Publications

2007

• Acoustic wave propagation and internal fields in rigid frame macroscopically inhomogeneous porous media
  
  • de Ryck Laurent
  • Lauriks Walter
  • Fellah Zine El Abiddine
  • Wirgin Armand
  • Groby J.-P.
  • Leclaire Philippe
  • Dépollier Claude
  Journal of Applied Physics, American Institute of Physics, 2007, 102 (2), pp.024910.1-024910.11. A wave propagation model in macroscopically inhomogeneous porous media is derived from the alternative Biot's theory of 1962. As a first application, the wave equation is reduced and solved in the case of rigid frame inhomogeneous porous materials. The pressure field, as well as the reflection and transmission coefficients, are obtained numerically using a wave splitting and “transmission” Green's functions approach (WS-TGF). To validate both the wave equation and the method of resolution at normal and oblique incidence, results obtained by the WS-TGF method are compared to those calculated by the classical transfer matrix method and to experimental measurements for a known two-layered porous material, considered as a single inhomogeneous layer. Discussions are then given of the reflection and transmission coefficients for various inhomogeneity profiles as well as of the internal pressure field. (10.1063/1.2752135)
  DOI : 10.1063/1.2752135
• The Dynamic Programming Equation for the Problem of Optimal Investment Under Capital Gains Taxes
  
  • Ben Tahar Imen
  • Touzi Nizar
  • Soner Mete H.
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2007, 46 (5), pp.1779-1801. This paper considers an extension of the Merton optimal investment problem to the case where the risky asset is subject to transaction costs and capital gains taxes. We derive the dynamic programming equation in the sense of constrained viscosity solutions. We next introduce a family of functions $(V_\varepsilon)_{\varepsilon>0}$, which converges to our value function uniformly on compact subsets, and which is characterized as the unique constrained viscosity solution of an approximation of our dynamic programming equation. (10.1137/050646044)
  DOI : 10.1137/050646044
• A large deviation approach to some transportation cost inequalities
  
  • Gozlan Nathael
  • Léonard Christian
  Probability Theory and Related Fields, Springer Verlag, 2007, 139, pp.235-283. New transportation cost inequalities are derived by means of elementary large deviation reasonings. Their dual characterization is proved; this provides an extension of a well-known result of S. Bobkov and F. Götze. Their tensorization properties are investigated. Sufficient conditions (and necessary conditions too) for these inequalities are stated in terms of the integrability of the reference measure. Applying these results leads to new deviation results: concentration of measure and deviations of empirical processes.
• Inverse spectral problem for radial Schrödinger operator on [0, 1]
  
  • Serier Frédéric
  Journal of Differential Equations, Elsevier, 2007, 235, pp.101-126. For a class of singular Sturm-Liouville equations on the unit interval with explicit singularity $a(a + 1)/x^2, a \in \mathbb{N}$, we consider an inverse spectral problem. Our goal is the global parametrization of potentials by spectral data noted by $\lambda^a$, and some norming constants noted by $\kappa^a$. For $a = 0$ and $a=1$, $\lambda^a\times \kappa^a$ was already known to be a global coordinate system on $\lr$. With the help of transformation operators, we extend this result to any non-negative integer $a$ and give a description of isospectral sets. (10.1016/j.jde.2006.12.014)
  DOI : 10.1016/j.jde.2006.12.014
• Generalized Impedance Boundary Condition At High Frequency For A Domain With Thin Layer: The Circular Case
  
  • Poignard Clair
  Applicable Analysis, Taylor & Francis, 2007, 86 (12), pp.1549-1568. Consider a disk surrounded by a thin conducting layer submitted to an electric field at the pulsation ω . The conductivity of the layer σm grows as ω 1-γ , for γ in [0,2/3), as the pulsation ω tends to infinity. Using a pseudodifferential approach on the torus, we build an equivalent boundary condition with the help of an appropriate factorization of Helmholtz operator in the layer. This generalized impedance condition approximates the thin membrane as ω tends to infinity and the thickness of the layer tends to zero. Error estimates are given and we illustrate our results with numerical simulations. This work extends, in the circular geometry, previous works of Lafitte and Lebeau, in which γ identically equals zero. (10.1080/00036810701714172)
  DOI : 10.1080/00036810701714172
• Editorial: Waves 2005 conference
  
  • Haddar Houssem
  • Hesthaven Jan S.
  Journal of Computational and Applied Mathematics, Elsevier, 2007, 204 (2), pp.197--198. (10.1016/j.cam.2006.05.021)
  DOI : 10.1016/j.cam.2006.05.021
• Identification of partially coated anisotropic buried objects using electromagnetic Cauchy data
  
  • Cakoni Fioralba
  • Haddar Houssem
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2007, 19 (3), pp.359--389. (10.1216/jiea/1190905491)
  DOI : 10.1216/jiea/1190905491
• MUSIC-type electromagnetic imaging of a collection of small 3-D bounded scatterers
  
  • Ammari Habib
  • Iakovleva Ekaterina
  • Lesselier Dominique
  • Perrusson Gaële
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2007, 29 (2), pp.674-709. In this paper we consider the localization of a collection of small, three-dimensional bounded homogeneous inclusions via time-harmonic electromagnetic means, typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization(s) at some distance from the collection, possibly also lying in the far field. The inclusions, somewhat apart or closely spaced, are buried within a homogeneous medium, and are of arbitrary contrast of permittivity, conductivity, and permeability vis-a-vis this embedding medium. The problem is formulated as an inverse scattering problem for the full Maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix. No specific application is studied at this stage, but characteriza- tion of obstacles in subsoils, nondestructive evaluation of man-made structures, and medical imaging are primary fields of application envisaged. The proposed approach uses a MUSIC (multiple signal classification)-type algorithm, and it yields fast numbering, accurate localization, and estimates of the electromagnetic and geometric parameters (polarization tensors) of the inclusions. The mathematical machinery is detailed first, some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes (for the latter known results are retrieved). Then, the viability of this algorithm--which would be easily extended to planarly layered environments by introduction of their Green's functions--is documented by a variety of numerical results from synthetic, noiseless, and severely noisy field data. (10.1137/050640655)
  DOI : 10.1137/050640655
• On the numerical solution of the heat equation I: Fast solvers in free space
  
  • Li Jing-Rebecca
  • Greengard Leslie
  Journal of Computational Physics, Elsevier, 2007, 226 (2), pp.1891--1901. We describe a fast solver for the inhomogeneous heat equation in free space, following the time evolution of the solution in the Fourier domain. It relies on a recently developed spectral approximation of the free-space heat kernel coupled with the non-uniform fast Fourier transform. Unlike finite difference and finite element techniques, there is no need for artificial boundary conditions on a finite computational domain. The method is explicit, unconditionally stable, and requires an amount of work of the order O(NMlogN), where N is the number of discretization points in physical space and M is the number of time steps. We refer to the approach as the fast recursive marching (FRM) method. (10.1016/j.jcp.2007.06.021)
  DOI : 10.1016/j.jcp.2007.06.021
• Isoperimetry between exponential and Gaussian
  
  • Barthe Franck
  • Cattiaux Patrick
  • Roberto Cyril
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2007, 12 (none). We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. (10.1214/EJP.v12-441)
  DOI : 10.1214/EJP.v12-441
• Random modulation of solitons for the stochastic Korteweg–de Vries equation
  
  • de Bouard Anne
  • Debussche Arnaud
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire, EMS, 2007, 24 (2), pp.251-278. We study the asymptotic behavior of the solution of a Korteweg–de Vries equation with an additive noise whose amplitude ε tends to zero. The noise is white in time and correlated in space and the initial state of the solution is a soliton solution of the unperturbed Korteweg–de Vries equation. We prove that up to times of the order of 1/ε2, the solution decomposes into the sum of a randomly modulated soliton, and a small remainder, and we derive the equations for the modulation parameters. We prove in addition that the first order part of the remainder converges, as ε tends to zero, to a Gaussian process, which satisfies an additively perturbed linear equation (10.1016/j.anihpc.2006.03.009)
  DOI : 10.1016/j.anihpc.2006.03.009
• Interpolation between logarithmic Sobolev and Poincaré inequalities
  
  • Arnold Anton
  • Bartier Jean-Philippe
  • Dolbeault Jean
  Communications in Mathematical Sciences, International Press, 2007, 5 (4), pp.971-979. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. Cette note est consacrée à des inégalités intermédiaires qui interpolent entre les inégalités de Sobolev logarithmiques et les inégalités de Poincaré. Pour de telles inégalités de Poincaré généralisées, nous améliorons les constantes données dans la littérature.
• New cardiovascular indices based on nonlinear spectral analysis of arterial blood pressure waveforms
  
  • Laleg Taous-Meriem
  • Médigue Claire
  • Papelier Yves
  • Crépeau Emmanuelle
  • Sorine Michel
  , 2007. A new method for analyzing arterial blood pressure is presented in this article. The technique is based on the scattering transform and consists in solving the spectral problem associated to a one-dimensional Schrödinger operator with a potential depending linearly upon the pressure. This potential is then expressed with the discrete spectrum which includes negative eigenvalues and corresponds to the interacting components of an N-soliton. The approach is similar to a nonlinear Fourier transform where the solitons play the role of sine and cosine components. The method provides new cardiovascular indices that seem to have meaningful physiological information, especially about the stroke volume and the ventricular contractility. We first show how to reconstruct the arterial blood pressure waves and separate its systolic and diastolic phases using this approach. Then we analyse the parameters computed from this technique in two physiological conditions, the head-up 60 degrees tilt test and the isometric handgrip test, widely used for studying short term cardiovascular control. Promising results are obtained.
• Interactive evolution for cochlear implants fitting
  
  • Legrand Pierrick
  • Bourgeois-Republique Claire
  • Pean Vincent
  • Harboun-Cohen Esther
  • Lévy Véhel Jacques
  • Frachet Bruno
  • Lutton Evelyne
  • Collet Pierre
  Genetic Programming and Evolvable Machines, Springer Verlag, 2007, 8 (4), pp.319-354. Cochlear implants are devices that become more and more sophisticated and adapted to the need of patients, but at the same time they become more and more difficult to parameterize. After a deaf patient has been surgically implanted, a specialised medical practitioner has to spend hours during months to precisely fit the implant to the patient. This process is a complex one implying two intertwined tasks: the practitioner has to tune the parameters of the device (optimisation) while the patient's brain needs to adapt to the new data he receives (learning). This paper presents a study that intends to make the implant more adaptable to environment (auditive ecology) and to simplify the process of fitting. Real experiments on volunteer implanted patients are presented, that show the efficiency of interactive evolution for this purpose. This work has partially been funded by the French ANR - RNTS HEVEA project 04T550.
• New Cardiovascular Indices Based on a Nonlinear Spectral Analysis of Arterial Blood Pressure Waveforms
  
  • Laleg Taous-Meriem
  • Médigue Claire
  • Papelier Yves
  • Crépeau Emmanuelle
  • Sorine Michel
  , 2007. A new method for analyzing arterial blood pressure is presented in this report. The technique is based on the scattering transform and consists in solving the spectral problem associated to a one-dimensional Schrödinger operator with a potential depending linearly upon the pressure. This potential is then expressed with the discrete spectrum which includes negative eigenvalues and corresponds to the interacting components of an N-soliton. The approach is similar to a nonlinear Fourier transform where the solitons play the role of sine and cosine components. The method provides new cardiovascular indices that seem to contain relevant physiological information. We first show how to use this approach to decompose the arterial blood pressure pulse into elementary waves and to reconstruct it or to separate its systolic and diastolic phases. Then we analyse the parameters computed from this technique in two physiological conditions, the head-up 60 degrees tilt test and the isometric handgrip test, widely used for studying short term cardiovascular control. Promising results are obtained.
• Optimisation évolutionnaire (rédigé par Marc Schoenauer)
  
  • Schoenauer Marc
  , 2007, pp.221-264. Ce chapitre présente une méthode de résolution de problèmes d’optimisation topologique de formes radicalement différente de toutes les précédentes. Cette méthode est basée sur l’utilisation des algorithmes dits “évolutionnaires”, algorithmes stochastiques d’optimisation globale. Une première partie introduit ces algorithmes, hors de tout contexte d’optimisation de formes, cependant que la deuxième partie présente leur application à l’optimisation topologique des structures. (10.1007/978-3-540-36856-4_8)
  DOI : 10.1007/978-3-540-36856-4_8
• Sur quelques problèmes d'homogénéisation non locale et de fluides en milieu poreux : une contribution de Abdelhamid Ziani
  
  • Amirat Youcef
  • Hamdache Kamel
  Annales Mathématiques Blaise Pascal, Université Blaise-Pascal - Clermont-Ferrand, 2007, 14, pp.149-186.
• Numerical methods for sensitivity analysis of Feynman-Kac models
  
  • Coquelin Pierre-Arnaud
  • Deguest Romain
  • Munos Rémi
  , 2007. The aim of this work is to provide efficient numerical methods to estimate the gradient of a Feynman-Kac flow with respect to a parameter of the model. The underlying idea is to view a Feynman-Kac flow as an expectation of a product of potential functions along a canonical Markov chain, and to use usual techniques of gradient estimation in Markov chains. Combining this idea with the use of interacting particle methods enables us to obtain two new algorithms that provide tight estimations of the sensitivity of a Feynman-Kac flow. Each algorithm has a linear computational complexity in the number of particles and is demonstrated to be asymptotically consistent. We also carefully analyze the differences between these new algorithms and existing ones. We provide numerical experiments to assess the practical efficiency of the proposed methods and explain how to use them to solve a parameter estimation problem in Hidden Markov Models. To conclude we can say that these algorithms outperform the existing ones in terms of trade-off between computational complexity and estimation quality.
• Weak logarithmic Sobolev inequalities and entropic convergence
  
  • Cattiaux Patrick
  • Gentil Ivan
  • Guillin Arnaud
  Probability Theory and Related Fields, Springer Verlag, 2007, 139 (3-4), pp.563-603. In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.
• A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media
  
  • Cakoni Fioralba
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2007, 1 (3), pp.443--456. (10.3934/ipi.2007.1.443)
  DOI : 10.3934/ipi.2007.1.443
• A strictly hyperbolic equilibrium phase transition model
  
  • Allaire Grégoire
  • Faccanoni Gloria
  • Kokh Samuel
  Comptes Rendus de l'Academie des Sciences. Série IV, Physique, Astronomie, Elsevier, 2007, 344, pp.135-140. This Note is concerned with the strict hyperbolicity of the compressible Euler equations equipped with an equation of state that describes the thermodynamical equilibrium between the liquid phase and the vapor phase of a uid. The proof is valid for a very wide class of uids. The argument only relies on smoothness assumptions and on the classical thermodynamical stability assumptions, that requires a de nite negative Hessian matrix for each phase entropy as a function of the speci c volume and internal energy. (10.1016/j.crma.2006.11.008)
  DOI : 10.1016/j.crma.2006.11.008
• A mathematical model for the Fermi weak interactions
  
  • Amour Laurent
  • Grébert Benoît
  • Guillot Jean-Claude
  Cubo, a Mathematical Journal, Temuco Departamento de Matemática y Estadística, Facultad de Ciencias e Ingeniería, Universidad de La Frontera, 2007, 9, pp.37-57. We consider a mathematical model of the Fermi theory of weak interactions as patterned according to the well-known current-current coupling of quantum electrodynamics. We focuss on the example of the decay of the muons into electrons, positrons and neutrinos but other examples are considered in the same way. We prove that the Hamiltonian describing this model has a ground state in the fermionic Fock space for a sufficiently small coupling constant. Furthermore we determine the absolutely continuous spectrum of the Hamiltonian and by commutator estimates we prove that the spectrum is absolutely continuous away from a small neighborhood of the thresholds of the free Hamiltonian. For all these results we do not use any infrared cutoff or infrared regularization even if fermions with zero mass are involved.
• Asymptotic expansion of the optimal control under logarithmic penalty: worked example and open problems
  
  • Alvarez Felipe
  • Bonnans J. Frederic
  • Laurent-Varin Julien
  , 2007. We discuss the problem of expansion of optimal control, state and costate when a logarithmic penalty is applied to constraints. We show that, in a simple case, that the variation of (a regular) junction point, and of the optimal control, state and costate is of order $\eps\log \eps$, where $\eps$ is the penalty parameter.
• Singular arcs in the generalized Goddard's Problem
  
  • Bonnans J. Frederic
  • Martinon Pierre
  • Trélat Emmanuel
  , 2007, pp.25. We investigate variants of Goddard's problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this report, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control.
• Bandit Algorithms for Tree Search
  
  • Coquelin Pierre-Arnaud
  • Munos Rémi
  , 2007, pp.20. Bandit based methods for tree search have recently gained popularity when applied to huge trees, e.g. in the game of go (Gelly et al., 2006). The UCT algorithm (Kocsis and Szepesvari, 2006), a tree search method based on Upper Confidence Bounds (UCB) (Auer et al., 2002), is believed to adapt locally to the effective smoothness of the tree. However, we show that UCT is too ``optimistic'' in some cases, leading to a regret O(exp(exp(D))) where D is the depth of the tree. We propose alternative bandit algorithms for tree search. First, a modification of UCT using a confidence sequence that scales exponentially with the horizon depth is proven to have a regret O(2^D \sqrt{n}), but does not adapt to possible smoothness in the tree. We then analyze Flat-UCB performed on the leaves and provide a finite regret bound with high probability. Then, we introduce a UCB-based Bandit Algorithm for Smooth Trees which takes into account actual smoothness of the rewards for performing efficient ``cuts'' of sub-optimal branches with high confidence. Finally, we present an incremental tree search version which applies when the full tree is too big (possibly infinite) to be entirely represented and show that with high probability, essentially only the optimal branches is indefinitely developed. We illustrate these methods on a global optimization problem of a Lipschitz function, given noisy data.