Publications

2009

• Perturbative numeric approach in microwave imaging
  
  • Rozanova-Pierrat Anna
  , 2009. In this paper, we show that using measurements for different frequencies, and using ultrasound localized perturbations it is possible to extend the method of the imaging by elastic deformation developed by Ammari and al. [Electrical Impedance Tomography by Elastic Deformation SIAM J. Appl. Math. , 68(6), (2008), 1557–1573.] to problems for the Helmholtz equations with Neumann boundary conditions, and to reconstruct by a perturbation method both the conductivity and the permittivity, provided that the conductivity function is coercive and the wave number in the Helmholtz equation is not a resonant frequency.
• The linear sampling method revisited
  
  • Arens Tilo
  • Lechleiter Armin
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2009, 21 (2), pp.179--202. (10.1216/JIE-2009-21-2-179)
  DOI : 10.1216/JIE-2009-21-2-179
• Fluid-structure interaction and multi-body contact. Application to aortic valves
  
  • Astorino Matteo
  • Gerbeau Jean-Frédéric
  • Pantz Olivier
  • Traore Karim-Frédéric
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2009, 198 (45-46), pp.3603-3612. In this article we present a partitioned procedure for fluid-structure interaction problems in which contacts among different deformable bodies can occur. A typical situation is the movement of a thin valve (e.g. the aortic valve) immersed in an incompressible viscous fluid (e.g. the blood). In the proposed strategy the fluid and structure solvers are considered as independent black-boxes that exchanges forces and displacements; the structure solvers are moreover not supposed to manage contact by themselves. The hypothesis of non-penetration among solid objects defines a non-convex optimization problem. To solve the latter, we use an internal approximation algorithm that is able to directly manage the cases of thin structures and self-contacts. A numerical simulation on an idealized aortic valve is finally realized with the aim of illustrating the proposed scheme. (10.1016/j.cma.2008.09.012)
  DOI : 10.1016/j.cma.2008.09.012
• Homogenization of a conductive and radiative heat transfer problem
  
  • Allaire Grégoire
  • El Ganaoui K.
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2009, 7 (3), pp.1148-1170. This paper is devoted to the homogenization of a heat conductionproblem in a periodically perforated domain with a nonlinear andnonlocal boundary condition modeling radiative heat transfer inthe perforations. Because of the critical scaling considered it isessential to use a method of two-scale asymptotic expansionsinside the variational formulation of the problem. We obtain anonlinear homogenized problem of heat conduction with effectivecoefficients which are computed via a cell problem featuring aradiative heat transfer boundary condition. We rigorously justifythis homogenization process for the linearized problem by usingtwo-scale convergence. We perform numerical simulations in twodimensions: we reconstruct an approximate temperature field byadding to the homogenized temperature a corrector term. Thecomputed numerical errors agree with the theoretical predictederrors and prove the effectiveness of our method for multiscalesimulation of conductive and radiative heat transfer problems inperiodically perforated domains. (10.1137/080714737)
  DOI : 10.1137/080714737
• Control problems with mixed constraints and application to an optimal investment problem
  
  • Bonnans J. Frederic
  • Tiba Dan
  Mathematical Reports, Romanian Academy of Sciences, 2009, 4, pp.293-306.
• Comparison principle for a Generalized Fast Marching Method
  
  • Forcadel Nicolas
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.pp. 1923-1951. In \cite{CFFM06}, the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in \cite{CFFM06} for which we state a general comparison principle. We also prove the convergence of the new algorithm.
• Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher
  
  • Martinon Pierre
  • Bonnans J. Frederic
  • Laurent-Varin Julien
  • Trélat Emmanuel
  Journal of Guidance, Control, and Dynamics, American Institute of Aeronautics and Astronautics, 2009, 32 (1), pp.51--55. We consider a flight mission to the geostationary transfer orbit (GTO) for an Ariane 5 launcher, while maximizing the payload or, as a variant, minimizing the fuel consumption. We first solve the complete flight sequence up to the final orbit, assuming a maximal thrust for all propulsion systems. Then we focus on the tmospheric ascent phase, which has been studied for instance in [1, 2, 3]. We are more specifically interested in optimal tra jectories with singular arcs (flight phases with a non maximal thrust) for the boosters. Due to the presence of tabulated data in the physical model, the exact expression of the singular control cannot be obtained from the time derivatives of the switching function. An alternate way to compute the singular control is provided, and numerical experiments are carried out for for several launcher variants.
• Music for extended scatterers as an instance of the factorization method
  
  • Arens Tilo
  • Lechleiter Armin
  • Luke D. Russell
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2009, 70 (4), pp.1283--1304. (10.1137/080737836)
  DOI : 10.1137/080737836
• Dynamics of red blood cells in 2D
  
  • Bui Cuc
  • Lleras Vanessa
  • Pantz Olivier
  , 2009, 28, pp.182--194. (10.1051/proc/2009046)
  DOI : 10.1051/proc/2009046
• Generalized Impedance Boundary Conditions for Thin Dielectric Coatings with Variable Thickness
  
  • Aslanyurek Birol
  • Haddar Houssem
  • Sahinturk Hulya
  , 2009. We derive so-called Generalized Impedance Boundary Conditions (GIBC) that model thin dielectric coatings with variable width. We treat the 2-D electromagnetic problem for both TM and TE polarizations. The expressions of the GIBCs are explicited up to the third order (with respect to the coating width). The order of convergence is numerically validated through various numerical examples. A particular attention is given to the cases where the inner boundary has corner singularities.
• Lyapunov conditions for logarithmic Sobolev and Super Poincaré inequality
  
  • Cattiaux Patrick
  • Guillin Arnaud
  • Wang Feng-Yu
  • Wu Liming
  Journal of Functional Analysis, Elsevier, 2009, 256 (6), pp.1821-1841. We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...).
• The operator equations of Lippmann-Schwinger type for acoustic and electromagnetic scattering problems in $L^2$
  
  • Kirsch A.
  • Lechleiter A.
  Applicable Analysis, Taylor & Francis, 2009, 88 (6), pp.807--830. (10.1080/00036810903042125)
  DOI : 10.1080/00036810903042125
• Editorial: Special Issue on Inverse Scattering Problems dedicated to D. Colton and R. Kress
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Piana Michele
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2009, 3 (2), pp.i-i. This special issue is dedicated to Professors David Colton and Rainer Kress in honor of their contribution and leadership in the area of direct and inverse scattering theory for more then 30 years. The papers in this special issue were solicited from the invited speakers at the International Conference on Inverse Scattering Problems organized in honor of the 65th birthdays of David Colton and Rainer Kress held in the seaside resort of Sestry Levante, Italy, May 8-10, 2008. (10.3934/ipi.2009.3.2i)
  DOI : 10.3934/ipi.2009.3.2i
• The computation of lower bounds for the norm of the index of refraction in an anisotropic media from far field data
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2009, 21, pp.203--227. We consider the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous, anisotropic dielectric medium and show that under certain as- sumptions a lower bound on the norm of the (matrix) index of refraction can be obtained from a knowledge of the smallest transmission eigenvalue corresponding to the medium. Nu- merical examples are given showing the efficaciousness of our estimates. (10.1216/JIE-2009-21-2-203)
  DOI : 10.1216/JIE-2009-21-2-203
• Some convergence results for Howard's algorithm
  
  • Bokanowski Olivier
  • Maroso Stefania
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (4), pp.3001--3026. This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in \cA} (B^a x - b^a)=0$ where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and $\cA$ is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem. (10.1007/s00245-006-0865-2)
  DOI : 10.1007/s00245-006-0865-2
• Two Asymptotic Models for Arrays of Underground Waste Containers
  
  • Allaire Grégoire
  • Briane M.
  • Brizzi R.
  • Capdeboscq Y.
  Applicable Analysis, Taylor & Francis, 2009, 88, pp.1445-1467. We study the homogenization of two models of an underground nuclear waste repository. The nuclear waste cells are periodically stored in the middle of a geological layer and are the only source terms in a parabolic evolution problem. The diffusion constants have a very large contrast between the fuel repository and the soil. It is thus a combined problem of homogenization and singular perturbation. For two different asymptotic contrasts we give the homogenized limit problem which is rigorously justified by using two-scale convergence. Eventually we perform 2-d numerical computations to show the effectiveness of using the limit model instead of the original one.
• A post-treatment of the homogenization method in shape optimization
  
  • Trabelsi Karim
  • Pantz Olivier
  , 2009.
• Central Limit Theorems for the Brownian motion on large unitary groups
  
  • Benaych-Georges Florent
  , 2009. In this paper, we are concerned with the large N limit of linear combinations of the entries of a Brownian motion on the group of N by N unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distribution are concerned, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a quite short proof of the asymptotic Gaussian feature of the linear combinations of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al.
• Two-dimensional almost-Riemannian structures with tangency points
  
  • Agrachev Andrei
  • Boscain Ugo
  • Ghezzi Roberta
  • Charlot Grégoire
  • Sigalotti Mario
  , 2009, pp.4340-4345.
• Numerical approximation for a superreplication problem under gamma constraints
  
  • Bruder Benjamin
  • Bokanowski Olivier
  • Maroso Stefania
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.2289-2320. We study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized finite difference scheme is used since basic finite differences cannot work in our case. Numerical tests illustrate the validity of our approach. (10.1137/080725222)
  DOI : 10.1137/080725222
• Approximate transmission conditions through a weakly oscillating thin layer
  
  • Poignard Clair
  Mathematical Methods in the Applied Sciences, Wiley, 2009, 32 (4), pp.435-453. We study the behavior of the electro-quasistatic voltage potentials in a material composed by a bidimensional medium surrounded by a weakly oscillating thin layer and embedded in an ambient medium. We build approximate transmission conditions in order to replace the layer by these conditions on the boundary of the interior material. We deal with a weakly oscillating thin layer: the period of the oscillations is greater than the square root of the thinness. Our approach is essentially geometric and based on a suitable change of variable in the layer. This paper extends previous works of the former author, in which the layer had constant thickness. (10.1002/mma.1045)
  DOI : 10.1002/mma.1045
• Quasi-stationary distributions and diffusion models in population dynamics
  
  • Cattiaux Patrick
  • Collet Pierre
  • Lambert Amaury
  • Martinez Servet
  • Méléard Sylvie
  • San Martín Jaime
  The Annals of Probability, Institute of Mathematical Statistics, 2009, 37 (5), pp.1926-1969. (10.1214/09-AOP451)
  DOI : 10.1214/09-AOP451
• Discrete-Time Approximation of BSDEs and Probabilistic Schemes for Fully Nonlinear PDEs
  
  • Bouchard Bruno
  • Elie Romuald
  • Touzi Nizar
  Radon Series Comp. Appl. Math., Advanced Financial Modelling, 2009, 8, pp.91-124.
• Estimation de Volatilité en Présence de Bruit de Microstructure Endogène
  
  • Robert Christian Yann
  • Rosenbaum Mathieu
  , 2009. Ce papier considère des procédures statistiques facilement implémentables pour l'estimation de mesures de volatilité haute fréquence pour des actifs financiers. Le modèle de microstructure sous-jascent se base sur un prix efficient de type semi-martingale continue et permet de reproduire les principales caractéristiques empiriques des données ultra haute fréquence. Dans ce modèle, le bruit de microstructure est endogène mais ne dépend pas uniquement du prix efficient. En utilisant les prix de transaction observés, nous développons une nouvelle approche permettant d'approximer les valeurs du prix efficient à certains instants aléatoires. En se basant sur ces valeurs approchées, on construit un estimateur de la volatilité intégrée et on fournit sa théorie asymptotique. On donne aussi un estimateur consistant de la co-volatilité intégrée dans le cas où deux actifs (asynchrones par construction du modèle) sont observés.
• Existence of solutions for a model describing the dynamics of junctions between dislocations
  
  • Forcadel Nicolas
  • Monneau Régis
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2009, 40 (6), pp.pp. 2517-2535. We study a dynamical version of a multi-phase field model of Koslowski and Ortiz for planar dislocation networks. We consider a two-dimensional vector field which describes phase transitions between constant phases. Each phase transition corresponds to a dislocation line, and the vectorial field description allows the formation of junctions between dislocations. This vector field is assumed to satisfy a non-local vectorial Hamilton-Jacobi equation with non-zero viscosity. For this model, we prove the existence for all time of a weak solution. (10.1137/070710925)
  DOI : 10.1137/070710925