Publications

2006

• Error estimates for stochastic differential game: the adverse stopping case
  
  • Bonnans Frédéric
  • Maroso Stefania
  • Zidani Hasnaa
  IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2006, 28, pp.188-212. (10.1093/imanum/dri034)
  DOI : 10.1093/imanum/dri034
• On Wagner-Magyarik Cryptosystem
  
  • Levy-Dit-Vehel Françoise
  • Perret Ludovic
  Lecture Notes in Computer Science, Springer, 2006, 3969, pp.316-329. We investigate a monoid variant of the scheme based on the word problem on groups proposed by Wagner and Magyarik at Crypto'84, that has the advantage of being immune to reaction attacks so far. We study the security of this variant. Our main result is a complexity-theoretic one: we show that the problem underlying this cryptosystem, say WM, is NP-hard. We also present an algorithm for solving WM. Its complexity permits to shed light on the size of the parameters to choose to reach a given level of security. (10.1007/11779360_25)
  DOI : 10.1007/11779360_25
• Une présentation mathématique de la méthode de Cagniard-de Hoop Partie II En dimension trois
  
  • Diaz Julien
  • Joly Patrick
  , 2006, pp.93. Dans ce rapport nous présentons l'extension de la méthode de Cagniard-de Hoop, que nous avons étudiée dans la première partie en dimension deux, à la dimension trois. Comme dans la première partie nous effectuons une présentation mathématique détaillée de la méthode, nous regroupons les résultats déjà connus et nous éclaircissons certaines difficultés mathématiques qui ne semblent pas avoir été traités jusqu'à maintenant .
• Une présentation mathématique de la méthode de Cagniard-de Hoop Partie I En dimension deux
  
  • Diaz Julien
  • Joly Patrick
  , 2006, pp.89. Dans ce rapport, nous effectuons une présentation mathématique détaillée de la méthode de Cagniard-de Hoop en dimension deux. Nous regroupons les résultats déjà connus et nous éclaircissons certaines difficultés mathématiques qui ne semblent pas avoir été traités jusqu'à maintenant .
• Un problème de Laplace non standard en milieu non borné
  
  • Tordeux Sébastien
  , 2006, pp.13. Dans le cadre des problèmes elliptiques en dimension deux, nous nous intéressons à un domaine constitué d'un demi-espace connecté à une bande infinie. Un résultat d'existence et unicité est obtenu pour un problème de Laplace inhomogène muni de comportements asymptotiques à l'infini.
• The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains
  
  • Ciarlet Patrick
  • Jung Beate
  • Kaddouri Samir
  • Labrunie Simon
  • Zou Jun
  Numerische Mathematik, Springer Verlag, 2006, 102, pp.583-610. This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used. (10.1007/s00211-005-0664-8)
  DOI : 10.1007/s00211-005-0664-8
• A time domain analysis of PML models in acoustics.
  
  • Diaz Julien
  • Joly Patrick
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2006, 195 (29-32), pp.3820-3853. We present in this paper a time-domain analysis of PML's for non-advective and advective acoustics. We focus our attention on time-stability and error estimates (with respect to the parameters of the layers). The main new technical tool is the Cagniard-de Hoop method. Our theoretical results are validated and illustrated by various numerical results. (10.1016/j.cma.2005.02.031)
  DOI : 10.1016/j.cma.2005.02.031
• A mathematical analysis of the resonance of the finite thin slots
  
  • Clausel Marianne
  • Duruflé Marc
  • Joly Patrick
  • Tordeux Sébastien
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2006, 56 (10-11), pp.1432-1449. In this article we are interested in the asymptotic behaviour of the solution of a wave propagation problem in a domain including a thin slot. Some rate of convergences are obtained and illustrated by numerical results.
• An application of PL continuation methods to singular arcs problems
  
  • Gergaud Joseph
  • Martinon Pierre
  Lecture notes in economics and mathematical systems, 2006, 563, pp.163-186.
• Conservative space-time mesh refinement methods for the FDTD solution of Maxwell's equations
  
  • Collino Francis
  • Fouquet Thierry
  • Joly Patrick
  Journal of Computational Physics, Elsevier, 2006, 211 (1), pp.9-35. A new variational space-time mesh refinement method is proposed for the FDTD solution of Maxwell's equations. The main advantage of this method is to guarantee the conservation of a discrete energy that implies that the scheme remains L2 stable under the usual CFL condition. The only additional cost induced by the mesh refinement is the inversion, at each time step, of a sparse symmetric positive definite linear system restricted to the unknowns located on the interface between coarse and fine grid. The method is presented in a rather general way and its stability is analyzed. An implementation is proposed for the Yee scheme. In this case, various numerical results in 3-D are presented in order to validate the approach and illustrate the practical interest of space-time mesh refinement methods. (10.1016/j.jcp.2005.03.035)
  DOI : 10.1016/j.jcp.2005.03.035
• A justification of Peek's empirical law in electrostatics [Justification de la loi de Peek en électrostatique]
  
  • Ciarlet Patrick
  • Kaddouri Samir
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2006, 343 (10), pp.671-674. We consider the computation of the electrostatic charge density at the tip of a rounded corner. The relation between the curvature radius and the electrostatic field is given by Peek's empirical law which is valid only for thin, cylindrical or spherical, geometries. In this Note, we justify mathematically this law and extend it to other geometries. With the help of multiscaled asymptotic expansions, we derive an expression for the charge density for geometries which coincide at infinity with a cone. A numerical illustration is provided. To cite this article: P. Ciarlet Jr., S. Kaddouri, C. R. Acad. Sci. Paris, Ser. I 343 (2006). © 2006 Académie des sciences. (10.1016/j.crma.2006.10.009)
  DOI : 10.1016/j.crma.2006.10.009
• Typer la dé-sérialisation sans sérialiser les types
  
  • Henry Grégoire
  • Mauny Michel
  • Chailloux Emmanuel
  , 2006, pp.133-146. In this paper, we propose a way of assigning static type information to unmarshalling functions and we describe a verification technique for unmarshalled data that preserves the execution safety provided by static type checking. This technique, whose correctness is proven, relies on singleton types whose values are transmitted to unmarshalling routines at runtime, and on an efficient checking algorithm able to deal with sharing and cycles.
• Exact boundary conditions for periodic waveguides containing a local perturbation
  
  • Joly Patrick
  • Li Jing-Rebecca
  • Fliss Sonia
  Communications in Computational Physics, Global Science Press, 2006, 1 (6), pp.945-973. We consider the solution of the Helmholtz equation $-\Delta u({\bf x}) - n({\bf x})^2\omega^2 u({\bf x}) = f({\bf x})$, ${\bf x}=(x,y)$, in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f({\bf x})$ is supported in $\Omega^0:={{\bf x}\in {\Omega} \; | a^
• Raccordement de développements asymptotiques pour la propagation des ondes dans les milieux comportant des fentes
  
  • Joly Patrick
  • Tordeux Sébastien
  , 2006. 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.
• Discontinuous Galerkin methods for Maxwell's equations in the time domain
  
  • Cohen Gary
  • Ferrieres Xavier
  • Pernet Sébastien
  Comptes Rendus. Physique, Académie des sciences (Paris), 2006, 7, pp.494-500. In this article, we describe a new high-order Discontinuous Galerkin approach to Maxwell's equations in the time domain. This approach is based on hexahedral meshes and uses a mass-lumping technique. Thanks to the orthogonality of the basis functions and a judicious choice of the approximation spaces, it provides an efficient solver for these equations in terms of storage and CPU time. (10.1016/j.crhy.2006.03.004)
  DOI : 10.1016/j.crhy.2006.03.004
• Gravity, dimension, equilibrium, and thermodynamics
  
  • Perez Jérôme
  Comptes rendus de l'académie des sciences de physique, 2006, 7 (3-4), pp.406-413. It is actually possible to interpret gravitation as a property of space in a purely classical way. We note that an extended self-gravitating system equilibrium depends directly on the number of dimensions of the space in which it evolves. Given these precisions, we review the principal thermodynamical knowledge in the context of classical gravity with arbitrary dimension of space. Stability analyses for bounded 3D systems, namely the Antonov instability paradigm, are then associated to some amazing properties of globular clusters and galaxies. (10.1016/j.crhy.2006.01.011)
  DOI : 10.1016/j.crhy.2006.01.011
• High Order Generalized Impedance Boundary Conditions in Electromagnetic Scattering Problems
  
  • Duruflé Marc
  • Haddar Houssem
  • Joly Patrick
  Comptes Rendus. Physique, Académie des sciences (Paris), 2006, 7, pp.533-542. We briefly review the use and the derivation of Generalized Impedance Boundary Conditions (GIBC) in the case of thin dielectric coating and in the case of strongly absorbing medium, within the context of electromagnetic scattering problem at a fixed frequency. We then numerically test the validity and accuracy of these boundary conditions in the case of high absorption. A numerical treatment of the corner singularity is proposed to recover the accuracy of the GIBC for singular geometries.
• Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots
  
  • Joly Patrick
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2006, 40 (1), pp.63--97. The authors study the asymptotic properties of the solution to the Helmoholtz equation with Neumann boundary conditions in a dumbbell-type domain in the regime when the "handle'' is thin and tightening to a curve. A mathematical analysis is done for the model problem posed in the half-plane with an infinite thin straight channel. It is proved that the solution of such a perturbed problem converges to the solution of the limiting problem for the same equation posed in the whole half-plane. Optimal estimates for the convergence rate are obtained in various norms. The authors also construct one more approximation for the solution of the perturbed problem which takes into account the presence of the channel. It is shown that this approximation is better in the sense that the estimates for the difference between this approximation and the "perturbed'' solution are smaller in order than similar estimates for the limiting solutions. The authors also conjecture that the last mentioned estimates are optimal; this conjecture is supported by a series of numerical results. (10.1051/m2an:2006008)
  DOI : 10.1051/m2an:2006008
• Etude d'un problème modèle pour la diffraction par des fils minces par développements asymptotiques raccordés Cas 2D
  
  • Claeys Xavier
  • Haddar Houssem
  • Joly Patrick
  , 2006, pp.52. Dans ce rapport, nous analysons un problème modèle pour l'étude de la diffraction d'une onde par des fils minces. Nous nous intéressons, en deux dimensions, à la solution sortante de l'équation de Helmholtz à l'extérieur d'un obstacle de petit diamètre (vis-à-vis de la longueur d'onde) sur la frontière duquel est imposée une condition de Dirichlet homogène ou une condition de Neumann homogène. Un développement à tout ordre de cette solution par rapport au diamètre de l'obstacle est obtenu.
• Perfectly matched layers for time-harmonic acoustics in the presence of a uniform flow
  
  • Bécache Eliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2006, 44 (3), pp.1191-1217. This paper is devoted to the resolution of the time-harmonic linearized Galbrun 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 absorption 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. © 2006 Society for Industrial and Applied Mathematics. (10.1137/040617741)
  DOI : 10.1137/040617741
• Matching of asymptotic expansions for wave propagation in media with thin slots. I. The asymptotic expansion
  
  • Joly Patrick
  • Tordeux Sébastien
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2006, 5 (1), pp.304--336 (electronic). In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter ε/λ, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of ε/λ: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary value problems. The second one will be devoted to establishing error estimates for truncated series. (10.1137/05064494X)
  DOI : 10.1137/05064494X