. We advance the understanding of existing techniques first through a thorough review. We propose desiderata for non-parametric basis function model approximations, which we use to assess the existing approximations. Following this, we perform an in-depth empirical investigation of two popular approximations (VFE and FITC). Based on the insights gained, we propose a new inter-domain Gaussian process approximation, which can be used to increase the sparsity of the approximation, in comparison to regular inducing point approximations. This allows GP models to be stored and communicated more compactly. Next, we show that inter-domain approximations can also allow the use of models which would otherwise be impractical, as opposed to improving existing approximations. We introduce an inter-domain approximation for the Convolutional Gaussian process - a model that makes Gaussian processes suitable to image inputs, and which has strong relations to convolutional neural networks. This same technique is valuable for approximating Gaussian processes with more general invariance properties. Finally, we revisit the derivation of the Gaussian process State Space Model, and discuss some subtleties relating to their approximation. We hope that this thesis illustrates some benefits of non-parametric models and their approximation in a non-parametric fashion, and that it provides models and approximations that prove to be useful for the development of more complex and performant models in the future.

Orientador: Samuel Conceição de Oliveira / Resumo: A determinação da estratégia de controle adequada para bioprocessos batelada e batelada alimentada é uma questão prática importante devido ao alto valor agregado de alguns bioprodutos. Desde que é altamente desejável otimizar a produção de bioprodutos, vários métodos têm sido propostos para esse objetivo. Uma vez dispondo de um modelo matemático adequado para o bioprocesso, o problema de otimização pode ser formulado no âmbito do princípio do máximo de Pontryagin e da teoria de controle ótimo para determinar a melhor trajetória de controle para certas variáveis manipuladas, como temperatura, pH e taxa de alimentação do substrato. Neste estudo, duas aplicações dessas técnicas baseadas em modelos matemáticos para otimizar e controlar bioprocessos de produção de antibióticos são revisadas e novos aspectos são enfatizados. Os casos analisados incluem a otimização da taxa de alimentação de substrato em um reator batelada alimentada e da temperatura em um reator batelada durante fermentações penicilínicas. Os principais resultados obtidos neste estudo foram: (i) a constatação de que métodos numéricos simples (Runge-Kutta, Newton-Raphson) podem ser aplicados para resolver satisfatoriamente os problemas de valor no contorno propostos; (ii) a demonstração de que a operação não isotérmica é mais produtiva em antibiótico do que a operação sob temperatura constante; (iii) a necessidade de adoção de um modelo matemático apropriado para o bioprocesso visando à resolução do problema de cont... (Resumo completo, clicar acesso eletrônico abaixo) / Mestre

Les propriétés mécaniques des alliages de fer sont largement conditionnées par leur microstructure et la population de défauts locaux qu’elle contient. Comme il s'agit d'un moment de forte interaction entre ces deux éléments, l'étude des transformations α-γ et γ-α du fer mérite une attention particulière. Outre le passage d'une structure cristalline cubique centrée à une structure cubique faces centrées – donc de compacités différentes –,elles sont associées à une réduction des paramètres de maille. Le changement de volume correspondant est responsable de déformations mécaniques locales autour des sites de germination. La technique de corrélation d'images numériques(CIN) s'est montrée fiable pour ce qui est de calculer des champs cinématiques à l'échelle de quelques grains. Dans ce travail, elle est adaptée à la capture de localisations de la déformation pendant les transformations allotropiques. Un banc expérimental est conçu pour obtenir des images haute résolution avec un contrôle fin des sollicitations thermiques. Des essais de validation sont d'abord effectués sur du fer industriel. Puis des échantillons de fer haute pureté sont ensuite soumis à des cycles de transformation α-γ-α et les champs de déformation correspondant sont calculés par CIN. Associés à l'acquisition des orientations initiales et finales, ils sont utilisés pour valider les mécanismes de transformation proposés dans la littérature. En parallèle, un modèle, écrit en petites déformations, est construit en incorporant des composants liés à la transformation dans une fonctionnelle dont les conditions de stationnarité sont équivalentes au problème thermomécanique à résoudre. Les incréments des variables internes, incluant glissements plastiques et fractions volumique transformées, sont obtenus en minimisant cette fonctionnelle. / Mechanical properties of iron-based alloys are largely conditioned by their microstructure and the population of local defects inside this microstructure. As it is a moment of massive interplay between these two elements, the study of the α-γ and γ-α transformations in iron is of particular interest. Besides a change from a body-centered cubic to a face-centered cubic crystal structure – and thus a change in compacity –, they lead to a lattice parameter reduction. The corresponding change in volume is responsible for local mechanical deformations around transformation sites. Digital Image Correlation (DIC) technique has been proven reliable to compute kinematic fields at the scale of a few grains. In the present work, an adaptation of this technique to the observation of strain localizations induced by the formation of a new phase is proposed. A home-made device is designed to obtain high resolution images and to control heating and cooling. Tests are first conducted on industrial iron to assess the viability of the procedure. High-purity iron samples are then submitted to α-γ-α transformation cycles and the associated strain fields are computed. In combination with the acquisition of initial and final orientations they are used to validate transformation mechanisms proposed in the literature. In parallel, a model, written under the small strain format, is built by incorporating transformation related components into a power functional whose stationarity conditions are equivalent to the thermomechanical problem. In accordance with variational principles, the evolution of internal variables,including plastic slip increments and fraction of the material locally transformed, are computed through the minimization of the functional.

La perception des robots joue un rôle crucial dans l’interaction homme-robot (HRI). Le système de perception fournit les informations au robot sur l’environnement, ce qui permet au robot de réagir en consequence. Dans un scénario de conversation, un groupe de personnes peut discuter devant le robot et se déplacer librement. Dans de telles situations, les robots sont censés comprendre où sont les gens, ceux qui parlent et de quoi ils parlent. Cette thèse se concentre sur les deux premières questions, à savoir le suivi et la diarisation des locuteurs. Nous utilisons différentes modalités du système de perception du robot pour remplir cet objectif. Comme pour l’humain, l’ouie et la vue sont essentielles pour un robot dans un scénario de conversation. Les progrès de la vision par ordinateur et du traitement audio de la dernière décennie ont révolutionné les capacités de perception des robots. Dans cette thèse, nous développons les contributions suivantes : nous développons d’abord un cadre variationnel bayésien pour suivre plusieurs objets. Le cadre bayésien variationnel fournit des solutions explicites, rendant le processus de suivi très efficace. Cette approche est d’abord appliqué au suivi visuel de plusieurs personnes. Les processus de créations et de destructions sont en adéquation avec le modèle probabiliste proposé pour traiter un nombre variable de personnes. De plus, nous exploitons la complémentarité de la vision et des informations du moteur du robot : d’une part, le mouvement actif du robot peut être intégré au système de suivi visuel pour le stabiliser ; d’autre part, les informations visuelles peuvent être utilisées pour effectuer l’asservissement du moteur. Par la suite, les informations audio et visuelles sont combinées dans le modèle variationnel, pour lisser les trajectoires et déduire le statut acoustique d’une personne : parlant ou silencieux. Pour expérimenter un scenario où l’information visuelle est absente, nous essayons le modèle pour la localisation et le suivi des locuteurs basé sur l’information acoustique uniquement. Les techniques de déréverbération sont d’abord appliquées, dont le résultat est fourni au système de suivi. Enfin, une variante du modèle de suivi des locuteurs basée sur la distribution de von-Mises est proposée, celle-ci étant plus adaptée aux données directionnelles. Toutes les méthodes proposées sont validées sur des bases de données specifiques à chaque application. / Robot perception plays a crucial role in human-robot interaction (HRI). Perception system provides the robot information of the surroundings and enables the robot to give feedbacks. In a conversational scenario, a group of people may chat in front of the robot and move freely. In such situations, robots are expected to understand where are the people, who are speaking, or what are they talking about. This thesis concentrates on answering the first two questions, namely speaker tracking and diarization. We use different modalities of the robot’s perception system to achieve the goal. Like seeing and hearing for a human-being, audio and visual information are the critical cues for a robot in a conversational scenario. The advancement of computer vision and audio processing of the last decade has revolutionized the robot perception abilities. In this thesis, we have the following contributions: we first develop a variational Bayesian framework for tracking multiple objects. The variational Bayesian framework gives closed-form tractable problem solutions, which makes the tracking process efficient. The framework is first applied to visual multiple-person tracking. Birth and death process are built jointly with the framework to deal with the varying number of the people in the scene. Furthermore, we exploit the complementarity of vision and robot motorinformation. On the one hand, the robot’s active motion can be integrated into the visual tracking system to stabilize the tracking. On the other hand, visual information can be used to perform motor servoing. Moreover, audio and visual information are then combined in the variational framework, to estimate the smooth trajectories of speaking people, and to infer the acoustic status of a person- speaking or silent. In addition, we employ the model to acoustic-only speaker localization and tracking. Online dereverberation techniques are first applied then followed by the tracking system. Finally, a variant of the acoustic speaker tracking model based on von-Mises distribution is proposed, which is specifically adapted to directional data. All the proposed methods are validated on datasets according to applications.

Ingeniero Civil Mecánico / Dentro del campo de la ingeniería mecánica, una de las áreas que más crecimiento ha mostrado en los últimos años es la de la gestión de activos físicos y confiabilidad. Junto con la capacidad de construir máquinas y sistemas más complejos, el problema de la detección temprana de fallas en elementos mecánicos se vuelve de suma importancia. Al mismo tiempo, el incremento en la disponibilidad de tecnología sensitoria ha dado a los ingenieros la capacidad de medir una gran cantidad de variables operacionales, como por ejemplo presión, temperatura o emisiones acústicas, a frecuencias de muestreo altísimas. Es ese aspecto, se vuelve un desafío en sí mismo el poder procesar esa cantidad de datos de una manera eficiente, con tal de extrar información útil a partir de ellos. Una metodología para enfrentar este problema es el desarrollo de técnicas de reducción de dimensionalidad, las cuales, si son implementadas de forma correcta, pueden generar una mejor representación de los datos con el fin de mejorar el diagnóstico posterior de los modos de falla presentes. La motivación principal de este trabajo de título es la necesidad de desarrollar modelos confiables para el diagnóstico de modos de falla en elementos mecánicos utilizando técnicas de Aprendizaje de Máquinas. Estos modelos pueden resultar en grandes beneficios para los sectores industriales, tanto en términos de ahorros monetarios como seguridad operacional. El principal objetivo de esta tesis es desarrollar modelos para el diagnóstico de fallas en elementos mecánicos basados en una reducción de dimensionalidad usando un Auto Encoder Variacional (VAE), y luego evaluar y comparar los resultados obtenidos con un modelo similar que usa Análisis de Componentes Principales (PCA) como método de reducción de dimensionalidad y un tercer modelo que no genera una reducción. La metodología usada para este trabajo consiste principalmente de cinco etapas. Primero, una revisión del estado del arte respecto a metodologías existentes para el diagnóstico de fallas es desarrollada. Luego, la adquisición y preprocesamiento de datos operacionales que serán utilizados para entrenar y evaluar los modelos desarrollados. Tercero, el modelo que usa PCA y el modelo que no realiza reducción de dimensionalidad es implementado. Cuarto, el modelo que utiliza VAE es desarrollado e implementado. Por último, el modelo que usa VAE es comparado con los otros dos modelos para extraer conclusiones sobre su aplicabilidad. La principal conclusión de este trabajo es que el modelo que utiliza VAE es mejor en el diagnóstico de modos de falla que el que utiliza PCA para situaciones donde la cantidad de datos etiquetados es escasa, o para los casos cuando una reducción de dimensionalidad muy drástica es requerida. Tambien, el modelo que utiliza VAE casi siempre presenta mejores resultados que el modelo que no genera reducción en los datos, mostrando la importancia de reducir la dimensionalidad de los datos previo a una operación de diagnóstico o clasificación.

Interest in the variation of fundamental constants has recently been stimulated by claims that the fine structure constant, α, was smaller in the past. Physicists are investigating whether α is currently varying using a number of methods including atomic clock experiments and quasar absorption spectra. To date atomic clock experiments have not reached the same level of precision as the quasar results but the precision to which transition frequencies are being measured is increasing dramatically and very soon atomic clock experiments based on Earth will be able to rival or surpass the quasar results. In order to relate the change in transition frequencies to a variation of α accurate calculations of relativistic effects in atoms and their dependence upon α are needed. Other effects, such as the small shift of transition frequencies due to blackbody radiation also need to be accounted for. In this thesis we perform accurate calculations of the dependence of transition frequencies in two-valence-electron atoms and ions on a variation of α. The relativistic Hartree-Fock method is used with many-body perturbation theory and configuration interaction methods to calculate transition frequencies. We also consider transitions with an enhanced sensitivity to α variation. In particular, narrow lines that correspond to atomic transitions between close lying, long-lived atomic states of different configurations. The small transition frequency, coupled with differences in the electron structure ensures a strong enhancement of the relative frequency change compared to a possible change in α . We also show that using the modified form of the Dirac Hamiltonian, as suggested by Bekenstein, does not affect the analysis of the quasar data pertaining to a measurement of α variation, nor does it affect atomic clock experiments. Finally we have performed calculations of the size of the frequency shift induced by a static electric field on the clock transition frequencies of the hyperfine splitting in Y b+, Rb, Cs, Ba+, and Hg+. The calculations are used to find the frequency shifts due to blackbody radiation which are needed for accurate frequency measurements and improvements of the limits on variation of α. Our result for Cs [??v/=E2 = -2:26(2) x 10-10Hz/(V/m)2] is in good agreement with early measurements and ab initio calculations. We present arguments against recent claims that the actual value might be smaller. The difference (~ 10%) is due to the continuum spectrum in the sum over intermediate states.

Mathematical program with equilibrium constraints (MPEC)has extensive applications in practical areas such as traffic control, engineering design, and economic modeling. Some generalized stationary points of MPEC are studied to better describe the limiting points produced by interior point methods for MPEC.A primal-dual interior point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or linear independence constraint qualification. Under very general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limiting point of the generated sequence is a piece-wise stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are satisfactory, which include a case analyzed by Leyffer for which the penalty interior point algorithm failed to find a stationary solution. / Singapore-MIT Alliance (SMA)

We introduce basic ideas of a nonsmooth Newton’s method and its application in solving semidefinite optimization (SDO) problems. In particular, the method can be used to solve both linear and nonlinear semidefinite complementarity problems. We also survey recent theoretical results in matrix functions and stability of SDO that are stemed from the research on the matrix form of the nonsmooth Newton’s method. / Singapore-MIT Alliance (SMA)

This paper studies the problem of multi-period pricing for perishable products in a competitive (oligopolistic) market. We study non cooperative Nash equilibrium policies for sellers. At the beginning of the time horizon, the total inventories are given and additional production is not an available option. The analysis for periodic production-review models, where production decisions can be made at the end of each period at some production cost after incurring holding or backorder costs, does not extend to this model. Using results from game theory and variational inequalities we study the existence and uniqueness of equilibrium policies. We also study convergence results for an algorithm that computes the equilibrium policies. The model in this paper can be used in a number of application areas including the airline, service and retail industries. We illustrate our results through some numerical examples. / Singapore-MIT Alliance (SMA)

The low-level interpretation of images provides constraints on 3D surface shape at multiple resolutions, but typically only at scattered locations over the visual field. Subsequent visual processing can be facilitated substantially if the scattered shape constraints are immediately transformed into visible-surface representations that unambiguously specify surface shape at every image point. The required transformation is shown to lead to an ill-posed surface reconstruction problem. A well-posed variational principle formulation is obtained by invoking 'controlled continuity,' a physically nonrestrictive (generic) assumption about surfaces which is nonetheless strong enough to guarantee unique solutions. The variational principle, which admits an appealing physical interpretation, is locally discretized by applying the finite element method to a piecewise, finite element representation of surfaces. This forms the mathematical basis of a unified and general framework for computing visible-surface representations. The computational framework unifies formal solutions to the key problems of (i) integrating multiscale constraints on surface depth and orientation from multiple visual sources, (ii) interpolating these scattered constraints into dense, piecewise smooth surfaces, (iii) discovering surface depth and orientation discontinuities and allowing them to restrict interpolation appropriately, and (iv) overcoming the immense computational burden of fine resolution surface reconstruction. An efficient surface reconstruction algorithm is developed. It exploits multiresolution hierarchies of cooperative relaxation processes and is suitable for implementation on massively parallel networks of simple, locally interconnected processors. The algorithm is evaluated empirically in a diversity of applications.