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.
• 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
• 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.
• Dynamic Numerical Investigation of Random Packing for Spherical and Nonconvex Particles
  
  • Faure Sylvain
  • Lefebvre-Lepot Aline
  • Semin Benoît
  ESAIM: Proceedings, EDP Sciences, 2009, 28, pp.13-32. We simulate the sedimentation in a parallelepipedic container of spheres and nonconvex particles constituted by two overlapping spheres. We use the self-written code SCoPI. Thanks to an efficient handling of contacts between particles, it allowed us to consider up to 100, 000 spheres and 10, 000 nonconvex particles. The packing fraction (in bulk and close to a wall) as well as the mean value and the distribution of contacts of the final packings are reported. The results obtained for the classical case of spherical particles (packing fraction: 63.7%, mean number of contacts: 6) are in agreement with previous studies and validate the algorithm. The packing fraction for nonconvex particles increases and then decreases with respect to the aspect ratio, which is similar to the ellipsoid (convex) case. The number of contacts is different from the number of neighbours, which is of course never the case for spherical particles (convex particles). The number of contacts is discontinuous when slightly increasing the aspect ratio from the spherical case: it is equal to 6 in the spherical case and to 10 in the nonconvex case. These values correspond to the isocounting values, i.e. the number of contacts is twice the number of degrees of freedom. This contrasts with the ellipsoid case, where it sharply but continuously increases. Concerning the number of neighbours, it continuously increases for small aspect ratio (which is similar to the convex particle case), but decreases for higher aspect ratio. (10.1051/proc/2009037)
  DOI : 10.1051/proc/2009037
• Spectral theory for a mathematical model of the weak interactions: The decay of the intermediate vector bosons W+/-. I
  
  • Barbaroux Jean-Marie
  • Guillot J. -C.
  Advances in Mathematical Physics, Hindawi Publishing Corporation, 2009, 2009, pp.978903. We consider a Hamiltonian with cutoffs describing the weak decay of spin one massive bosons into the full family of leptons. The Hamiltonian is a self-adjoint operator in an appropriate Fock space with a unique ground state. We prove a Mourre estimate and a limiting absorption principle above the ground state energy and below the first threshold for a sufficiently small coupling constant. As a corollary, we prove absence of eigenvalues and absolute continuity of the energy spectrum in the same spectral interval. (10.1155/2009/978903)
  DOI : 10.1155/2009/978903
• Forgetting of the initial distribution for Hidden Markov Models
  
  • Douc Randal
  • Fort Gersende
  • Moulines Éric
  • Priouret Pierre
  Stochastic Processes and their Applications, Elsevier, 2009, 119 (4), pp.1235--1256. The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the non-linear state space model and the stochastic volatility model.
• On iterative reconstruction in the nonlinearized polarization tomography
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2009, 25 (11), pp.115010 (18pp). We give uniqueness theorem and reconstruction algorithm for the nonlinearized problem of finding the dielectric anisotropy f of the medium from non-overdetermined polarization tomography data. We assume that the medium has uniform background parameters and that the anisotropic (dielectric permeability) perturbation is described by symmetric and sufficiently small matrix-function f . On a pure mathematical level this article contributes to the theory of non-abelian Radon transforms and to iterative methods of inverse scattering.
• GAUSSIAN MODEL SELECTION WITH AN UNKNOWN VARIANCE
  
  • Baraud Yannick
  • Giraud Christophe
  • Huet Sylvie
  Annals of Statistics, Institute of Mathematical Statistics, 2009, 37 (2), pp.630-672.. Let Y be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean μ of Y by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_{m},m\in\mathcal{M}\}$ of linear subspaces of ℝn and associate to each of these the least-squares estimator of μ on Sm. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the performance of estimators associated to classical criteria such as FPE, AIC, BIC and AMDL. Our second objective is to propose better penalties that are versatile enough to take into account both the complexity of the collection $\mathcal{S}$ and the sample size. Then we apply those to solve various statistical problems such as variable selection, change point detections and signal estimation among others. Our results are based on a nonasymptotic risk bound with respect to the Euclidean loss for the selected estimator. Some analogous results are also established for the Kullback loss.
• The factorization method is independent of transmission eigenvalues
  
  • Lechleiter Armin
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2009, 3 (1), pp.123--138. (10.3934/ipi.2009.3.123)
  DOI : 10.3934/ipi.2009.3.123
• Planning reinforcement on gas transportation networks with optimization methods
  
  • Bonnans Joseph Frederic
  • André Jean
  • Cornibert Laurent
  European Journal of Operational Research, Elsevier, 2009, 197 (3), pp.1019-1027.
• 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
• Efficient solution of a wave equation with fractional order dissipative terms
  
  • Haddar Houssem
  • Li Jing-Rebecca
  • Matignon Denis
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.2003-2010. (10.1016/j.cam.2009.08.051)
  DOI : 10.1016/j.cam.2009.08.051
• Homogenization of a conductive and radiative heat transfer problem
  
  • Allaire Grégoire
  • El Ganaoui K.
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2009, 7 (3), pp.1148-1170. This paper is devoted to the homogenization of a heat conductionproblem in a periodically perforated domain with a nonlinear andnonlocal boundary condition modeling radiative heat transfer inthe perforations. Because of the critical scaling considered it isessential to use a method of two-scale asymptotic expansionsinside the variational formulation of the problem. We obtain anonlinear homogenized problem of heat conduction with effectivecoefficients which are computed via a cell problem featuring aradiative heat transfer boundary condition. We rigorously justifythis homogenization process for the linearized problem by usingtwo-scale convergence. We perform numerical simulations in twodimensions: we reconstruct an approximate temperature field byadding to the homogenized temperature a corrector term. Thecomputed numerical errors agree with the theoretical predictederrors and prove the effectiveness of our method for multiscalesimulation of conductive and radiative heat transfer problems inperiodically perforated domains. (10.1137/080714737)
  DOI : 10.1137/080714737
• Control problems with mixed constraints and application to an optimal investment problem
  
  • Bonnans J. Frederic
  • Tiba Dan
  Mathematical Reports, Romanian Academy of Sciences, 2009, 4, pp.293-306.
• Comparison principle for a Generalized Fast Marching Method
  
  • Forcadel Nicolas
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.pp. 1923-1951. In \cite{CFFM06}, the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in \cite{CFFM06} for which we state a general comparison principle. We also prove the convergence of the new algorithm.
• Music for extended scatterers as an instance of the factorization method
  
  • Arens Tilo
  • Lechleiter Armin
  • Luke D. Russell
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2009, 70 (4), pp.1283--1304. (10.1137/080737836)
  DOI : 10.1137/080737836