Publications

2012

• Temps de transitions métastables pour des systèmes dynamiques stochastiques fini et infini-dimensionnels
  
  • Barret Florent
  , 2012. Dans cette thèse, nous nous sommes intéressés à la métastabilité de certains systèmes dynamiques stochastiques. Plus précisément, nous avons étudié des équations différentielles ou des équations aux dérivées partielles perturbées par un bruit blanc additif dans l'asymptotique du bruit faible. Nous avons donné l'expression et le calcul de l'espérance de temps des transitions métastables pour certains types de modèles (formule dite d'Eyring-Kramers). Dans un premier temps, nous avons généralisé des résultats connus pour des diffusions d'Itô dont la dérive est le gradient d'un potentiel. Nous donnons une équivalence entre la géométrie du paysage décrit par le potentiel et des circuits électriques qui nous permet de donner des expressions simples pour le calcul des temps de transition entre des minima du potentiel. Nous utilisons la théorie du potentiel et les capacités dans le calcul de ces temps. Le principal résultat de cette thèse concerne des équations aux dérivées partielles stochastiques scalaires, paraboliques, semi-linéaires et perturbées par un bruit blanc espace-temps sur un intervalle borné réel comme l'équation d'Allen-Cahn. Ce modèle constitue un analogue infini-dimensionnel aux diffusions en dimension finie. Nous avons considéré deux types de conditions au bord, Dirichlet et Neumann, et discutons le cas des conditions périodiques. Sous certaines hypothèses, nous donnons l'expression, analogue à la dimension finie, des temps transitions. La preuve utilise une discrétisation par différence finie de l'équation et un couplage nous permettant d'appliquer les estimations pour la dimension finie. Il a fallu notamment contrôler uniformément ces estimations en fonction de la dimension pour passer à la limite et récupérer le système infini-dimensionnel.
• Une interpolation entre un rectangle et un tore
  
  • Colonna Jean-François
  , 2012. An interpolation between a rectangle and a torus (Une interpolation entre un rectangle et un tore)
• A non-iterative sampling approach using noise subspace projection for EIT
  
  • Bellis Cédric
  • Constantinescu Andrei
  • Coquet Thomas
  • Jaravel Thomas
  • Lechleiter Armin
  Inverse Problems, IOP Publishing, 2012. This study concerns the problem of the reconstruction of inclusions embedded in a conductive medium in the context of electrical impedance tomography (EIT), which is investigated within the framework of a non-iterative sampling approach. This type of identification strategy relies on the construction of a special indicator function that takes, roughly speaking, small values outside the inclusion and large values inside. Such a function is constructed in this paper from the projection of a fundamental singular solution onto the space spanned by the singular vectors associated with some of the smallest singular values of the data-to-measurement operator. The behavior of the novel indicator function is analyzed. For a subsequent implementation in a discrete setting, the quality of classical finite-dimensional approximations of the measurement operator is discussed. The robustness of this approach is also analyzed when only noisy spectral information is available. Finally, this identification method is implemented numerically and experimentally, and its efficiency is discussed on a set of, partly experimental, examples. (10.1088/0266-5611/28/7/075015)
  DOI : 10.1088/0266-5611/28/7/075015
• Spectral volumetric integral equation methods for acoustic medium scattering in a 3D waveguide
  
  • Lechleiter Armin
  • Nguyen Dinh Liem
  IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2012, 32 (3), pp.813-844. The Lippmann–Schwinger integral equation describes the scattering of acoustic waves from an inhomogeneous medium. For scattering problems in free space, Vainikko proposed a fast spectral solution method exploiting the convolution structure of this equation's integral operator and the fast Fourier transform. Although the integral operator of the Lippmann–Schwinger integral equation for scattering in a planar three-dimensional waveguide is not a convolution, we show in this paper that the separable structure of the kernel allows to construct fast spectral collocation methods. The numerical analysis of this method requires smooth material parameters; for discontinuous materials there is no theoretical convergence statement. Therefore, we construct a Galerkin variant of Vainikko's method avoiding this drawback. For several distant scattering objects inside the three-dimensional waveguide this discretization technique would lead to a computational domain consisting of one large box containing all scatterers and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects. (10.1093/imanum/drr036)
  DOI : 10.1093/imanum/drr036
• Reachability of Delayed Hybrid Systems Using Level-set Methods
  
  • Granato Giovanni
  , 2012. This study proposes an algorithm to synthesize controllers for the power management on board hybrid vehicles that allows the vehicle to reach its maximum range along a given route. The algorithm stems from a level-set approach that computes the reachable set of the system, i.e., the collection of states reachable from a certain initial condition via the computation of the value function of an optimal control problem. The discrete-time vehicle model is one of a particular class of hybrid vehicles, namely, range extender electric vehicles (REEV). This kind of hybridization departures from a full electric vehicle that has an additional module -- the range extender (RE) -- as an extra energy source in addition to its main energy source -- a high voltage battery. As an important feature, our model allows for the switching on and off of the range extender and includes a decision lag constraint, i.e., imposes two consecutive switches to be separated by a positive time interval. The approach consists in the introduction of an adequate optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a dynamic programming algorithm. This algorithm is implemented and some numerical examples are presented.
• A simple preconditioned domain decomposition method for electromagnetic scattering problems
  
  • Alouges François
  • Bourguignon-Mirebeau Jennifer
  • Levadoux David P.
  , 2012. We present a domain decomposition method (DDM) devoted to the iterative solution of time-harmonic electromagnetic scattering problems, involving large and resonant cavities. This DDM uses the electric field integral equation (EFIE) for the solution of Maxwell problems in both interior and exterior subdomains, and we propose a simple preconditioner for the global method, based on the single layer operator restricted to the fictitious interface between the two subdomains.
• Fast Relaxation Solvers for Hyperbolic-Elliptic Phase Transition Problems
  
  • Chalons Christophe
  • Coquel Frédéric
  • Engel Patrick
  • Rohde Christian
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (3), pp.http://dx.doi.org/10.1137/110848815. Phase transition problems in compressible media can be modelled by mixed hyperbolicelliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. In recent years several tracking-type algorithms have been suggested for numerical approximation. Typically a core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation at the location of phase boundaries. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel sub-iteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kinetic relation where exact Riemann solvers are available. For a nonlinear kinetic relation and a thermoelastic system we use the new method to gain information on the Riemann problem. Up to our knowledge an exact solution for arbitrary Riemann data is currently not available in these cases.
• A class of two-fluid two-phase flow models
  
  • Coquel Frédéric
  • Hérard Jean-Marc
  • Saleh Khaled
  • Seguin Nicolas
  , 2012. We introduce a class of two-fluid models that complies with a few theoretical requirements that include: (i) hyperbolicity of the convective subset, (ii) entropy inequality, (iii) uniqueness of jump conditions for non-viscous flows. These specifications are necessary in order to compute relevant approximations of unsteady flow patterns. It is shown that the Baer-Nunziato model belongs to this class of two-phase flow models, and the main properties of the model are given, before showing a few numerical experiments. (10.2514/6.2012-3356)
  DOI : 10.2514/6.2012-3356
• Papillon quaternionique avec arithmétique étendue
  
  • Colonna Jean-François
  , 2012. Quaternionic butterfly with extended arithmetics (Papillon quaternionique avec arithmétique étendue)
• Papillon quaternionique avec arithmétique étendue
  
  • Colonna Jean-François
  , 2012. Quaternionic butterfly with extended arithmetics (Papillon quaternionique avec arithmétique étendue)
• Papillon quaternionique avec arithmétique étendue
  
  • Colonna Jean-François
  , 2012. Quaternionic butterfly with extended arithmetics (Papillon quaternionique avec arithmétique étendue)
• Papillon quaternionique avec arithmétique étendue
  
  • Colonna Jean-François
  , 2012. Quaternionic butterfly with extended arithmetics (Papillon quaternionique avec arithmétique étendue)
• Nonlinear historical superprocess approximations for population models with past dependence
  
  • Méléard Sylvie
  • Tran Viet Chi
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2012, 17 (47), pp.1-32. We are interested in the evolving genealogy of a birth and death process with trait structure and ecological interactions. Traits are hereditarily transmitted from a parent to its offspring unless a mutation occurs. The dynamics may depend on the trait of the ancestors and on its past and allows interactions between individuals through their lineages. We define an interacting historical particle process describing the genealogies of the living individuals; it takes values in the space of point measures on an infinite dimensional càdlàg path space. This individual-based process can be approximated by a nonlinear historical superprocess, under the assumptions of large populations, small individuals and allometric demographies. Because of the interactions, the branching property fails and we use martingale problems and fine couplings between our population and independent branching particles. Our convergence theorem is illustrated by two examples of current interest in biology. The first one relates the biodiversity history of a population and its phylogeny, while the second treats a spatial model with competition between individuals through their past trajectories. (10.1214/EJP.v17-2093)
  DOI : 10.1214/EJP.v17-2093
• Homogénéisation et analyse numérique d'équations elliptiques et paraboliques dégénérées.
  
  • Thouroude Gilles
  , 2012. Cette thèse comporte deux parties. Dans un premier temps, nous allons faire un lien entre des solutions stationnaires de problèmes d'évolutions de frontières par courbure moyenne avec des champs extérieurs et l'existence de minimiseur globaux d'un problème de minimisation de périmètre avec une énergie. Ces solutions stationnaires permettent en outre de fournir des bornes pour les solutions non stationnaires du problème. De plus, en modifiant l'énergie, on montre que les résolutions successives des problème de périmètre permettent de calculer l'évolution d'un ensemble par courbure moyenne. Enfin, on présentera un algorithme permettant de calculer les solutions de viscosité d'un problème de Dirichlet portant sur le Laplacien Infini grâce aux équations d'Aronsson.
• Optimization of the HOTS score of a website's pages
  
  • Fercoq Olivier
  , 2012.
• Stochastic models for a chemostat and long time behavior
  
  • Collet Pierre
  • Martinez Servet
  • Meleard Sylvie
  • San Martin Jaime
  , 2012. We introduce two stochastic chemostat models consisting in a coupled population-nutrient process reflecting the interaction between the nutrient and the bacterias in the chemostat with finite volume. The nutrient concentration evolves continuously but depending on the population size, while the population size is a birth and death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long time behavior of the bacterial population conditioned to the non-extinction. We prove the global existence of the process and its almost sure extinction. The existence of quasi-stationary distributions is obtained based on a general fixed point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.
• Absence of sufficiently localized traveling wave solutions for the Novikov-Veselov equation at zero energy
  
  • Kazeykina Anna
  , 2012. We demonstrate that the Novikov.Veselov equation (a (2+1)-dimensional analog of KdV) at zero energy does not possess solitons with the space localization stronger than O(|x|^{-4}).
• Rotation autour de l'axe Y (vertical) du noeud '5-trèfle' torique sur son tore, qui peut être vue comme un ensemble de 4x3 stéréogrammes
  
  • Colonna Jean-François
  , 2012. Rotation about the Y (vertical) axis of the 5-foil torus knot on its torus, that can also be viewed as a set of 4x3 stereograms (Rotation autour de l'axe Y (vertical) du noeud '5-trèfle' torique sur son tore, qui peut être vue comme un ensemble de 4x3 stéréogrammes)
• Rotation autour de l'axe Y (vertical) du noeud '7-trèfle' torique sur son tore, qui peut être vue comme un ensemble de 4x3 stéréogrammes
  
  • Colonna Jean-François
  , 2012. Rotation about the Y (vertical) axis of the 7-foil torus knot on its torus, that can also be viewed as a set of 4x3 stereograms (Rotation autour de l'axe Y (vertical) du noeud '7-trèfle' torique sur son tore, qui peut être vue comme un ensemble de 4x3 stéréogrammes)
• Rotation autour de l'axe Y (vertical) du noeud '1-trèfle' torique sur son tore avec son ruban de Möbius associé, qui peut être vue comme un ensemble de 4x3 stéréogrammes
  
  • Colonna Jean-François
  , 2012. Rotation about the Y (vertical) axis of the 1-foil torus knot on its torus and its asociated Möbius strip, that can also be viewed as a set of 4x3 stereograms (Rotation autour de l'axe Y (vertical) du noeud '1-trèfle' torique sur son tore avec son ruban de Möbius associé, qui peut être vue comme un ensemble de 4x3 stéréogrammes)
• Rotation autour de l'axe Y (vertical) de la ligne u=v sur la triple bouteille de Bonan-Jeener-Klein et un ruban, qui peut être vue comme un ensemble de 4x3 stéréogrammes
  
  • Colonna Jean-François
  , 2012. Rotation about the Y (vertical) axis of the u=v line on the Bonan-Jeener's triple Klein bottle and a strip, that can also be viewed as a set of 4x3 stereograms (Rotation autour de l'axe Y (vertical) de la ligne u=v sur la triple bouteille de Bonan-Jeener-Klein et un ruban, qui peut être vue comme un ensemble de 4x3 stéréogrammes)
• Rotation autour de l'axe Y (vertical) du noeud '3-trèfle' torique sur son tore, qui peut être vue comme un ensemble de 4x3 stéréogrammes
  
  • Colonna Jean-François
  , 2012. Rotation about the Y (vertical) axis of the 3-foil torus knot on its torus, that can also be viewed as a set of 4x3 stereograms (Rotation autour de l'axe Y (vertical) du noeud '3-trèfle' torique sur son tore, qui peut être vue comme un ensemble de 4x3 stéréogrammes)
• Les anneaux borroméens étendus -un point de vue inhabituel
  
  • Colonna Jean-François
  , 2012. The extended Borromean Rings -an unusual point of view- (Les anneaux borroméens étendus -un point de vue inhabituel-)
• Les anneaux borroméens étendus
  
  • Colonna Jean-François
  , 2012. The extended Borromean Rings (Les anneaux borroméens étendus)
• Les anneaux borroméens
  
  • Colonna Jean-François
  , 2012. The Borromean Rings (Les anneaux borroméens)