In this thesis, we consider a robust state estimation problem for discrete-time, homogeneous, first-order, finite-state finite-alphabet hidden Markov models (HMMs). Based on Kolmogorov's Theorem on the existence of a process, we first present the Kolmogorov model for the HMMs under consideration. A new change of measure is introduced. The statistical properties of the Kolmogorov representation of an HMM are discussed on the canonical probability space. A special Kolmogorov measure is constructed. Meanwhile, the ergodicity of two expanded Markov chains is investigated. In order to describe the uncertainty of HMMs, we study probability distance problems based on the Kolmogorov model of HMMs. Using a change of measure technique, the relative entropy and the relative entropy rate as probability distances between HMMs, are given in terms of the HMM parameters. Also, we obtain a new expression for a probability distance considered in the existing literature such that we can use an information state method to calculate it. Furthermore, we introduce regular conditional relative entropy as an a posteriori probability distance to measure the discrepancy between HMMs when a realized observation sequence is given. A representation of the regular conditional relative entropy is derived based on the Radon-Nikodym derivative. Then a recursion for the regular conditional relative entropy is obtained using an information state method. Meanwhile, the well-known duality relationship between free energy and relative entropy is extended to the case of regular conditional relative entropy given a sub-[special character]-algebra. Finally, regular conditional relative entropy constraints are defined based on the study of the probability distance problem. Using a Lagrange multiplier technique and the duality relationship for regular conditional relative entropy, a finite horizon robust state estimator for HMMs with regular conditional relative entropy constraints is derived. A complete characterization of the solution to the robust state estimation problem is also presented.

La Reconnaissance Automatique de la Parole (RAP) est affectée par les nombreuses variabilités présentes dans le signal de parole.<br />En dépit de l'utilisation de techniques sophistiquées, un système RAP seul n'est généralement pas en mesure de prendre en compte l'ensemble de ces variabilités. Nous proposons l'utilisation de diverses sources d'information acoustique pour augmenter la précision et la robustesse des systèmes. <br /><br />La combinaison de différents jeux de paramètres acoustiques repose sur l'idée que certaines caractéristiques du signal de parole sont davantage mises en avant par certains jeux de paramètres que par d'autres.<br />L'intérêt est donc d'exploiter les points forts de chacun.<br />Par ailleurs, les différentes partitions de l'espace acoustique opérées par les modèles acoustiques peuvent être mises à profit dans des techniques de combinaison bénéficiant de leur complémentarité.<br /><br />Le diagnostic est au coeur de ce travail. L'analyse des performances de chaque jeu de paramètres permet de dégager des contextes spécifiques dans lesquels la prédiction du résultat de reconnaissance est possible. Nous présentons une architecture de diagnostic dans laquelle le système RAP est vu comme un "canal de transmission" dont l'entrée correspond aux phonèmes et la sortie au résultat de reconnaissance. Cette architecture permet de séparer les sources d'ambiguïté au sein du système de reconnaissance. Les analyses ont permis d'intégrer des stratégies de combinaison post-décodage à un niveau segmental élevé (phonème ou mot).<br /><br />Des techniques de combinaison des probabilités a posteriori des états d'un modèle de Markov caché au niveau de la trame sont également proposées. Afin d'améliorer l'estimation de ces probabilités, les probabilités obtenues avec différents modèles acoustiques sont fusionnées. <br />Pour combiner les probabilités de manière cohérente, nous avons développé un protocole permettant d'entraîner des modèles de même topologie avec des paramètres acoustiques différents.

La partie 1 releve de l'Analyse Numerique. Partant de l'interpretation Element Finis Mixtes des schemas volumes finis classiques, l'estimation a posteriori de l'erreur est analysee dans la hierarchie des elements de Raviart-Thomas. Un estimateur calculable est explicite pour ces schemas volumes finis.<br />La partie 2 introduit, d'abord un maillage rectangulaire, puis un maillage structure, une famille de schemas volumes finis de type differences finies. Des essais numeriques sur des problemes modeles montrent que l'ordre prevu par l'analyse peut etre atteint.<br />La partie 3 presente l'application de ces schemas volumes finis a la simulation numerique du comportement d'un bloc de gomme en presence d'une fissure finie. Il s'agit d'un materiau hyperelastique compressible en grandes deformations et differents tenseurs de contraintes, avec tests en quasi-incompressible et des simulations d'endommagement.

Optimal parametric design of a system must be able to respond quickly to short term needs as well as long term conditions. To this end, we present an Assess-Predict-Optimize (APO) strategy which allows for easy modification of a system’s characteristics and constraints, enabling quick design adaptation. There are three components to the APO strategy: Assess - extract necessary information from given data; Predict - predict future behavior of system; and Optimize – obtain optimal system configuration based on information from the other components. The APO strategy utilizes three key mathematical ingredients to yield real-time results which would certainly conform to given constraints: dimension reduction of the model, a posteriori error estimation, and optimization methods. The resulting formulation resembles a bilevel optimization problem with an inherent nonconvexity in the inner level. Using a simple infiltration-evaporation model to simulate an irrigation system, we demonstrate the APO strategy’s ability to yield real-time optimal results. The linearized model, described by a coercive elliptic partial differential equation, is discretized by the reduced-basis output bounds method. A primal-dual interior point method is then chosen to solve the resulting APO problem. / Singapore-MIT Alliance (SMA)

The optimization, control, and characterization of engineering components or systems require fast, repeated, and accurate evaluation of a partial-differential-equation-induced input-output relationship. We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The method has three components: (i) rapidly convergent reduced{basis approximations; (ii) a posteriori error estimation; and (iii) off-line/on-line computational procedures. These components -- integrated within a special network architecture -- render partial differential equation solutions truly "useful": essentially real{time as regards operation count; "blackbox" as regards reliability; and directly relevant as regards the (limited) input-output data required. / Singapore-MIT Alliance (SMA)

In this thesis we develop and analyze the performance ofadaptive finite element methods for multiphysics problems. Inparticular, we propose a methodology for deriving computable errorestimates when solving unidirectionally coupled multiphysics problemsusing segregated finite element solvers.  The error estimates are of a posteriori type and are derived using the standard frameworkof dual weighted residual estimates.  A main feature of themethodology is its capability of automatically estimating thepropagation of error between the involved solvers with respect to anoverall computational goal. The a posteriori estimates are used todrive local mesh refinement, which concentrates the computationalpower to where it is most needed.  We have applied and numericallystudied the methodology to several common multiphysics problems usingvarious types of finite elements in both two and three spatialdimensions. Multiphysics problems often involve convection-diffusion equations for whichstandard finite elements can be unstable. For such equations we formulatea robust discontinuous Galerkin method of optimal order with piecewiseconstant approximation. Sharp a priori and a posteriori error estimatesare proved and verified numerically. Fractional step methods are popular for simulating incompressiblefluid flow. However, since they are not genuine Galerkin methods, butrather based on operator splitting, they do not fit into the standardframework for a posteriori error analysis. We formally derive an aposteriori error estimate for a prototype fractional step method byseparating the error in a functional describing the computational goalinto a finite element discretization residual, a time steppingresidual, and an algebraic residual.

The problem of reconstruction of digital images from their degraded measurements is regarded as a problem of central importance in various fields of engineering and imaging sciences. In such cases, the degradation is typically caused by the resolution limitations of an imaging device in use and/or by the destructive influence of measurement noise. Specifically, when the noise obeys a Poisson probability law, standard approaches to the problem of image reconstruction are based on using fixed-point algorithms which follow the methodology proposed by Richardson and Lucy in the beginning of the 1970s. The practice of using such methods, however, shows that their convergence properties tend to deteriorate at relatively high noise levels (which typically takes place in so-called low-count settings). This work introduces a novel method for de-noising and/or de-blurring of digital images that have been corrupted by Poisson noise. The proposed method is derived using the framework of MAP estimation, under the assumption that the image of interest can be sparsely represented in the domain of a properly designed linear transform. Consequently, a shrinkage-based iterative procedure is proposed, which guarantees the maximization of an associated maximum-a-posteriori criterion. It is shown in a series of both computer-simulated and real-life experiments that the proposed method outperforms a number of existing alternatives in terms of stability, precision, and computational efficiency.

The problem of reconstruction of digital images from their degraded measurements is regarded as a problem of central importance in various fields of engineering and imaging sciences. In such cases, the degradation is typically caused by the resolution limitations of an imaging device in use and/or by the destructive influence of measurement noise. Specifically, when the noise obeys a Poisson probability law, standard approaches to the problem of image reconstruction are based on using fixed-point algorithms which follow the methodology proposed by Richardson and Lucy in the beginning of the 1970s. The practice of using such methods, however, shows that their convergence properties tend to deteriorate at relatively high noise levels (which typically takes place in so-called low-count settings). This work introduces a novel method for de-noising and/or de-blurring of digital images that have been corrupted by Poisson noise. The proposed method is derived using the framework of MAP estimation, under the assumption that the image of interest can be sparsely represented in the domain of a properly designed linear transform. Consequently, a shrinkage-based iterative procedure is proposed, which guarantees the maximization of an associated maximum-a-posteriori criterion. It is shown in a series of both computer-simulated and real-life experiments that the proposed method outperforms a number of existing alternatives in terms of stability, precision, and computational efficiency.

Images are liable to being corrupted by noise when they are processed for many applications such as sampling, storage and transmission. In this thesis, we propose a method of image restoration for image corrupted by a white Gaussian noise. This method is based upon Gauss-Markov random field model combined with a technique of image segmentation. As a result, the image can be restored by MAP estimation. In the approach of Gauss-Markov random field model, the image is restored by MAP estimation implemented by simulated annealing or deterministic search methods. By image segmentation, the region parameters and the power of generating noise can be obtained for every region. The above parameters are important for MAP estimation of the Gauss-Markov Random field model. As a summary, we first segment the image to find the important region parameters and then restore the image by MAP estimation with using the above region parameters. Finally, the intermediate image is restored again by the conventional Gauss-Markov random field model method. The advantage of our method is the clear edges by the first restoration and deblured images by the second restoration.

The restoration of degraded images is one important application of image processing. The classical approach of image restoration, such as low-pass filter method, is usually stressed on the numerical error but with a disadvantage in visual quality of blurred texture. Therefore, a new method of image restoration, based upon image model by Compound Gauss-Markov(CGM) Random Fields, using MAP(maximum a posteriori probability) approach focused on image texture effect has been proved to be helpful. However, the contour of the restored image and numerical error for the method is poor because the conventional CGM model uses fixed global parameters for the whole image. To improve these disadvantages, we adopt the adjustable parameters method to estimate model parameters and restore the image. But the parameter estimation for the CGM model is difficult since the CGM model has 80 interdependent parameters. Therefore, we first adopt the parameter reduction approach to reduce the complexity of parameter estimation. Finally, the initial value set of the parameters is important. The different initial value might produce different results. The experiment results show that the proposed method using adjustable parameters has good numerical error and visual quality than the conventional methods using fixed parameters.