Generalized solutions for the Euler equations in one and two dimensions

International audienceIn this paper we study generalized solutions (in the Brenier's sense) for the Euler equations. We prove that uniqueness holds in dimension one whenever the pressure field is smooth, while we show that in dimension two uniqueness is far from being true. In the case of the two-dimensional disc we study solutions to Euler equations where particles located at a point $x$ go to $-x$ in a time $\pi$, and we give a quite general description of the (large) set of such solutions. As a byproduct, we can construct a new class of classical solutions to Euler equations in the disc

Transport optimal et irrigation

This thesis models irrigation structures such as leaves venation, blood veins, lungs, etc. A model generalizing Gilbert-Steiner problem is given. We then study existence properties, stability and regularity. Algorithms are then given for simulation.L'objet de cette thèse est de modéliser et d'étudier des structures d'irrigation telles les nervures des feuilles, réseau sanguin, poumons,etc. Un modèle généralisant le problème de Gilbert Steiner est introduit ; on étudie alors les propriétés d'existence, de stabilité et régularité. Des algorithmes sont alors proposés pour la simulation

Synchronized traffic plans and stability of optima

The irrigation problem is the problem of finding an efficient way to transport a measure μ+ onto a measure μ-. By efficient, we mean that a structure that achieves the transport (which, following [Bernot, Caselles and Morel, Publ. Mat. 49 (2005) 417–451], we call traffic plan) is better if it carries the mass in a grouped way rather than in a separate way. This is formalized by considering costs functionals that favorize this property. The aim of this paper is to introduce a dynamical cost functional on traffic plans that we argue to be more realistic. The existence of minimizers is proved in two ways: in some cases, we can deduce it from a classical semicontinuity argument; the other cases are treated by studying the link between our cost and the one introduced in [Bernot, Caselles and Morel, Publ. Mat. 49 (2005) 417–451]. Finally, we discuss the stability of minimizers with respect to specific variations of the cost functional

Model Discrepancy in Robotic Calibration: Its Influence on the Experimental Parameter Identification of a Parallel Space Telescope

International audienceThe model of a robot may not be able to consider all the physical phenomena influencing the manipulator performances since they are too numerous and/or difficult to measure: this is model discrepancy. For a highly-accurate active space telescope, an important source of inaccuracy was measured using photogrammetry: the deformation of its mobile platform. This deformation cannot be directly measured in space and needs to be properly modeled in order to enable the telescope calibration with the available measurements. Two incremental models are proposed and the parameter observability is discussed. After experimental calibration, a micrometer accuracy can be reached. The influence of model discrepancy on the experimental parameter identification is finally discussed

Optimal transportation networks: models and theory

The transportation problem can be formalized as the problem of finding the optimal way to transport a given measure into another with the same mass. In contrast to the Monge-Kantorovitch problem, recent approaches model the branched structure of such supply networks as minima of an energy functional whose essential feature is to favour wide roads. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electrical power supply systems and in natural counterparts such as blood vessels or the branches of trees. These lectures provide mathematical proof of several existence, structure and regularity properties empirically observed in transportation networks. The link with previous discrete physical models of irrigation and erosion models in geomorphology and with discrete telecommunication and transportation models is discussed. It will be mathematically proven that the majority fit in the simple model sketched in this volume

Non-Parametric Online Change-Point Detection with Kernel LMS By Relative Density Ratio Estimation

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