Publications

2009

• Structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2009. Tridimensional fractal structure (Structure fractale tridimensionnelle)
• Structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2009. Tridimensional fractal structure (Structure fractale tridimensionnelle)
• Une interpolation entre deux réseaux fractals
  
  • Colonna Jean-François
  , 2009. An interpolation between two fractal networks (Une interpolation entre deux réseaux fractals)
• Effet Larsen
  
  • Colonna Jean-François
  , 2009. Larsen effect (Effet Larsen)
• Réseau
  
  • Colonna Jean-François
  , 2009. Network (Réseau)
• Réseau fractal
  
  • Colonna Jean-François
  , 2009. Fractal network (Réseau fractal)
• 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
• 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.
• 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
• 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.
• 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.
• Homogenization of fully overdamped Frenkel-Kontorova models
  
  • Forcadel Nicolas
  • Imbert Cyril
  • Monneau Régis
  Journal of Differential Equations, Elsevier, 2009, 246, pp.pp 1057--1097. In this paper, we consider the fully overdamped Frenkel-Kontorova model. This is an infinite system of coupled first order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel-Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions. (10.1016/j.jde.2008.06.034)
  DOI : 10.1016/j.jde.2008.06.034
• On the existence of transmission eigenvalues in an inhomogeneous medium
  
  • Cakoni Fioralba
  • Haddar Houssem
  Applicable Analysis, Taylor & Francis, 2009, 88 (4), pp.475-493. We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783-753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough. (10.1080/00036810802713966)
  DOI : 10.1080/00036810802713966
• Acoustic response of a rigid-frame porous medium plate with a periodic set of inclusions
  
  • Groby Jean-Philippe
  • Wirgin Armand
  • de Ryck L.
  • Lauriks W.
  • Gilbert R. P.
  • Xu Y. S.
  Journal of the Acoustical Society of America, Acoustical Society of America, 2009, 126 (2), pp.685-693. The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption. (10.1121/1.3158936)
  DOI : 10.1121/1.3158936
• Sensitivity analysis in HMMs with application to likelihood maximization
  
  • Coquelin Pierre-Arnaud
  • Deguest Romain
  • Munos Rémi
  , 2009. This paper considers a sensitivity analysis in Hidden Markov Models with continuous state and observation spaces. We propose an Infinitesimal Perturbation Analysis (IPA) on the filtering distribution with respect to some parameters of the model. We describe a methodology for using any algorithm that estimates the filtering density, such as Sequential Monte Carlo methods, to design an algorithm that estimates its gradient. The resulting IPA estimator is proven to be asymptotically unbiased, consistent and has computational complexity linear in the number of particles. We consider an application of this analysis to the problem of identifying unknown parameters of the model given a sequence of observations. We derive an IPA estimator for the gradient of the log-likelihood, which may be used in a gradient method for the purpose of likelihood maximization. We illustrate the method with several numerical experiments.
• Continuous limits of discrete perimeters
  
  • Chambolle Antonin
  • Giacomini Alessandro
  • Lussardi Luca
  , 2009. We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula, and study their limit in the continuum.
• Second-order Analysis for Optimal Control Problems with Pure State Constraints and Mixed Control-State Constraints
  
  • Bonnans Joseph Frederic
  • Hermant Audrey
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2009, 26 (2), pp.561-598.
• 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.
• 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