Publications

2009

• 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.
• 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.
• 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.
• 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.
• 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
• 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.
• 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.
• 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.
• 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
• 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
• 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
• 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