Publications

2003

• Estimating deformations of stationary processes
  
  • Clerc Maureen
  • Mallat Stéphane
  Annals of Statistics, Institute of Mathematical Statistics, 2003, 31 (6), pp.pp. 1772-1821. This paper studies classes of nonstationary processes, such as warped processes and frequency-modulated processes, that result from the deformation of stationary processes. Estimating deformations can often provide important information about an underlying physical phenomenon. A computational harmonic analysis viewpoint shows that the deformed autocovariance satisfies a transport equation at small scales, with a velocity proportional to a deformation gradient. We derive an estimator of the deformation from a single realization of the deformed process, with a proof of consistency under appropriate assumptions. (10.1214/aos/1074290327)
  DOI : 10.1214/aos/1074290327
• Log-infinitely divisible multifractal processes
  
  • Bacry Emmanuel
  • Muzy J. F.
  Communications in Mathematical Physics, Springer Verlag, 2003, 236,num.3, pp.449-475. We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal \"Multifractal Random Walk\" processes (MRW) and the log-Poisson \"product of cynlindrical pulses\". Their construction involves some ``continuous stochastic multiplication\'\' from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.
• A criterion for Talagrand's quadratic transportation cost inequality
  
  • Cattiaux Patrick
  • Guillin Arnaud
  , 2003. We show that the quadratic transportation cost inequality $T_2$ is equivalent to both a Poincaré inequality and a strong form of the Gaussian concentration property. The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux \cite{BL97}, that are shown to hold as soon as a Poincaré inequality is satisfied.
• The linear sampling method in inverse electromagnetic scattering theory
  
  • Colton David
  • Haddar Houssem
  • Piana Michele
  Inverse Problems, IOP Publishing, 2003, 19 (6), pp.S105--S137. (10.1088/0266-5611/19/6/057)
  DOI : 10.1088/0266-5611/19/6/057
• Option pricing and hedging with minimum local expected shortfall
  
  • Pochart Benoît
  • Bouchaud Jean-Philippe
  , 2003. We propose a versatile Monte-Carlo method for pricing and hedging options when the market is incomplete, for an arbitrary risk criterion (chosen here to be the expected shortfall), for a large class of stochastic processes, and in the presence of transaction costs. We illustrate the method on plain vanilla options when the price returns follow a Student-t distribution. We show that in the presence of fat-tails, our strategy allows to significantly reduce extreme risks, and generically leads to low Gamma hedging. Similarly, the inclusion of transaction costs reduces the Gamma of the optimal strategy.
• A long exact sequence for symplectic Floer cohomology
  
  • Seidel Paul
  Topology, Elsevier, 2003, 42, pp.1003-1063. The long exact sequence describes how the Floer cohomology of two Lagrangian submanifolds changes if one of them is modified by applying a Dehn twist. We give a proof in the simplest case (no bubbling). The paper contains a certain amount of material that may be of interest independently of the exact sequence: in particular, chapter 1 covers "symplectic Picard-Lefschetz theory" in some detail, and chapter 2 contains a generalization of the usual TQFT setup for Floer cohomology.
• Exponential meshes and three-dimensional computation of a magnetic field
  
  • Laminie Jacques
  • Alouges François
  • Mefire Séraphin
  Numerical Methods for Partial Differential Equations, Wiley, 2003, 19 (5), pp.595-637. We describe the simulation of an exterior problem using a magnetic field deriving from magnetostatics, with a numerical method mixing the approaches of C. I. Goldstein and L.-A. Ying. This method is based on the use of a graded mesh obtained by gluing homothetic layers in the exterior domain. On this mesh, we use an edge elements discretization and a recently proposed mixed formulation. In this paper, we provide both a theoretical and numerical study of the method. We establish an error estimate, describe the implementation, propose some preconditioning techniques and show the numerical results. We also compare these results with those obtained from an equivalent boundary elements approach. In this way, we retain that our method leads to a practical numerical implementation, a saving of storage, and turns out to be an alternative to the classical boundary elements method. (10.1002/num.10064)
  DOI : 10.1002/num.10064
• Numerical and analytical studies of the linear sampling method in electromagnetic inverse scattering problems
  
  • Collino Francis
  • Fares M'Barek
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2003, 19 (6), pp.1279--1298. We present in this study some three-dimensional numerical results that validate the use of the linear sampling method as an inverse solver in electromagnetic scattering problems. We recall that this method allows the reconstruction of the shape of an obstacle from the knowledge of multi-static radar data at a fixed frequency. It does not require any a priori knowledge of the physical properties of the scatterer nor any nonlinear optimization scheme. This study also contains some analytical results in the simplified case of a spherical scatterer that somehow make the link between known abstract theoretical results and the numerical scheme. Special attention has been given to pointing out the influence of the frequency on the inversion accuracy. (10.1088/0266-5611/19/6/004)
  DOI : 10.1088/0266-5611/19/6/004
• Homogenization and localization with an interface
  
  • Allaire Grégoire
  • Capdeboscq Yves
  • Piatnitski Andrey
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2003, 52 (6), pp.1413--1446. We consider the homogenization of a spectral problem for a diffusion equation posed in a singularly perturbed periodic medium. Denoting by ε the period, the diffusion coefficients are scaled as ε 2. The domain is composed of two periodic medium separated by a planar interface, aligned with the periods. Three different situations arise when ε goes to zero. First, there is a global homogenized problem as if there were no interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition on the interface. Third, there is an exponential localization near the interface of the first eigenfunc-tion. (10.1512/iumj.2003.52.2352)
  DOI : 10.1512/iumj.2003.52.2352
• Strongly consistent marching schemes for the wave equation
  
  • Li Jing-Rebecca
  • Greengard Leslie
  Journal of Computational Physics, Elsevier, 2003, 188 (1), pp.194--208. In this paper, we consider a class of explicit marching schemes first proposed in [1] for solving the wave equation in complex geometry using an embedded Cartesian grid. These schemes rely on an integral evolution formula for which the numerical domain of dependence adjusts automatically to contain the true domain of dependence. Here, we refine and analyze a subclass of such schemes, which satisfy a condition we refer to as strong u-consistency. This requires that the evolution scheme be exact for a single-valued approximation to the solution at the previous time steps. We provide evidence that many of these strongly u-consistent schemes are stable and converge at very high order even in the presence of small cells in the grid, while taking time steps dictated by the uniform grid spacing.
• A Webster-Lokshin model for waves with viscothermal losses and impedance boundary conditions: strong solutions
  
  • Haddar Houssem
  • Matignon Denis
  • Hélie Thomas
  , 2003, pp.66--71.
• On the validation of the linear sampling method in electromagnetic inverse scattering problems
  
  • Collino Francis
  • Fares M'Barek
  • Haddar Houssem
  , 2003, pp.649--654.
• Determination of the stabilized response of a structure undergoing cyclic thermal-mechanical loads by a direct cyclic method
  
  • Nguyen-Tajan Thi Mac-Lan
  • Pommier Benjamin
  • Maitournam Habibou
  • Mechkour Houari
  • Verger Loick
  • Du Z. Z.
  • Snyman M.
  , 2003.
• Harmonic Decomposition of Audio Signals with Matching Pursuit
  
  • Gribonval Rémi
  • Bacry Emmanuel
  IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2003, 51 (1), pp.101--111. We introduce a dictionary of elementary waveforms, called harmonic atoms, that extends the Gabor dictionary and fits well the natural harmonic structures of audio signals. By modifying the "standard" matching pursuit, we define a new pursuit along with a fast algorithm, namely, the fast harmonic matching pursuit, to approximate N-dimensional audio signals with a linear combination of M harmonic atoms. Our algorithm has a computational complexity of O(MKN), where K is the number of partials in a given harmonic atom. The decomposition method is demonstrated on musical recordings, and we describe a simple note detection algorithm that shows how one could use a harmonic matching pursuit to detect notes even in difficult situations, e.g., very different note durations, lots of reverberation, and overlapping notes. (10.1109/TSP.2002.806592)
  DOI : 10.1109/TSP.2002.806592