L'étude théorique de la transition vers la turbulence d'écoulements en tuyau de fluides non newtoniens rhéofluidifiants (fluides de Carreau) est menée, avec l'approche consistant à calculer des «~structures très cohérentes~» sous la forme d'«~ondes non linéaires~». Pour cela un code pseudo-spectral de type Petrov-Galerkin, permettant de suivre des solutions ondes non linéaires tridimensionnelles dans l'espace des paramètres par continuation, est développé. Ce code est validé par comparaison à des résultats existants en fluide newtonien, et grâce à un test de consistance en fluide non newtonien. Une convergence spectrale exponentielle est obtenue dans tous les cas. Ce code est utilisé pour chercher (guidé par des résultats expérimentaux récents) de nouvelles solutions de nombre d'onde azimutal fondamental égal à 1, sans succès pour l'instant. Par contre des solutions de nombre d'onde azimutal fondamental égal à 2 ou 3 sont obtenues par continuation à partir du cas newtonien. La rhéofluidification induit, en termes de nombres de Reynolds critiques, un retard à l'apparition de ces ondes par rapport au cas newtonien. Ce retard est caractérisé, et le parallèle est fait avec divers résultats expérimentaux qui montrent un retard à l'apparition de bouffées turbulentes en fluides non newtoniens / The transition to turbulence in pipe flows of shear-thinning fluids is studied theoretically. The method used is the computation of `exact coherent structures' that are tridimensional nonlinear waves. For this purpose a pseudo-spectral Petrov-Galerkin code is developped, which also allows to follow solution branches in the parameter space with continuation methods. This code is validated by recovering already published results in the Newtonian case, and by a consistency test in the non-Newtonian case. A spectral exponential convergence is obtained in all cases. This code is used to seek (guided by recent experimental results) new solutions of fundamental azimuthal wavenumber equal to 1,without success at the time being. On the contrary solutions with a fundamental azimuthal wavenumber equal to 2 and 3 are obtained by continuation from the Newtonian case. The shear-thinning effects induce, in terms of critical Reynolds numbers, a delay for the onset of these waves, as compared with the Newtonian case. This delay is characterized. An analogy is made with various experimental results that show a delay in the transition to turbulence, more precisely, in the onset of `puffs', in non-Newtonian fluids

L'objectif de ce travail de thèse est de développer un modèle mathématique pour l'étude et la compréhension de la dynamique des populations d'un insecte ravageur, l'Eudémis de la vigne (Lobesia botrana Den. & Schiff.), dans son écosystème. Le modèle proposé est un système d'équations aux dérivées partielles de type hyperbolique qui décrit les variations numériques au cours du temps de la population en fonction des stades de développement, du sexe des individus et des conditions environnementales. La ressource alimentaire, la température, l'humidité et la prédation sont les principaux facteurs environnementaux du modèle expliquant les fluctuations du nombre d'individus au cours du temps. Les différences de développement qui existent dans une cohorte d'Eudémis sont aussi modélisées pour affiner les prédictions du modèle. A partir de données expérimentales obtenues par les entomologistes de l'INRA, situé à Bordeaux, les paramètres du modèle sont estimés. Ce modèle ainsi ajusté nous permet alors d’étudier quelques aspects biologiques et écologiques de l’insecte comme par exemple l'impact de scénarios climatiques sur la ponte des femelles ou sur la dynamique d’attaque de la vigne par les jeunes larves. Les analyses mathématique et numérique du modèle mathématique et des problèmes d'estimation des paramètres sont développées dans cette thèse. / The objective of the thesis is to develop a mathematical model for studying the population dynamics of the European grapevine moth (Lobesia botrana Den. & Schiff.) in its ecosystem. The model proposed is a system of hyperbolic equations that describe the numerical variations in time of the population with respect to developmental stage, the gender and the environmental conditions. The food, the temperature, the humidity and the predation are the main environmental factors of the model that explain the fluctuations of the population in time. The differences in growth inside a cohort are modeled in order to precise the model simulations. We use experimental data obtained by entomologists of the National Research Institut of Agronomy to estimate the parameters of the model. This ajusted model allows us to study some biological and ecological aspects of this pest like for example the impact of climate change on the female laying or on the young larvae dynamic, main actors in the depredation of the Vine. The mathematical analysis and the numerical analysis of the mathematical model and of the parameters estimation problems are presented in this thesis.

Ce travail de thèse concerne la mécanique numérique du contact entre solides déformables. Il s'agit de contribuer à l'amélioration de la performance, de la fiabilité et de la robustesse des algorithmes et des modèles numériques utilisés dans les codes éléments finis en particulier Code_Aster qui est un code libre développé par Électricité De France (EDF) pour ses besoins en ingénierie. L'objectif final est de traiter les problèmes industriels de grande dimension avec un temps de calcul optimisé. Pour parvenir à ces objectifs, les algorithmes et formulations doivent prendre en compte les difficultés liées à la mécanique non régulière à cause des lois de Signorini-Coulomb ainsi que la gestion des non linéarités dûes aux grandes déformations et aux comportements des matériaux étudiés.Le premier axe de ce travail est dédié à une meilleure compréhension de la formulation dite de « Lagrangien stabilisé » initialement implémentée dans le code. Il a été démontré l'équivalence entre cette formulation et la formulation bien connue de « Lagrangien augmenté ». Les caractéristiques mathématiques liées aux opérateurs discrets ont été précisées et une écriture énergétique globale a été trouvée. / This work deals with computational contact mechanics between deformable solids. The aim of this work is to improve the performance, the reliability and the robustness of the algorithms and numerical models set in Code_Aster which is finite element code developped by Électricité De France (EDF) for its engineering needs. The proposed algorithms are used to solve high dimensional industrial problems in order to optimize the computational running times. Several solutions techniques are available in the field of computational contact mechanics but they must take into account the difficulties coming from non-smooth aspects due to Signorini-Coulomb laws coupled to large deformations of bodies and material non linearities. Firstly the augmented Lagrangian formulation so-called « stabilized Lagrangian » is introduced. Successively, the mathematical properties of the discrete operators are highlighted and furthermore a novel energetic function is presented. Secondly the kinematical condition with regard to the normal unknowns are reinforced through unconstrained optimization techniques which result to a novel formulation which is so-called « non standard augmented Lagrangian formulation ». Three types of strategies are implemented in the code. The generalized Newton method is developped : it is a method in which all the non linearities are solved in one loop of iterations. The partial Newton method is an hybrid technique between the generalized Newton one and a fixed point method.

Este trabalho tem por objetivo estudar e analisar a influência do parâmetro de barreira e de seu fator de correção no processo de convergência dos métodos de pontos interiores primal-dual, primal-dual barreira modificada e primal-dual barreira modificada com as técnicas preditor-corretor e Newton composto. A grande motivação para o desenvolvimento desta pesquisa está relacionada com a busca de métodos eficientes para resolver problemas de otimização de programação não-linear, existentes na área de engenharia elétrica mais especificamente na operação de sistemas elétricos de potência. Esses métodos foram aplicados a um problema de programação não-linear e aos sistemas elétricos de três e de trinta barras para analisar a sensibilidade em relação ao parâmetro de barreira e ao seu fator de correção. / This work has for objective to study and to analyze the influence of the barrier parameter and its correction factor in the convergence process of the methods primal-dual interior point, primal-dual modified barrier and primal-dual barrier modified with the techniques predictor-corrector and composed Newton. The great motivation for the development of this research is related with the search of efficient methods to solve nonlinear programming optimization problems, existent in the area of electric engineering more specifically in the operation of power systems. Those methods were applied to a nonlinear programming problem and the electric systems of three and thirty buses to analyze the sensibility in relation to the barrier parameter and its correction factor.

Este trabalho tem por objetivo estudar e analisar a influência do parâmetro de barreira e de seu fator de correção no processo de convergência dos métodos de pontos interiores primal-dual, primal-dual barreira modificada e primal-dual barreira modificada com as técnicas preditor-corretor e Newton composto. A grande motivação para o desenvolvimento desta pesquisa está relacionada com a busca de métodos eficientes para resolver problemas de otimização de programação não-linear, existentes na área de engenharia elétrica mais especificamente na operação de sistemas elétricos de potência. Esses métodos foram aplicados a um problema de programação não-linear e aos sistemas elétricos de três e de trinta barras para analisar a sensibilidade em relação ao parâmetro de barreira e ao seu fator de correção. / This work has for objective to study and to analyze the influence of the barrier parameter and its correction factor in the convergence process of the methods primal-dual interior point, primal-dual modified barrier and primal-dual barrier modified with the techniques predictor-corrector and composed Newton. The great motivation for the development of this research is related with the search of efficient methods to solve nonlinear programming optimization problems, existent in the area of electric engineering more specifically in the operation of power systems. Those methods were applied to a nonlinear programming problem and the electric systems of three and thirty buses to analyze the sensibility in relation to the barrier parameter and its correction factor.

Cette thèse est consacrée à la recherche des zéros d'un opérateur maximal monotone structuré, à l'aide de systèmes dynamiques dissipatifs continus et discrets. Les solutions sont obtenues comme limites des trajectoires lorsque le temps t tend vers l'infini. On s'intéressera principalement aux dynamiques obtenues par régularisation de type Levenberg-Marquardt de la méthode de Newton. On décrira aussi les approches basées sur des dynamiques voisines.Dans un cadre Hilbertien, on s'intéresse à la recherche des zéros de l'opérateur maximal monotone structuré M = A + B, où A est un opérateur maximal monotone général et B est un opérateur monotone Lipschitzien. Nous introduisons des dynamiques continues et discrètes de type Newton régularisé faisant intervenir d'une façon séparée les résolvantes de l'opérateur A (implicites), et des évaluations de B (explicites). A l'aide de la représentation de Minty de l'opérateur A comme une variété Lipschitzienne, nous reformulons ces dynamiques sous une forme relevant du théorème de Cauchy-Lipschitz. Nous nous intéressons au cas particulier où A est le sous différentiel d'une fonction convexe, semi-continue inférieurement, et propre, et B est le gradient d'une fonction convexe, différentiable. Nous étudions le comportement asymptotique des trajectoires. Lorsque le terme de régularisation ne tend pas trop vite vers zéro, et en s'appuyant sur une analyse asymptotique de type Lyapunov, nous montrons la convergence des trajectoires. Par ailleurs, nous montrons la dépendance Lipschitzienne des trajectoires par rapport au terme de régularisation.Puis nous élargissons notre étude en considérant différentes classes de systèmes dynamiques visant à résoudre les inclusions monotones gouvernées par un opérateur maximal monotone structuré M = $partialPhi$+ B, où $partialPhi$ désigne le sous différentiel d'une fonction convexe, semicontinue inférieurement, et propre, et B est un opérateur monotone cocoercif. En s'appuyant sur une analyse asymptotique de type Lyapunov, nous étudions le comportement asymptotique des trajectoires de ces systèmes. La discrétisation temporelle de ces dynamiques fournit desalgorithmes forward-backward (certains nouveaux ).Finalement, nous nous intéressons à l'étude du comportement asymptotique des trajectoires de systèmes dynamiques de type Newton régularisé, dans lesquels on introduit un terme supplémentaire de viscosité évanescente de type Tikhonov. On obtient ainsi la sélection asymptotique d'une solution de norme minimale. / This thesis is devoted to finding zeroes of structured maximal monotone operators, by using discrete and continuous dissipative dynamical systems. The solutions are obtained as the limits of trajectories when the time t tends towards infinity.We pay special attention to the dynamics that are obtained by Levenberg-Marquardt regularization of Newton's method. We also revisit the approaches based on some related dynamical systems.In a Hilbert framework, we are interested in finding zeroes of a structured maximal monotone operator M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. We introduce discrete and continuous dynamical systems which are linked to Newton's method. They involve separately B and the resolvents of A, and are designed to splitting methods. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem. We focus on the particular case where A is the subdifferential of a convex lower semicontinuous proper function, and B is the gradient of a convex, continuously differentiable function. We study the asymptotic behavior of trajectories. When the regularization parameter does not tend to zero too rapidly, and by using Lyapunov asymptotic analysis, we show the convergence of trajectories. Besides, we show the Lipschitz continuous dependence of the solution with respect to the regularization term.Then we extend our study by considering various classes of dynamical systems which aim at solving inclusions governed by structured monotone operators M = $partialPhi$+ B, where $partialPhi$ is the subdifferential of a convex lower semicontinuous function, and B is a monotone cocoercive operator. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splittingmethods (some new).Finally, we focus on the study of the asymptotic behavior of trajectories of the regularized Newton dynamics, in which we introduce an additional vanishing Tikhonov-like viscosity term.We thus obtain the asymptotic selection of the solution of minimal norm.

<p>The focus of this thesis is on numerical algorithms for efficient solution of QTL analysis problem in genetics.</p><p>Firstly, we consider QTL mapping problems where a standard least-squares model is used for computing the model fit. We develop optimization methods for the local problems in a hybrid global-local optimization scheme for determining the optimal set of QTL locations. Here, the local problems have constant bound constraints and may be non-convex and/or flat in one or more directions. We propose an enhanced quasi-Newton method and also implement several schemes for constrained optimization. The algorithms are adopted to the QTL optimization problems. We show that it is possible to use the new schemes to solve problems with up to 6 QTLs efficiently and accurately, and that the work is reduced with up to two orders magnitude compared to using only global optimization.</p><p>Secondly, we study numerical methods for QTL mapping where variance component estimation and a REML model is used. This results in a non-linear optimization problem for computing the model fit in each set of QTL locations. Here, we compare different optimization schemes and adopt them for the specifics of the problem. The results show that our version of the active set method is efficient and robust, which is not the case for methods used earlier. We also study the matrix operations performed inside the optimization loop, and develop more efficient algorithms for the REML computations. We develop a scheme for reducing the number of objective function evaluations, and we accelerate the computations of the derivatives of the log-likelihood by introducing an efficient scheme for computing the inverse of the variance-covariance matrix and other components of the derivatives of the log-likelihood.</p>

The focus of this thesis is on numerical algorithms for efficient solution of QTL analysis problem in genetics. Firstly, we consider QTL mapping problems where a standard least-squares model is used for computing the model fit. We develop optimization methods for the local problems in a hybrid global-local optimization scheme for determining the optimal set of QTL locations. Here, the local problems have constant bound constraints and may be non-convex and/or flat in one or more directions. We propose an enhanced quasi-Newton method and also implement several schemes for constrained optimization. The algorithms are adopted to the QTL optimization problems. We show that it is possible to use the new schemes to solve problems with up to 6 QTLs efficiently and accurately, and that the work is reduced with up to two orders magnitude compared to using only global optimization. Secondly, we study numerical methods for QTL mapping where variance component estimation and a REML model is used. This results in a non-linear optimization problem for computing the model fit in each set of QTL locations. Here, we compare different optimization schemes and adopt them for the specifics of the problem. The results show that our version of the active set method is efficient and robust, which is not the case for methods used earlier. We also study the matrix operations performed inside the optimization loop, and develop more efficient algorithms for the REML computations. We develop a scheme for reducing the number of objective function evaluations, and we accelerate the computations of the derivatives of the log-likelihood by introducing an efficient scheme for computing the inverse of the variance-covariance matrix and other components of the derivatives of the log-likelihood.

We present a new method for the computation of low rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low rank property of the right hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Hence, it provides a possibility to tackle large scale control problems. Besides the efficient solution of the matrix equation itself, a thorough integration of the method into several control algorithms can improve their performance to a high degree. This is demonstrated for algorithms for model reduction and optimal control. Furthermore, we propose a heuristic for determining a set of suboptimal ADI shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters.

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