Publications

2004

• On the Fortet-Mourier metric for the stability of Stochastic Optimization Problems, an example
  
  • Strugarek Cyrille
  Stochastic Programming E-Print Series (SPEPS), 2004, 2004 (25). #HTML We consider the use of the Fortet-Mourier metric between two probability measures to bound the error term made by an approximated solution of a stochastic program. After a short analysis of usual stability arguments, we propose a simple example of stochastic program which enlightens the importance of the information structure. As a conclusion, we underline the need to take into account both the probability measure and the information structure in the discretization of a stochastic program.
• On the long-time behavior of unsplit Perfectly Matched Layers
  
  • Bécache Eliane
  • Petropoulos Peter
  • Gedney Stephen
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2004, 52 (5). This paper shows how to eliminate an undesirable long-time linear growth of the electromagnetic field in a class of unsplit perfectly matched layers (PML) typically used as absorbing boundary conditions in computational electromagnetics codes. For the new PML equations, we give energy arguments that show the fields in the layer are bounded by a time-independent constant, hence they are long-time stable. Numerical experiments confirm the elimination of the linear growth, and the long-time boundedness of the fields. (10.1109/TAP.2004.827253)
  DOI : 10.1109/TAP.2004.827253
• Nonhomogeneous nilpotent approximations for systems with singularities
  
  • Vendittelli Marilena
  • Oriolo Giuseppe
  • Jean Frédéric
  • Laumond Jean-Paul
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2004, 49 (2), pp.261-266. Nilpotent approximations are a useful tool for analyzing and controlling systems whose tangent linearization does not preserve controllability, such as nonholonomic mechanisms. However, conventional homogeneous approximations exhibit a drawback: in the neighborhood of singular points (where the system growth vector is not constant) the vector fields of the approximate dynamics do not vary continuously with the approximation point. The geometric counterpart of this situation is that the sub-Riemannian distance estimate provided by the classical Ball-Box Theorem is not uniform at singular points. With reference to a specific family of driftless systems, we show how to build a nonhomogeneous nilpotent approximation whose vector fields vary continuously around singular points. It is also proven that the privileged coordinates associated to such an approximation provide a uniform estimate of the distance. (10.1109/TAC.2003.822872)
  DOI : 10.1109/TAC.2003.822872
• Mathematical justification of simplified models for acoustics wave in media including thin slots
  
  • Tordeux Sébastien
  , 2004.
• Attacks on Public-Key Cryptosystems Based on Free Partially Commutative Monoids and Groups
  
  • Levy-Dit-Vehel Françoise
  • Perret Ludovic
  Progress in Cryptology - INDOCRYPT 2004, 2004, 3348, pp.275-289. At indocrypt 2003, Abisha, Thomas and Subramanian have proposed a public key encryption scheme and a zero-knowledge authentication protocol based on the word problem on monoids, as well as a group variant of these systems. We here present a total break attack on each of the two encryption schemes. The complexity bounds of our algorithms show that these schemes are insecure for practical parameter sizes. In the monoid setting, we go one step further by proposing an algorithm that breaks the NP-hard problem underlying both the encryption scheme and the zero-knowledge protocol, as well as an upper bound on its complexity. (10.1007/978-3-540-30556-9_22)
  DOI : 10.1007/978-3-540-30556-9_22
• Perfectly matched layers for the convected Helmholtz equation
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2004, 42 (1), pp.409-433. In this paper, we propose and analyze perfectly matched absorbing layers for a problem of time-harmonic acoustic waves propagating in a duct in the presence of a uniform flow. The absorbing layers are designed for the pressure field, satisfying the convected scalar Helmholtz equation. A difficulty, compared to the Helmholtz equation, comes from the presence of so-called inverse upstream modes which become unstable, instead of evanescent, with the classical Bérenger's perfectly matched layers (PMLs). We investigate here a PML model, recently introduced for time-dependent problems, which makes all outgoing waves evanescent. We then analyze the error due to the truncation of the domain and prove that the convergence is exponential with respect to the size of the layers for both the classical and the new PML models. Numerical validations are finally presented. © 2004 Society for Industrial and Applied Mathematics. (10.1137/s0036142903420984)
  DOI : 10.1137/s0036142903420984
• Sur la reconstruction des polynômes linéaires : un nouvel algorithme de décodage des codes de Gabidulin
  
  • Loidreau Pierre
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2004.
• Mathematical justification of simplified models for acoustics wave in media including thin slots
  
  • Tordeux Sébastien
  , 2004.
• Raccordement de développements asymptotiques pour la propagation des ondes dans les milieux comportant des fentes
  
  • Tordeux Sébastien
  • Joly Patrick
  , 2004. 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.
• Homogenization of L-infinity functionals
  
  • Briani Ariela
  • Garroni Adriana
  • Prinari Francesca
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2004, 14 (12), pp.1761-1784. We study, via $\Gamma$ convergence, the homogenization in L-infinity of supremal functionals of the form $$ F\varepsilon (u) =ess sup{A} f( x \varepsilon, Du). $$ We prove the homogenized problem is still a supremal and its energy density is given by a cell problem formula.
• Another approach to linearized elasticity and Korn's inequality
  
  • Ciarlet Patrick
  • Ciarlet Philippe G.
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2004, 339 (4), pp.307-312. We describe and analyze an approach to the pure traction problem of three-dimensional linearized elasticity, whose novelty consists in considering the linearized strain tensor as the 'primary' unknown, instead of the displacement itself as is customary. This approach leads to a well-posed minimization problem, constrained by a weak form of the St Venant compatibility conditions. It also provides a new proof of Korn's inequality. (10.1016/j.crma.2004.06.021)
  DOI : 10.1016/j.crma.2004.06.021
• A fast algorithm for the two dimensional HJB equation of stochastic control
  
  • Bonnans Frédéric
  • Ottenwaelter Elisabeth
  • Zidani Hasnaa
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2004, 38 (4), pp.723-735. This paper analyses the implementation of the generalized finite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani, SIAM J. Numer. Anal. 41 (2003) 1008-1021]. The computation of coefficients needs to solve at each point of the grid (and for each control) a linear programming problem. We show here that, for two dimensional problems, this linear programming problem can be solved in O(p max) operations, where p max is the size of the stencil. The method is based on a walk on the Stern-Brocot tree, and on the related filling of the set of positive semidefinite matrices of size two. (10.1051/m2an:2004034)
  DOI : 10.1051/m2an:2004034
• Modélisation de la propagation d'ondes dans les milieux viscoélastiques linéaires II : Analyse numérique
  
  • Bécache Eliane
  • Ezziani Abdelaâziz
  • Joly Patrick
  , 2004. Nous nous intéressons à la modélisation de la propagation d'ondes dans des milieux viscoélastiques. Un premier rapport concernait l'analyse mathématique des modèles continus. Le second rapport est consacré à leur approximation numérique. Nous présentons une méthode d'éléments finis mixtes pour approcher les équations viscoélastiques dans des milieux anisotropes hétérogènes. Cette méthode permet de faire la condensation de masse et ainsi d'obtenir un schéma explicite centré pour la discrétisation en temps. Nous démontrons, pour le schéma ainsi obtenu, un résultat de décroissance d'une énergie discrète et donnons une condition suffisante de stabilité. Pour simuler la propagation dans les milieux ouverts, nous adaptons la technique de couches absorbantes parfaitement adaptées aux ondes viscoélastiques. Nous présentons finalement des validations de la méthode ainsi que des simulations d'expériences réalistes.
• Simulation of muffler's transmission losses by a homogenized finite element method
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Drissi Dora
  • Gmati Nabil
  Journal of Computational Acoustics, World Scientific Publishing, 2004, 12 (3), pp.447-474. In this work, we are interested in the modelling of the acoustic attenuation of exhaust mufflers including perforated ducts, and its numerical computation. The study is worked out in harmonic time regime, for the two-dimensional case. The hole diameter and the center-to-center distance between consecutive holes are supposed of same order, and small compared to the size of the muffler. The formulation is derived by using multiscale techniques and matching the asymptotic expansions. The numerical method couples finite elements in the muffler with modal decomposition in the inlet and the outlet of the duct. (10.1142/s0218396x04002304)
  DOI : 10.1142/s0218396x04002304