Publications

2009

• 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
• Stability Analysis of Optimal Control Problems with a Second-order State Constraint
  
  • Hermant Audrey
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 20 (1), pp.104-129. This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak second-order sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the $L^2$ norm, and Hölder continuous in the $L^\infty$ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 1998]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a ``primitive'' of the state constraint multiplier. (10.1137/070707993)
  DOI : 10.1137/070707993
• Diffractive behavior of the wave equation in periodic media: weak convergence analysis
  
  • Allaire Grégoire
  • Palombaro M.
  • Rauch J.
  Annali di Matematica Pura ed Applicata, Springer Verlag, 2009, 188, pp.561-590. We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider inital data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schr\"{o}dinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schr\"{o}dinger equation. (10.1007/s10231-008-0089-y)
  DOI : 10.1007/s10231-008-0089-y
• 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
• 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
• 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.
• 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
• 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.
• 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.
• 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.
• Continuous limits of discrete perimeters
  
  • Chambolle Antonin
  • Giacomini Alessandro
  • Lussardi Luca
  , 2009. We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula, and study their limit in the continuum.
• Second-order Analysis for Optimal Control Problems with Pure State Constraints and Mixed Control-State Constraints
  
  • Bonnans Joseph Frederic
  • Hermant Audrey
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2009, 26 (2), pp.561-598.
• Sensitivity analysis in HMMs with application to likelihood maximization
  
  • Coquelin Pierre-Arnaud
  • Deguest Romain
  • Munos Rémi
  , 2009. This paper considers a sensitivity analysis in Hidden Markov Models with continuous state and observation spaces. We propose an Infinitesimal Perturbation Analysis (IPA) on the filtering distribution with respect to some parameters of the model. We describe a methodology for using any algorithm that estimates the filtering density, such as Sequential Monte Carlo methods, to design an algorithm that estimates its gradient. The resulting IPA estimator is proven to be asymptotically unbiased, consistent and has computational complexity linear in the number of particles. We consider an application of this analysis to the problem of identifying unknown parameters of the model given a sequence of observations. We derive an IPA estimator for the gradient of the log-likelihood, which may be used in a gradient method for the purpose of likelihood maximization. We illustrate the method with several numerical experiments.
• On the existence of transmission eigenvalues in an inhomogeneous medium
  
  • Cakoni Fioralba
  • Haddar Houssem
  Applicable Analysis, Taylor & Francis, 2009, 88 (4), pp.475-493. We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783-753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough. (10.1080/00036810802713966)
  DOI : 10.1080/00036810802713966
• Homogenization of fully overdamped Frenkel-Kontorova models
  
  • Forcadel Nicolas
  • Imbert Cyril
  • Monneau Régis
  Journal of Differential Equations, Elsevier, 2009, 246, pp.pp 1057--1097. In this paper, we consider the fully overdamped Frenkel-Kontorova model. This is an infinite system of coupled first order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel-Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions. (10.1016/j.jde.2008.06.034)
  DOI : 10.1016/j.jde.2008.06.034
• Acoustic response of a rigid-frame porous medium plate with a periodic set of inclusions
  
  • Groby Jean-Philippe
  • Wirgin Armand
  • de Ryck L.
  • Lauriks W.
  • Gilbert R. P.
  • Xu Y. S.
  Journal of the Acoustical Society of America, Acoustical Society of America, 2009, 126 (2), pp.685-693. The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption. (10.1121/1.3158936)
  DOI : 10.1121/1.3158936
• 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.
• 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 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
• 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
• 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.
• 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
• Two-dimensional almost-Riemannian structures with tangency points
  
  • Agrachev Andrei
  • Boscain Ugo
  • Ghezzi Roberta
  • Charlot Grégoire
  • Sigalotti Mario
  , 2009, pp.4340-4345.