Publications

2005

• Singular trajectories of driftless and control-affine systems
  
  • Chitour Yacine
  • Jean Frédéric
  • Trélat Emmanuel
  , 2005, pp.6 pages. We establish generic properties for singular trajectories, first for driftless, and then for control-affine systems. We show that, generically -- for the Whitney topology -- nontrivial singular trajectories are of minimal order and of corank one. As a consequence, if the number of vector fields of the system is greater than or equal to 3, then there exists generically no singular minimizing trajectory. (10.1109/cdc.2005.1582278)
  DOI : 10.1109/cdc.2005.1582278
• Résolution des équations de Maxwell avec des éléments finis de Galerkin continus
  
  • Jamelot Erell
  , 2005. Les equations de Maxwell se resolvent aisement lorsque le domaine d'etude est regulier, mais lorsqu'il existe des singularites geometriques (coins rentrants en 2D, coins et aretes rentrants en 3D), le champ electromagnetique est localement non borne au voisinage de ces singularites. Nous nous interessons a la resolution des equations de Maxwell dans des domaines bornes, singuliers, a l'aide de methodes d'elements finis continus. En pratique, cela permet de modeliser des instruments de telecommunication comme les guides d'onde, les filtres a stubs. Nous analysons tout d'abord le probleme quasi-electrostatique 2D, afin de maitriser la discretisation en espace. Nous presentons trois methodes de calcul (formulations augmentees mixtes) qui donnent des resultats numeriques tres convaincants : - Une version epuree de la methode du complement singulier (conditions aux limites essentielles). - La methode de regularisation a poids : on introduit un poids qui depend des distances aux singularites geometriques (conditions aux limites essentielles). - La methode avec conditions aux limites naturelles. Nous etudions ensuite la generalisation de ces methodes aux domaines 3D. Nous detaillons la resolution des equations de Maxwell instationnaires en domaines singuliers 3D par la methode de regularisation a poids, et nous donnons des resultats numeriques inedits.
• A Stochastic Gradient Type Algorithm for Closed Loop Problems
  
  • Barty Kengy
  • Roy Jean-Sébastien
  • Strugarek Cyrille
  Stochastic Programming E-Print Series (SPEPS), 2005, 2005 (14). We focus on solving closed-loop stochastic problems, and propose a perturbed gradient algorithm to achieve this goal. The main hurdle in such problems is the fact that the control variables are infinite dimensional, and have hence to be represented in a finite way in order to numerically solve the problem. In the same way, the gradient of the criterion is itself an infinite dimensional object. Our algorithm replaces this exact (and unknown) gradient by a perturbed one, which consists in the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point. Proceeding this way, we explore the whole space iteration after iteration through random points. Since each kernel function is perfectly known by a finite (and small) number of parameters, say N, the control at iteration k is perfectly known as an infinite dimensional object by at most N x k parameters. The main strength of this method is that it avoids any discretization of the underlying space, provided that we can draw as many points as needed in this space. Hence, we can take into account in a new way the possible measurability constraints of the problem. Moreover, the randomization of this algorithm implies that the most probable parts of the space are the most explored ones, what is a priori an interesting feature. In this paper, we first show a convergence result of this algorithm in the general case, and then give a few numerical examples showing the interest of this method for solving practical stochastic optimization problems.
• Dynamics of Anisotropic Universes
  
  • Perez Jérôme
  , 2006, Proceedings of the Albert Einstein Century International Conference. We present a general study of the dynamical properties of Anisotropic Bianchi Universes in the context of Einstein General Relativity. Integrability results using Kovalevskaya exponents are reported and connected to general knowledge about Bianchi dynamics. Finally, dynamics toward singularity in Bianchi type VIII and IX universes are showed to be equivalent in some precise sence. HOMOGENEOUS UNIVERSE AND BIANCHI MODELS Considering the usual synchronous frame of General Relativity 1 ds 2 = g µν dx µ dx ν = ˜ g i j dx i dx j − dt 2. A Universe is said homogeneous when there exists an isometry group wich preserves the infinitesimal spacial lenght dl 2 = ˜ g i j dx i dx j A characterization of the isometry group is possible writing structure constants C (10.1063/1.2399648)
  DOI : 10.1063/1.2399648
• Relaxation lagrangienne et filtrage par coûts réduits appliqués à la production d'éléctricité.
  
  • Benoist Thierry
  • Diamantini Maurice
  • Rottembourg Benoît
  , 2005. Le problème UCP (Unit Commitment Problem) consiste à planifier la production d'un parc de centrales électriques de manière à satisfaire un besoin prévisionnel donné sur une échelle de temps discrétisée (besoin horaire sur 24 heures). L'objectif consiste à définir à moindre coût <ul> <li> d'une part, l'ordonnancement d'allumage/extinction de chaque centrale sur toute la période considérée, </li> <li> et d'autre part, la production de chaque centrale pour toute date où elle est allumée ; </li> </ul> de façon à satisfaire l'ensemble des **deux contraintes globales** (demande prévisionnelle et réserve de 10% modélisant l'incertitude de la prévision) et de **trois contraintes techniques** propres à chaque générateur: puissance bornée, temps minimum d'arrêt avant redémarrage, temps minimum de fonctionnement avant extinction. La fonction de coût d'un générateur comprend un coût de fonctionnement légèrement quadratique auquel s'ajoute un coût de démarrage dépendant de la durée d'arrêt d'une centrale que l'on allume. Nous réalisons une relaxation lagrangienne en dualisant les contraintes globales, et nous résolvons le problème ainsi relâché par programmation dynamique après avoir précalculé pour chaque date la production optimale connaissant les multiplicateurs de Lagrange. La programmation dynamique permet également le calcul du coût réduit nécessaire pour compenser le viol de certaines contraintes pour litération suivante. Ceci nous fournit une **borne duale additive** que nous exploitons dune part pour améliorer la borne inférieure, et d'autre part pour filtrer des variables par Programmation Par Contraintes en cours du processus dénumération implicite. Les coûts réduits sont également exploités pour guider le choix des couples (variable valeur) dans la phase de séparation du Branch and Bound.
• A density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians
  
  • Briani Ariela
  • Rampazzo Franco
  Nonlinear Differential Equations and Applications, Springer Verlag, 2005, 12 (1), pp.71-91. In 1985 H. Ishii [Is85] proposed a generalization of the notion of (continuous) viscosity solution for an Hamilton-Jacobi equation with a t-measurable Hamiltonian--that is, a Hamiltonian which is measurable in time and continuous in the other variables. This notion turned out to agree with natural applications, like Control and Differential Games Theory. Since then, several improvements have been achieved for the standard situation when the Hamiltonian is continuous. It is someway an accepted general idea that parallel improvements are likely for t-measurable Hamiltonians as well, though such a job might appear a bit tedious because of the necessarily involved technicalities. In this paper we show that Ishii's definition of viscosity solution coincides with the one which would arise by extending by density the standard definition. Namely, we regard a t-measurable Hamiltonian H as an element of the closure (for suitable topologies) of a class of continuous Hamiltonians. On the other hand, we show that the set of Ishii's (sub-, super-) solutions for H is nothing but the limit set of the (sub-, super-) solutions corresponding to continuous Hamiltonians approaching H. This put us in the condition of establishing comparison, existence, and regularity results by deriving them from the analogous results for the case of continuous Hamiltonians. (10.1007/s00030-004-2030-4)
  DOI : 10.1007/s00030-004-2030-4
• Temporal difference learning with kernels for pricing american-style options
  
  • Barty Kengy
  • Roy Jean-Sébastien
  • Strugarek Cyrille
  Optimization Online, 2005. We propose in this paper to study the problem of estimating the cost-to-go function for an infinite-horizon discounted Markov chain with possibly continuous state space. For implementation purposes, the state space is typically discretized. As soon as the dimension of the state space becomes large, the computation is no more practicable, a phenomenon referred to as the curse of dimensionality. The approximation of dynamic programming problems is therefore of major importance. A powerful method for dynamic programming, often referred to as neuro-dynamic programming, consists in representing the Bellman function as a linear combination of a priori defined functions, called neurons. In this article, we propose an alternative approach very similar to temporal differences, based on functional gradient descent and using an infinite kernel basis.Furthermore, our algorithm, though aimed at infinite dimensional problems, is implementable in practice. We prove the convergence of this algorithm, and show applications on e.g. bermudan option pricing.
• Consistency of a simple multidimensional scheme for Hamilton-Jacobi-Bellman equations
  
  • Munos Remi
  • Zidani Hasnaa
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2005, 340 (7), pp.499-502. This Note presents an approximation scheme for second-order Hamilton-Jacobi-Bellman equations arising in stochastic optimal control. The scheme is based on a Markov chain approximation method. It is easy to implement in any dimension. The consistency of the scheme is proved, which guarantees its convergence. To cite this article: R. Munos, H. Zidani, C. R. Acad. Sci. Paris, Ser. I 340 (2005). (10.1016/j.crma.2005.02.001)
  DOI : 10.1016/j.crma.2005.02.001
• A Perturbed Gradient Algorithm in Hilbert Spaces
  
  • Barty Kengy
  • Roy Jean-Sébastien
  • Strugarek Cyrille
  Optimization Online, 2005. We propose a perturbed gradient algorithm with stochastic noises to solve a general class of optimization problems. We provide a convergence proof for this algorithm, under classical assumptions on the descent direction, and new assumptions on the stochastic noises. Instead of requiring the stochastic noises to correspond to martingale increments, we only require these noises to be asymptotically so. Furthermore, the variance of these noises is allowed to grow infinitely under the control of a decreasing sequence linked with the gradient stepsizes. We then compare this new approach and assumptions with classical ones in the stochastic approximation literature. As an application of this general setting, we show how the algorithm to solve infinite dimensional stochastic optimization problems recently developped by the authors in another paper is a special case of the following perturbed gradient with stochastic noises.
• Approches analytiques et numériques de problèmes de transmission en propagation d'ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées
  
  • Diaz Julien
  , 2005. Dans la première partie nous présentons deux méthodes numériques non conformes espace-temps pour la propagation d'ondes en interaction fluide-structure. Ces méthodes, robustes et précises, sont basées sur deux formulations mixtes dites duale-duale et primale-primale. Elles sont explicites, sauf à l'interface, et conservatives, ce qui en assure la stabilité. Nous les validons à l'aide de solutions analytiques calculées par la méthode de Cagniard-de Hoop (CdH). Dans la deuxième partie nous obtenons, via la méthode CdH, des estimations d'erreur pour l'utilisation de conditions aux limites absorbantes (CLA) ou couches absorbantes parfaitement adaptées (PML) pour la résolution de l'équation des ondes dans le demi-espace. La troisième partie est consacrée aux PMLs pour l'acoustique en écoulement: analyse (par CdH) de l'instabilité des PMLs classiques et construction de PMLs stabilisées. La dernière partie consiste en une présentation mathématique détaillée de la méthode CdH.
• Modélisation mathématique et numérique de la propagation d'ondes dans les milieux viscoélastiques et poroélastiques
  
  • Ezziani Abdelaâziz
  , 2005. Nous nous intéressons à la modélisation de la propagation d'ondes dans le sous sol. Nous présentons deux modèles de propagation : (i) une généralisation du modèle de Zener pour les milieux viscoélastiques, (ii) le modèle de Biot pour les milieux poroélastiques. Nous menons une analyse mathématique complète de ces modèles : résultat d'existence, d'unicité et de décroissance de l'énergie. Pour la résolution numérique nous construisons une méthode spécifique à chaque modèle, basée sur des approches variationnelles, une approximation par éléments finis mixtes en espace et différences finies en temps. Nous montrons pour chaque schéma, un résultat de décroissance d'énergie discrète qui conduit à une condition suffisante de stabilité. Pour simuler la propagation d'ondes dans les milieux ouverts, nous adaptons la technique de couches absorbantes parfaitement adaptées aux ondes viscoélastiques et poroélastiques. Enfin, nous présentons des validations numériques des méthodes développées.
• Augmented formulations for solving Maxwell equations
  
  • Ciarlet Patrick
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2005, 194 (2-5), pp.559-586. We consider augmented variational formulations for solving the static or time-harmonic Maxwell equations. For that, a term is added to the usual H (curl) conforming formulations. It consists of a (weighted) L2 scalar product between the divergence of the EM and the divergence of test fields. In this respect, the methods we present are H (curl, div) conforming. We also build mixed, augmented variational formulations, with either one or two Lagrange multipliers, to dualize the equation on the divergence and, when applicable, the relation on the tangential or normal trace of the field. It is proven that one can derive formulations, which are equivalent to the original static or time-harmonic Maxwell equations. In the latter case, spurious modes are automatically excluded. Numerical analysis and experiments will be presented in the forthcoming paper [Augmented formulations for solving Maxwell equations: numerical analysis and experiments, in preparation]. (10.1016/j.cma.2004.05.021)
  DOI : 10.1016/j.cma.2004.05.021
• Stabilité et résonances pour le problème du mouvement sur la houle
  
  • Hazard Christophe
  • Lenoir Marc
  , 2005.
• Generalized Impedance Impedance Boundary Conditions For Scattering by Strongly Absorbing Obstacles: The Scalar Case
  
  • Haddar Houssem
  • Joly Patrick
  • Nguyen Hoai-Minh
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (08), pp.1273-1300. (10.1142/S021820250500073X)
  DOI : 10.1142/S021820250500073X
• A globally convergent steering algorithm for regular nonholonomic systems
  
  • Jean Frédéric
  • Oriolo G.
  • Vendittelli M.
  , 2005, pp.7514-7519. (10.1109/CDC.2005.1583374)
  DOI : 10.1109/CDC.2005.1583374
• Matching of asymptotic expansions for the wave propagation in media with thin slot
  
  • Tordeux Sébastien
  • Joly Patrick
  , 2005. This talk concerns the modelizing of scattering in the harmonic regime in two dimensional domains with thin slots. We use the technique of matching asymptotic expansions to obtain and justify the asymptotic expansion of the solution to any order with respect to the width of the slot.
• Mathematical analysis of the acoustic diffraction by a muffler containing perforated ducts
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Drissi Dora
  • Gmati Nabil
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (7), pp.1059-1090. We consider the three-dimensional scalar problem of acoustic propagation in a muffler. We develop and analyze a Fredholm-type formulation for a stationary fluid in the timeharmonic setting. We prove a homogenization result for a muffler containing periodically perforated ducts. Essentially, the perforated boundaries become completely transparent when the period of perforations, which is assumed to be of the same order as the size of perforations, tends to zero. We also derive a homogenized impedance condition when the perforated duct is coated by an absorbing material. We present numerical examples in two dimensions, obtained from coupling finite elements in the muffler with modal decompositions in the inlet and outlet ducts, which demonstrate the limiting validity of the theoretical results. © World Scientific Publishing Company. (10.1142/s0218202505000649)
  DOI : 10.1142/s0218202505000649
• Perfectly matched layers for time-harmonic acoustics in the presence of a uniform flow
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  , 2005, pp.35. This paper is devoted to the resolution of the time-harmonic linearized Galbrun's equation, which models, via a mixed Lagrangian-Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting amplitude process, we characterize the outgoing solution radiated by a compactly supported source. Then, we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included.
• Singular electromagnetic fields: Inductive approach [Singularités électromagnétiques: Une approche inductive]
  
  • Assous Franck
  • Ciarlet Patrick
  • Garcia Emmanuelle
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2005, 341 (10), pp.605-610. In a non-convex polyhedral domain, we describe the local trace (i.e. defined on a face) of the normal derivative of an L2 function, with L2 Laplacian. We then provide generalized integration by parts formulae for the Laplace, divergence and curl operators. Finally, these results allow us to split electromagnetic fields into regular and singular parts, which can be characterized. © 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. (10.1016/j.crma.2005.09.034)
  DOI : 10.1016/j.crma.2005.09.034
• Diffraction of an acoustic wave by a plate in a uniform flow: A numerical approach
  
  • Job Stéphane
  • Lunéville Éric
  • Mercier Jean-François
  Journal of Computational Acoustics, World Scientific Publishing, 2005, 13 (4), pp.689-709. We study the diffraction in time harmonic regime of an acoustic wave by a rigid plate in the presence of a uniform flow in a duct. Contrary to prior analytical studies, using Wiener-Hopf techniques and thus restricted to semi-infinite plates, we use a, finite elements method which allows us to deal with plates of finite length. To take into account irrotational perturbations induced by the trailing edge of the plate, a potential formulation requires the introduction of a vortex sheet behind the plate. The key point of the method is to get access at the singular coefficient of the velocity potential near the trailing edge, in order to cancel it using the so-called Kutta-Joukowski condition. This approach leads to an efficient finite elements method, and numerical computations are presented: we show the amplitude of the vortex sheet versus the Mach number and the plate length and the dissipated acoustic power versus the Mach number and the frequency. This method is extended to the case of two aligned plates to analyze the influence of the choice of the boundary condition on the downstream plate which interacts with a vortex sheet. © IMACS. (10.1142/s0218396x05002840)
  DOI : 10.1142/s0218396x05002840
• A dual-primal coupling technique with local time step for wave propagation problems
  
  • Bécache Eliane
  • Joly Patrick
  • Rodriguez Jeronimo
  , 2005. We are interested in space-time refinement methods for linear wave propagation. In 1,2 , some stable numerical schemes using non-conforming grids in space and time have been proposed. These methods use a Lagrange multiplier to cope with the interface conditions. The choice of the discretization space of this additional unknown can be in some cases not trivial. In the present paper we propose an alternative method. The main new idea is to use different variational formulations in the fine and in the coarse grids. We present a time discretization that leads to the conservation of a discrete energy and provide a complete stability and error analysis in the case where the time step is twice smaller in one domain than in the other one.
• Realistic numerical modeling of human head tissues exposure to electromagnetic waves from mobiles phones
  
  • Scarella Gilles
  • Clatz Olivier
  • Lanteri Stéphane
  • Beaume Grégory
  • Oudot Steve
  • Pons Jean-Philippe
  • Piperno Serge
  • Joly Patrick
  • Wiart Joe
  , 2005. The ever-rising diffusion of cellular phones has brought about an increased concern for the possible consequences of electromagnetic radiation on human health. Possible thermal effects have been investigated, via experimentation or simulation, by several research projects in the last decade. Concerning numerical modeling, the power absorption in a user's head is generally computed using discretized models built from clinical MRI data. The vast majority of such numerical studies have been conducted using Finite Differences Time Domain methods, although strong limitations of their accuracy are due to heterogeneity, poor definition of the detailed structures of head tissues (staircasing effects), etc. In order to propose numerical modeling using Finite Element or Discontinuous Galerkin Time Domain methods, reliable automated tools for the unstructured discretization of human heads are also needed. Results presented in this article aim at filling the gap between human head MRI images and the accurate numerical modeling of wave propagation in biological tissues and its thermal effects.
• Matching of asymptotic expansions for the wave propagation in media with thin slot
  
  • Tordeux Sébastien
  • Joly Patrick
  , 2005. This talk concerns the modelizing of scattering in the harmonic regime in two dimensional domains with thin slots. We use the technique of matching asymptotic expansions to obtain and justify the asymptotic expansion of the solution to any order with respect to the width of the slot.
• Raccordement de développements asymptotiques pour la propagation des ondes dans les milieux comportant des fentes
  
  • Joly Patrick
  • Tordeux Sébastien
  , 2005. Cet exposé porte sur la modélisation de la diffraction d'ondes en régime harmonique dans des milieux bidimensionnels comportant des fentes minces. Nous utilisons la technique des développements asymptotiques raccordés pour obtenir et justifier le développement asymptotique de la solution à tout ordre en fonction de l'épaisseur de la fente.
• An efficient numerical method for the resolution of the Kirchhoff-Love dynamic plate equation
  
  • Bécache Eliane
  • Derveaux Grégoire
  • Joly Patrick
  Numerical Methods for Partial Differential Equations, Wiley, 2005, 21 (2), pp.323 - 348. We solve numerically the Kirchhoff-Love dynamic plate equation for an anisotropic heterogeneous material using a spectral method. A mixed velocity-moment formulation is proposed for the space approximation allowing the use of classical Lagrange finite elements. The benefit of using high order elements is shown through a numerical dispersion analysis. The system resulting from this spatial discretization is solved analytically. Hence this method is particularly efficient for long duration experiments. This time evolution method is compared with explicit and implicit finite differences schemes in terms of accuracy and computation time. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 (10.1002/num.20041)
  DOI : 10.1002/num.20041