Publications

2009

• 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.
• 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.
• Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+/-. I
  
  • Barbaroux Jean-Marie
  • Guillot J. -C.
  Advances in Mathematical Physics, Hindawi Publishing Corporation, 2009, 2009, pp.978903. We consider a Hamiltonian with cutoffs describing the weak decay of spin one massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval. (10.1155/2009/978903)
  DOI : 10.1155/2009/978903
• Efficient solution of a wave equation with fractional order dissipative terms
  
  • Haddar Houssem
  • Li Jing-Rebecca
  • Matignon Denis
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.2003-2010. (10.1016/j.cam.2009.08.051)
  DOI : 10.1016/j.cam.2009.08.051
• Forgetting of the initial distribution for Hidden Markov Models
  
  • Douc Randal
  • Fort Gersende
  • Moulines Éric
  • Priouret Pierre
  Stochastic Processes and their Applications, Elsevier, 2009, 119 (4), pp.1235--1256. The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the non-linear state space model and the stochastic volatility model.
• On iterative reconstruction in the nonlinearized polarization tomography
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2009, 25 (11), pp.115010 (18pp). We give uniqueness theorem and reconstruction algorithm for the nonlinearized problem of finding the dielectric anisotropy f of the medium from non-overdetermined polarization tomography data. We assume that the medium has uniform background parameters and that the anisotropic (dielectric permeability) perturbation is described by symmetric and sufficiently small matrix-function f . On a pure mathematical level this article contributes to the theory of non-abelian Radon transforms and to iterative methods of inverse scattering.
• GAUSSIAN MODEL SELECTION WITH AN UNKNOWN VARIANCE
  
  • Baraud Yannick
  • Giraud Christophe
  • Huet Sylvie
  Annals of Statistics, Institute of Mathematical Statistics, 2009, 37 (2), pp.630-672.. Let Y be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean μ of Y by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_{m},m\in\mathcal{M}\}$ of linear subspaces of ℝn and associate to each of these the least-squares estimator of μ on Sm. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the performance of estimators associated to classical criteria such as FPE, AIC, BIC and AMDL. Our second objective is to propose better penalties that are versatile enough to take into account both the complexity of the collection $\mathcal{S}$ and the sample size. Then we apply those to solve various statistical problems such as variable selection, change point detections and signal estimation among others. Our results are based on a nonasymptotic risk bound with respect to the Euclidean loss for the selected estimator. Some analogous results are also established for the Kullback loss.
• The factorization method is independent of transmission eigenvalues
  
  • Lechleiter Armin
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2009, 3 (1), pp.123--138. (10.3934/ipi.2009.3.123)
  DOI : 10.3934/ipi.2009.3.123
• Planning reinforcement on gas transportation networks with optimization methods
  
  • Bonnans Joseph Frederic
  • André Jean
  • Cornibert Laurent
  European Journal of Operational Research, Elsevier, 2009, 197 (3), pp.1019-1027.
• TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS
  
  • Allaire Grégoire
  • Briane Marc
  • Brizzi Robert
  • Capdeboscq Yves
  Applicable Analysis, Taylor & Francis, 2009, 88 (10-11), 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. (10.1080/00036810902922590)
  DOI : 10.1080/00036810902922590
• Dynamic Numerical Investigation of Random Packing for Spherical and Nonconvex Particles
  
  • Faure Sylvain
  • Lefebvre-Lepot Aline
  • Semin Benoît
  ESAIM: Proceedings, EDP Sciences, 2009, 28, pp.13-32. We simulate the sedimentation in a parallelepipedic container of spheres and nonconvex particles constituted by two overlapping spheres. We use the self-written code SCoPI. Thanks to an efficient handling of contacts between particles, it allowed us to consider up to 100, 000 spheres and 10, 000 nonconvex particles. The packing fraction (in bulk and close to a wall) as well as the mean value and the distribution of contacts of the final packings are reported. The results obtained for the classical case of spherical particles (packing fraction: 63.7%, mean number of contacts: 6) are in agreement with previous studies and validate the algorithm. The packing fraction for nonconvex particles increases and then decreases with respect to the aspect ratio, which is similar to the ellipsoid (convex) case. The number of contacts is different from the number of neighbours, which is of course never the case for spherical particles (convex particles). The number of contacts is discontinuous when slightly increasing the aspect ratio from the spherical case: it is equal to 6 in the spherical case and to 10 in the nonconvex case. These values correspond to the isocounting values, i.e. the number of contacts is twice the number of degrees of freedom. This contrasts with the ellipsoid case, where it sharply but continuously increases. Concerning the number of neighbours, it continuously increases for small aspect ratio (which is similar to the convex particle case), but decreases for higher aspect ratio. (10.1051/proc/2009037)
  DOI : 10.1051/proc/2009037
• 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
• 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
• 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
• 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
• 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 ...).
• 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
• Two-dimensional almost-Riemannian structures with tangency points
  
  • Agrachev Andrei
  • Boscain Ugo
  • Ghezzi Roberta
  • Charlot Grégoire
  • Sigalotti Mario
  , 2009, pp.4340-4345.
• 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.
• 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
• 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.
• 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.
• 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.
• A post-treatment of the homogenization method in shape optimization
  
  • Trabelsi Karim
  • Pantz Olivier
  , 2009.
• 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