Conservation laws for under determined systems of differential equations

This work extends the Ibragimov's conservation theorem for partial differential equations [{\it J. Math. Anal. Appl. 333 (2007 311-328}] to under determined systems of differential equations. The concepts of adjoint equation and formal Lagrangian for a system of differential equations whose the number of equations is equal to or lower than the number of dependent variables are defined. It is proved that the system given by an equation and its adjoint is associated with a variational problem (with or without classical Lagrangian) and inherits all Lie-point and generalized symmetries from the original equation. Accordingly, a Noether theorem for conservation laws can be formulated

An automatic procedure to select a block size in the continuous generalized extreme value model estimation

The block maxima approach is one of the main methodologies in extreme value theory to obtain a suitable distribution to estimate the probability of large values. In this approach, the block size is usually selected in order to reflect the possible intrinsic periodicity of the studied phenomenon. The generalization of this approach to data from non-seasonal phenomena is not straightforward. To address this problem, we propose an automatic data-driven method to identify the block size to use in the generalized extreme value (GEV) distribution for extrapolation. This methodology includes the validation of sufficient theoretical conditions ensuring that the maximum term converges to the GEV distribution. The selected GEV model can be different from the GEV model fitted on a sample of block maxima from arbitrary large block size. This selected GEV model has the special property to associate high values of the underlining variable with the corresponding smallest return periods. Such a model is useful in practice as it allows, for example, a better sizing of certain structures of protection against natural disasters. To illustrate the developed method, we consider two real datasets. The first dataset contains daily observations over several years from some meteorological variables while the second dataset contains data observed at millisecond time scale over several minutes from sensors in the field of vehicle engineering

Méthodes analytiques pour résoudre des modèles non linéaires et des problèmes variationnels

This work substantially deals with our contribution to the theory of nonlinear systems of partialdifferential equations and variational calculus.Direct methods and symmetry analysis for investigation of analytical exact or approximate solutions of linear as well as nonlinear models are performed. We start by presenting the factorization method for linear and nonlinear partial differential equations as well as for their systems and give necessary and sufficient conditions of factorization in the particular case of second order equations. Then, we provide with three classes of factorizable second order linear ordinary differential equations together with their solutions. We come up with the generalization of the technique of finding exact solutions to nonlinear differential equations as an expansion of well-known functions. Known previous results are recovered as particular cases. The invariance of some differential equations with respect to certain symmetry transformation groups is a property which plays a considerable role in the study of these equations. The procedure to find symmetry groups and to construct group invariant solutions for systems of differential equations is described in this document with application to the family of fifth order KdV equations. It is sometimes difficult to handle exact solutions for differential equations. In this case, analytical approximate solutions can be computed by known powerful procedures, among which the so-called Adomian decomposition method plays a significant role. We make a detailed description of the latter method and perform its extension to the class of under determined systems of nonlinear differential equations. An illustration to the famous GinzburgLandau system is probed.For partial differential equations admitting Lie symmetry groups, one can also construct their conserved quantities. Conservation laws play an important role in science. Another aim of this thesis is to provide an overview on this topic and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether theorem. This approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be int oduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations, nonlocal conservation laws can be transformed into local ones. We extend this approach, only developed in the existing literature for determined systems, to under determined systems of partial differential equations possessing symmetry groups. An alternative direct algebraic method of constructing, for nonlinear evolution partial differential equations, conservation laws that may depend not only on dependent variables and its derivatives but also explicitly on independent variables is also proposed. This procedure starts by attributing to the differential equation a scaling symmetry group and then judiciously exploits the EulerLagrange and the homotopy operators to compute conserved quantities (conserved density and its related flux). Several examples on constructing conservation laws for some well-known equations are provided.Differential equations also appear in the study of optimization problems defined by a variational integral. Classical solutions to minimization problems in the calculus of variations are prescribed by boundary value problems involving certain types of differential equations, known as the associated Euler-Lagrange equations. The mathematical techniques that have been developed to handle such optimization problems are fundamental in many areas of mathematics, physics, engineering, and other applications. We present the basic mathematical analysis of nonlinear minimization principles on function spaces. The solutions to the Euler-Lagrange equations may also include (local) maxima, as well as other non-extremum critical functions. Our investigation of the second variation of a functional leads to the construction of an associated matrix. We show that the second order necessary and sufficient conditions that a function should satisfy to be either a minimum or a maximum are, in general, characterized by the properties of this matrix.Ce travail présente substantiellement notre contribution à la théorie des équations aux dérivées partielles non linéaires et à la théorie du calcul des variations. Des méthodes directes et d’analyse des symétries pour l’étude des solutions analytiques exactes ou approchées des modèles non linéaires ont été élaborées. De manière plus précise, nous avons développé une méthode de factorisation des systèmes d’équations aux dérivées partielles non linéaires et donné les conditions nécessaires et suffisantes de factorisation dans le cas particulier des systèmes du second ordre. Ensuite, nous avons fourni, ensemble avec leurs solutions, trois classes d’équations différentielles ordinaires linéaires factorisables du second ordre. Une généralisation de la technique de construction des solutions des équations aux dérivées partielles non linéaires comme une expansion de fonctions bien connues a été aussi faite. La procédure pour calculer les groupes de symétrie des systèmes d’équations aux dérivées partielles et pour construire des solutions qui sont invariantes par ces groupes a été décrite dans ce document. Des solutions analytiques approchées ont été calculées par la méthode de décomposition d’Adomian. Une description détaillée de cette dernière méthode et son extension à la classe des systèmes d’équations différentielles sous déterminées ont été présentées et illustrées par le système de Ginzburg-Landau. Cette thèse présente aussi des méthodes de construction des lois de conservation utilisant la théorie des groupes de Lie. Cette nouvelle approche adoptée permet l’obtention des lois de conservation même pour des équations différentielles sans Lagrangien. Une méthode algébrique alternative de construction directe, pour des équations aux dérivées partielles nonlinéaires évolutives, des lois de conservations a été proposée. Nous avons construit à titre d’examples plusieurs lois de conservations pour des équations bien connues. Nous avons enfin montré que les conditions nécessaires et suffisantes du second ordre qu’une fonction critique doit satisfaire pour être soit un minimum, soit un maximum sont, en générale, caractérisées par les propriétés d’une matrice de type Hessienne

Détection d'objets de scène de trafic à l'aide de l'algorithme YOLO: Théorie et guide pratique

The main goal of this report is to build a detector of all objects present in the surrounding of an autonomous vehicle on roads. We achieved this goal by means of a deep learning algorithm called YOLO (You Only Look Once) in the field of computer vision. The basic concepts in object detection as well as the core of the YOLO algorithm are recalled in this study. Our machine learning modeling operations consist of the three following steps. Firstly, we collect a large dataset of images containing 35 classes of traffic scene objects. Secondly, we annotate the collected images in the YOLO format. Thirdly, we feed a model from the YOLO family with the annotated dataset in order to estimate its parameters. The obtained model has pretty good predictive performance and can be used to extract in real-time all information associated with the external driving environment from videos taken by cameras embedded in an autonomous vehicle. Traffic scene elements extracted by such a detector can act as covariates in reliability analysis of automated driving systems consisting to check whether a safety requirement is satisfied in an operational design domain

Estimation of the extrapolation range associated with extreme-value models: Application to the assessment of sensors reliability

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