Publications

2009

• 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
• 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
• 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
• 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.
• 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
• 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.
• 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
• 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.
• 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
• 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 ...).
• 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
• 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
• 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.
• 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
• 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.
• 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
• 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.
• 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
• A post-treatment of the homogenization method in shape optimization
  
  • Trabelsi Karim
  • Pantz Olivier
  , 2009.
• 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
• Restauration d'images floutées & bruitées par une variante originale de la variation totale
  
  • Jalalzai Khalid
  • Chambolle Antonin
  , 2009. Dans cet article, nous introduisons une nouvelle variante de la variation totale (TV ) dont l'objectif est de simplifier la restauration d'images à base de TV lorsque cellesci sont dégradées par un noyau qui se calcule facilement du côté Fourier (flou, transformée de Radon,...). L'idée est de remplacer simplement le terme TV par la norme L1 d'un certain champ de vecteur, pour lequel l'optimisation est beaucoup plus facile. Cette approche nous permet ainsi d'utiliser un algorithme récent et rapide pour restaurer entre autres des images bruitées et floutées. Nous comparons notre approche avec la méthode classique basée sur la variation totale et montrons sa supériorité.