Publications

2009

• 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
• On a stochastic Korteweg-de Vries equation with homogeneous noise
  
  • de Bouard Anne
  • Debussche Arnaud
  , 2009.
• Homogenization of variational problems in manifold valued BV-spaces
  
  • Babadjian Jean-François
  • Millot Vincent
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2009, 36 (1), pp.7-47. This paper extends the result of Babadjian and Millot (preprint, 2008) on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a $\Gamma$-convergence analysis, we identify the homogenized energy in the space of functions of bounded variation. It turns out to be finite for $BV$-maps with values in the manifold. The bulk and Cantor parts of the energy involve the tangential homogenized density introduced in Babadjian and Millot (preprint, 2008), while the jump part involves an homogenized surface density given by a geodesic type problem on the manifold. (10.1007/s00526-008-0220-3)
  DOI : 10.1007/s00526-008-0220-3
• Transportation-information inequalities for Markov processes
  
  • Guillin Arnaud
  • Léonard Christian
  • Wu Liming
  • Yao Nian
  Probability Theory and Related Fields, Springer Verlag, 2009, 144 (3-4), pp.669-695. In this paper, one investigates the following type of transportation-information $T_cI$ inequalities: $\alpha(T_c(\nu,\mu))\le I(\nu|\mu)$ for all probability measures $\nu$ on some metric space $(\XX, d)$, where $\mu$ is a given probability measure, $T_c(\nu,\mu)$ is the transportation cost from $\nu$ to $\mu$ with respect to some cost function $c(x,y)$ on $\XX^2$, $I(\nu|\mu)$ is the Fisher-Donsker-Varadhan information of $\nu$ with respect to $\mu$ and $\alpha: [0,\infty)\to [0,\infty]$ is some left continuous increasing function. Using large deviation techniques, it is shown that $T_cI$ is equivalent to some concentration inequality for the occupation measure of a $\mu$-reversible ergodic Markov process related to $I(\cdot|\mu)$, a counterpart of the characterizations of transportation-entropy inequalities, recently obtained by Gozlan and Léonard in the i.i.d.\! case . Tensorization properties of $T_cI$ are also derived.
• Interacting Multi-Class Transmissions in Large Stochastic Networks
  
  • Graham Carl
  • Robert Philippe
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2009, 19 (6), pp.2334-2361. The mean-field limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. In the same way as for the TCP algorithm, each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. Since several classes of connections going through the nodes of the network are considered, an original mean-field result in a multi-class context is established. It is shown that, as the number of connections goes to infinity, the behavior of the different classes of connections can be represented by the solution of an unusual non-linear stochastic differential equation depending not only on the sample paths of the process, but also on its distribution. Existence and uniqueness results for the solutions of these equations are derived. Properties of their invariant distributions are investigated and it is shown that, under some natural assumptions, they are determined by the solutions of a fixed point equation in a finite dimensional space. (10.1214/09-AAP614)
  DOI : 10.1214/09-AAP614
• Rectangular R-transform as the limit of rectangular spherical integrals
  
  • Benaych-Georges Florent
  , 2009. In this paper, we connect rectangular free probability theory and spherical integrals. In this way, we prove the analogue, for rectangular or square non-Hermitian matrices, of a result that Guionnet and Maida proved for Hermitian matrices in 2005. More specifically, we study the limit, as $n,m$ tend to infinity, of the logarithm (divided by $n$) of the expectation of $\exp[\sqrt{nm}\theta X_n]$, where $X_n$ is the real part of an entry of $U_n M_n V_m$, $\theta$ is a real number, $M_n$ is a certain $n\times m$ deterministic matrix and $U_n, V_m$ are independent Haar-distributed orthogonal or unitary matrices with respective sizes $n\times n$, $m\times m$. We prove that when the singular law of $M_n$ converges to a probability measure $\mu$, for $\theta$ small enough, this limit actually exists and can be expressed with the rectangular R-transform of $\mu$. This gives an interpretation of this transform, which linearizes the rectangular free convolution, as the limit of a sequence of log-Laplace transforms.
• Asymptotic models for scattering problems from unbounded media with high conductivity
  
  • Haddar Houssem
  • Lechleiter Armin
  , 2009, pp.29. We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no longer possible. Our new analysis is based on the use of Rellich-type estimates and boundedness of $L2$ solution operators. We also discuss numerical approximation of obtained GIBC (up to order 3) and numerically test theoretical convergence rates.