Publications

2009

• 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
• Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise
  
  • de Bouard Anne
  • Debussche Arnaud
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2009, 14, pp.1727-1744. We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the dynamics of the soliton of the KdV equation in the presence of this random perturbation, assuming that the amplitude of the perturbation is small. We estimate precisely the exit time of the perturbed solution from a neighborhood of the modulated soliton, and we obtain the modulation equations for the soliton parameters. We moreover prove a central limit theorem for the dispersive part of the solution, and investigate the asymptotic behavior in time of the limit process.
• 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.
• Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation
  
  • de Bouard Anne
  • Fukuizumi Reika
  Asymptotic Analysis, IOS Press, 2009, 63 (4), pp.189-235. We study the asymptotic behavior of the solution of a model equation for Bose- Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude ε tends to zero. The initial condition of the solution is a standing wave solution of the unperturbed equation. We prove that up to times of the order of ε−2, the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as ε goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale. (10.3233/ASY-2008-0931)
  DOI : 10.3233/ASY-2008-0931
• High order accurate methods for the evaluation of layer heat potentials
  
  • Li Jing-Rebecca
  • Greengard Leslie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2009, 31 (5), pp.3847--3860. (10.1137/080732389)
  DOI : 10.1137/080732389
• Revisiting the Analysis of Optimal Control Problems with Several State Constraints
  
  • Bonnans Joseph Frederic
  • Hermant Audrey
  Control and Cybernetics, Polish Academy of Sciences, 2009, 38 (4), pp.1021--1052.
• Modelling and Numerical Simulation of Liquid-Vapor Phase Transition
  
  • Faccanoni Gloria
  • Allaire Grégoire
  • Kokh Samuel
  , 2009. The present work is dedicated to the simulation of compressible two-phase flows with phase change for pool boiling type problems. The model we are concerned with involves scales that allow to distinguish the interface between both phases. The mass transfer is driven by assuming local and instantaneous equilibria with respect to phasic pressures, temperatures and chemical potentials, which enables dynamic generation of two-phase interfaces within a pure phase. We present a general numerical solver that allows to cope with any type of EOS and preliminary numerical results of nucleation with transition towards film boiling.
• The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices
  
  • Benaych-Georges Florent
  • Rao Raj
  , 2009. We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models. The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of `spiked' random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.
• Efficient thermal field computation in phase-field models
  
  • Li Jing-Rebecca
  • Calhoun Donna
  • Brush Lucien
  Journal of Computational Physics, Elsevier, 2009, 228 (24), pp.8945--8957. (10.1016/j.jcp.2009.08.022)
  DOI : 10.1016/j.jcp.2009.08.022
• A stokesian submarine
  
  • Lefebvre-Lepot Aline
  • Merlet Benoît
  ESAIM: Proceedings, EDP Sciences, 2009, 28, pp.150-161. We consider the problem of swimming at low Reynolds numbers. This is the relevant asymptotic for micro-and nano-robots needing to navigate in an aqueous medium. As a model, we propose a robot composed of three balls. The relative positions of these balls can change according to three degrees of freedom. We prove that this robot is able to navigate in a plane by modifying the conformation of its shape. Résumé. Nous considérons le problème de la nageà faible nombre de Reynolds. C'est l'asymptotique pertinente pour les micro-et nano-robots devant se déplacer dans un milieu aqueux. Nous proposons comme modèle un robot formé de trois boules qui peuvent se déplacer les unes par rapport aux autres selon trois degrés de liberté. Nous démontrons qu'en changeant sa conformation, ce robot peut effectivement naviguer dans un plan. (10.1051/proc/2009044)
  DOI : 10.1051/proc/2009044