L'algorithme de Levenberg-Marquardt (LM) est parmi les algorithmes les plus populaires pour la résolution des problèmes des moindres carrés non linéaire. Motivés par la structure des problèmes de l'assimilation de données, nous considérons dans cette thèse l'extension de l'algorithme LM aux situations dans lesquelles le sous problème linéarisé, qui a la forme min||Ax - b ||^2, est résolu de façon approximative, et/ou les données sont bruitées et ne sont précises qu'avec une certaine probabilité. Sous des hypothèses appropriées, on montre que le nouvel algorithme converge presque sûrement vers un point stationnaire du premier ordre. Notre approche est appliquée à une instance dans l'assimilation de données variationnelles où les modèles stochastiques du gradient sont calculés par le lisseur de Kalman d'ensemble (EnKS). On montre la convergence dans L^p de l'EnKS vers le lisseur de Kalman, quand la taille de l'ensemble tend vers l'infini. On montre aussi la convergence de l'approche LM-EnKS, qui est une variante de l'algorithme de LM avec l'EnKS utilisé comme solveur linéaire, vers l'algorithme classique de LM ou le sous problème est résolu de façon exacte. La sensibilité de la méthode de décomposition en valeurs singulières tronquée est étudiée. Nous formulons une expression explicite pour le conditionnement de la solution des moindres carrés tronqués. Cette expression est donnée en termes de valeurs singulières de A et les coefficients de Fourier de b. / The Levenberg-Marquardt algorithm (LM) is one of the most popular algorithms for the solution of nonlinear least squares problems. Motivated by the problem structure in data assimilation, we consider in this thesis the extension of the LM algorithm to the scenarios where the linearized least squares subproblems, of the form min||Ax - b ||^2, are solved inexactly and/or the gradient model is noisy and accurate only within a certain probability. Under appropriate assumptions, we show that the modified algorithm converges globally and almost surely to a first order stationary point. Our approach is applied to an instance in variational data assimilation where stochastic models of the gradient are computed by the so-called ensemble Kalman smoother (EnKS). A convergence proof in L^p of EnKS in the limit for large ensembles to the Kalman smoother is given. We also show the convergence of LM-EnKS approach, which is a variant of the LM algorithm with EnKS as a linear solver, to the classical LM algorithm where the linearized subproblem is solved exactly. The sensitivity of the trucated sigular value decomposition method to solve the linearized subprobems is studied. We formulate an explicit expression for the condition number of the truncated least squares solution. This expression is given in terms of the singular values of A and the Fourier coefficients of b.

Master’s thesis describes adaptive optimal controller design and it’s settings. Identification with principles of artificial intelligence and recursive least squares identification with exponential and directional forgetting are compared separately and as part of controller. Adaptive optimal controller is tested on physical model and compared with solidly adjusted PSD controller. Possibilities of implementation of adaptive optimal controller into programmable logic controller B&R are show and tested.

<p>A neural network model was developed to predict the E. Coli levels and classes in six (6) select Lake Michigan beaches. Water quality observations at the time of sampling and discharge information from two close tributaries were used as input to predict the E. coli. This research was funded by the Indiana Department of Environmental Management (IDEM). A user-friendly Excel Sheet based tool was developed based on the best model for making future predictions of E. coli classes. This tool will facilitate beach managers to take real-time decisions.</p> <p>The nowcast model was developed based on historical tributary flows and water quality measurements (physical, chemical and biological). The model uses experimentally available information such as total dissolved solids, total suspended solids, pH, electrical conductivity, and water temperature to estimate whether the E. Coli counts would exceed the acceptable standard. For setting up this model, field data collection was carried out during 2019 beachgoer’s season.</p> <p>IDEM recommends posting an advisory at the beach indicating swimming and wading are not recommended when E. coli counts exceed advisory standards. Based on the advisory limit, a single water sample shall not exceed an E. Coli count of 235 colony forming units per 100 milliliters (cfu/100ml). Advisories are removed when bacterial levels fall within the acceptable standard. However, the E. coli results were available after a time lag leading to beach closures from previous day results. Nowcast models allow beach managers to make real-time beach advisory decisions instead of waiting a day or more for laboratory results to become available.</p> <p>Using the historical data, an extensive experiment was carried out, to obtain the suitable input variables and optimal neural network architecture. The best feed-forward neural network model was developed using Bayesian Regularization Neural Network (BRNN) training algorithm. Developed ANN model showed an average prediction accuracy of around 87% in predicting the E. coli classes. </p>