Publications

2009

• Measurability of optimal transportation and convergence rate for Landau type interacting particle systems
  
  • Fontbona Joaquin
  • Guérin Hélène
  • Méléard Sylvie
  Probability Theory and Related Fields, Springer Verlag, 2009, 143 (3-4), pp.329-351. In this paper, we consider nonlinear diffusion processes driven by space-time white noises, which have an interpretation in terms of partial differential equations. For a specific choice of coefficients, they correspond to the Landau equation arising in kinetic theory. A particular feature is that the diffusion matrix of this process is a linear function the law of the process, and not a quadratic one, as in the McKean-Vlasov model. The main goal of the paper is to construct an easily simulable diffusive interacting particle system, converging towards this nonlinear process and to obtain an explicit pathwise rate. This requires to find a significant coupling between finitely many Brownian motions and the infinite dimensional white noise process. The key idea will be to construct the right Brownian motions by pushing forward the white noise processes, through the Brenier map realizing the optimal transport between the law of the nonlinear process, and the empirical measure of independent copies of it. A striking problem then is to establish the joint measurability of this optimal transport map with respect to the space variable and the parameters (time and randomness) making the marginals vary. We shall prove a general measurability result for the mass transportation problem in terms of the support of the transfert plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates. (10.1007/s00440-007-0128-4)
  DOI : 10.1007/s00440-007-0128-4
• 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.
• Weak solutions to the equations of motion for compressible magnetic fluids
  
  • Amirat Youcef
  • Hamdache Kamel
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2009, 91, pp.433-467.
• Strong solutions to the equations of a ferrofluid flow model
  
  • Amirat Youcef
  • Hamdache Kamel
  Journal of Mathematical Analysis and Applications, Elsevier, 2009, 353 (1), pp.271-294.
• 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
• 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
• 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
• 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
• 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.
• 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
• 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.
• 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
• 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
• 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.
• 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.
• 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
• 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
• 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
• 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 ...).
• 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
• 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.
• Two-dimensional almost-Riemannian structures with tangency points
  
  • Agrachev Andrei
  • Boscain Ugo
  • Ghezzi Roberta
  • Charlot Grégoire
  • Sigalotti Mario
  , 2009, pp.4340-4345.
• 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