Publications

2014

• La singularité voilée
  
  • Perez Jérôme
  • Alimi Jean-Michel
  Pour la science, Pour la Science, 2014, Dossier N°83, pp.p 123. La relativité générale prédit l'existence de points de densité infinie où les lois de la physique s'effondrent : les singularités. Certaines sont tapies au cœur des trous noirs et nous ne pouvons les observer. Qu'en est-il de la singularité initiale, le Big Bang ? Elle est, elle aussi, isolée, car la relativité générale masque cet étrange événement en introduisant le chaos à l'origine de l'Univers.
• Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations
  
  • Bayen Térence
  • Bonnans J. Frederic
  • Silva Francisco J.
  Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (4), pp.2063--2087. In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ growths quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions. (10.1090/S0002-9947-2013-05961-2)
  DOI : 10.1090/S0002-9947-2013-05961-2
• Les fameux points de Lagrange
  
  • Perez Jérôme
  Images des mathématiques, CNRS, 2014. (10.60868/gc04-t229)
  DOI : 10.60868/gc04-t229
• Optimal control problems on stratifiable state constraints sets.
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  , 2014. We consider an infinite horizon problem with state constraints K : inf Z 1 0 e t'(yx;u(t); u(t))dt u : [ 0 ;+1) ! A measurable yx;u(t) 2 K 8t 0 (P) : where > 0 is fixed and yx;u( ) is a trajectory of the control system ( y_ = f (y; u) a.e. t 0 y(0) = x 2 K We are mainly concerned with a characterization of the value function of (P) as the bilateral solution to a Hamilton-Jacobi-Bellman equation.
• Second-order sufficient conditions for strong solutions to optimal control problems
  
  • Bonnans Joseph Frederic
  • Dupuis Xavier
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (03), pp.704-724. In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem. (10.1051/cocv/2013080)
  DOI : 10.1051/cocv/2013080
• La clé du mystère de la lettre H ?
  
  • Perez Jérôme
  Images des mathématiques, CNRS, 2014. (10.60868/qps3-ke02)
  DOI : 10.60868/qps3-ke02
• T-coercivity for the Maxwell problem with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  Communications in Partial Differential Equations, Taylor & Francis, 2014. In this paper, we study the time-harmonic Maxwell problem with sign-changing permittivity and/or permeability, set in a domain of R^3. We prove, using the T-coercivity approach, that the well-posedness of the two canonically associated scalar problems, with Dirichlet and Neumann boundary conditions, implies the well-posedness of the Maxwell problem. This allows us to give simple and sharp criteria, obtained in the study of the scalar cases, to ensure that the Maxwell transmission problem between a classical dielectric material and a negative metamaterial is well-posed.
• Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control
  
  • Fabbri Giorgio
  • Russo Francesco
  , 2014. The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$, taking values in a Hilbert space $H$, is the sum of a local martingale and a suitable "orthogonal" process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of "mild solution" for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.
• Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian
  
  • Graber P. Jameson
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2014, 70 (2), pp.185-224. We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed. Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE, viscosity solutions, front propagation, mean field games (10.1007/s00245-014-9239-3)
  DOI : 10.1007/s00245-014-9239-3
• Numerical modeling of nonlinear acoustic waves in a tube connected with an array of Helmholtz resonators
  
  • Lombard Bruno
  • Mercier Jean-François
  Journal of Computational Physics, Elsevier, 2014, 259 (15). (10.1016/j.jcp.2013.11.036)
  DOI : 10.1016/j.jcp.2013.11.036
• Quadro-quadric cremona transformations in low dimensions via the JC-correspondence
  
  • Pirio Luc
  • Russo Francesco
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2014, 64 (1), pp.71-111. We apply the results of arXiv:1109.3573 to study quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension four and five. (10.5802/aif.2839)
  DOI : 10.5802/aif.2839
• Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds
  
  • Ghezzi Roberta
  • Jean Frédéric
  , 2014, 5, pp.201-218. (10.1007/978-3-319-02132-4_13)
  DOI : 10.1007/978-3-319-02132-4_13
• Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time Effects
  
  • Abdulle Assyr
  • Grote Marcus J.
  • Stohrer Christian
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2014, 12 (3), pp.1230–1257. A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our new FE-HMM-L method captures not only the short-time behavior of the wave field, well described by classical homogenization theory, but also more subtle long-time dispersive effects, both at a computational cost independent of the microscale. Optimal error estimates in the energy norm and the $L^2$-norm are proved over finite time intervals, which imply convergence to the solution from classical homogenization theory when both the macro- and the microscale are refined simultaneously. Numerical experiments illustrate the usefulness of the FE-HMM-L method and corroborate the theory. (10.1137/13094195X)
  DOI : 10.1137/13094195X
• Asymptotic behaviour of codes in rank metric over finite fields
  
  • Loidreau P
  Designs, Codes and Cryptography, Springer Verlag, 2014, 71 (1), pp.105-118. In this paper, we rst recall some basic facts about rank metric. We then derive an asymptotic equivalent of the minimum rank distance of codes that reach the rank metric GilbertVarshamov bound. We then derive an asymptotic equivalent of the average minimum rank distance of random codes. We show that random codes reach GV bound. Finally, we show that optimal codes in rank metric have a packing density which is bounded by functions depending only on the base eld and the minimum distance and show the potential interest in cryptographic applications. (10.1007/s10623-012-9716-0)
  DOI : 10.1007/s10623-012-9716-0
• On the absence of trapped modes in locally perturbed open waveguides
  
  • Hazard Christophe
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2014, pp.14. This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the three-dimensional scalar wave equation is considered. (10.1093/imamat/hxu046)
  DOI : 10.1093/imamat/hxu046
• Extraordinary transmission through subwavelength dielectric gratings in the microwave range
  
  • Ahmed Akarid
  • Ourir Abdelwaheb
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  Optics Letters, Optical Society of America - OSA Publishing, 2014, 39 (13), pp.3752-3755. We address the problem of the transmission through subwavelength dielectric gratings. Following Maurel et al. [Phys. Rev. B 88, 115416 (2013)], the problem is reduced to the transmission by an homogeneous slab, either anisotropic (for transverse magnetic waves, TM) or isotropic (for transverse electric waves, TE), and an explicit expression of the transmission coefficient is derived. The optimum angle realizing perfect impedance matching (Brewster angle) is shown to depend on the contrasts of the dielectric layers with respect to the air. Besides, we show that the Fabry–Perot resonances may be dependent on the incident angle, in addition to the dependence on the frequency. These facts depart from the case of metallic gratings usually considered; they are confirmed experimentally both for TE and TM waves in the microwave regime. (10.1364/OL.39.003752)
  DOI : 10.1364/OL.39.003752
• Improved multimodal method in varying cross section waveguides
  
  • Maurel Agnes
  • Mercier Jean-François
  • Pagneux Vincent
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 470, pp.20130448. An improved version of the multimodal admittance method in acoustic waveguides with varying cross sections is presented. This method aims at a better convergence with respect to the number of transverse modes that are taken into account. It is based on an enriched modal expansion of the pressure: the N first modes are the local transverse modes and a supplementary (N+1)th mode, called boundary mode, is a well-chosen transverse function orthogonal to the N first modes. This expansion leads to the classical form of the coupled mode equations where the component of the boundary mode is of evanescent character. Under this form, the multimodal admittance method based on the Riccati equation on the admittance matrix (the Dirichlet-to-Neumann operator) is straightforwardly implemented. With this supplementary mode, in addition to the improvement of the convergence of the pressure field, results show a superconvergence of the scattered field outside of the varying cross sections region. (10.1098/rspa.2013.0448)
  DOI : 10.1098/rspa.2013.0448
• Quick reachability and proper extension for problems with unbounded controls
  
  • Aronna Maria Soledad
  • Motta Monica
  • Rampazzo Franco
  , 2014. For a CONTROL SYSTEM of the form _ x = f (x; u; v) + Σm =1 g (x)u_ ; on [0;T]; (x; u)(0) = ( x; u); with x : [0;T] ! IRn; u : [0;T] ! U IRm; v : [0;T] ! V IRl ; we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc.
• Finite element computation of trapped and leaky elastic waves in open stratified waveguides
  
  • Treyssede Fabien
  • Nguyen Khac-Long
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  Wave Motion, Elsevier, 2014, 51 (7), pp.pp.1093-1107. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results. (10.1016/j.wavemoti.2014.05.003)
  DOI : 10.1016/j.wavemoti.2014.05.003
• An approximation scheme for an Eikonal Equation with discontinuous coefficient
  
  • Festa Adriano
  • Falcone Maurizio
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2014, 52 (1), pp.236-257.. We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a-priori error estimate for the scheme in an integral norm. The last section contains some applications to control and image processing problems. (10.1137/120901829)
  DOI : 10.1137/120901829
• Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations
  
  • Rao Zhiping
  • Siconolfi Antonio
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2014, 257 (11), pp.3978--4014. We establish a comparison principle for a Hamilton-Jacobi-Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of "-partition and minimal "-partition for intervals of definition of an integral trajectory. (10.1016/j.jde.2014.07.015)
  DOI : 10.1016/j.jde.2014.07.015
• On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chambeyron Colin
  • Legendre Guillaume
  Wave Motion, Elsevier, 2014, 51 (2), pp.266-283. An efficient method to compute the scattering of a guided wave by a localized defect, in an elastic waveguide of infinite extent and bounded cross section, is considered. It relies on the use of perfectly matched layers (PML) to reduce the problem to a bounded portion of the guide, allowing for a classical finite element discretization. The difficulty here comes from the existence of backward propagating modes, which are not correctly handled by the PML. We propose a simple strategy, based on finite-dimensional linear algebra arguments and using the knowledge of the modes, to recover a correct approximation to the solution with a low additional cost compared to the standard PML approach. Numerical experiments are presented in the two-dimensional case involving Rayleigh--Lamb modes. (10.1016/j.wavemoti.2013.08.001)
  DOI : 10.1016/j.wavemoti.2013.08.001
• Surface integral equations for electromagnetic testing: the low-frequency and high-contrast case
  
  • Vigneron Audrey
  • Demaldent Édouard
  • Bonnet Marc
  IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 2014, 50, pp.7002704. This study concerns boundary element methods applied to electromagnetic testing, for a wide range of frequencies and conductivities. The eddy currents approximation cannot handle all configurations, while the common Maxwell formulation suffers from numerical instabilities at low frequency or in presence of highly contrasted media. We draw on studies that overcome these problems for dielectric configurations to treat conductive bodies, and show how to link them to eddy current formulations under suitable assumptions. This is intended as a first step towards a generic formulation that can be modified in each sub-domain according to the corresponding medium. (10.1109/TMAG.2013.2283297)
  DOI : 10.1109/TMAG.2013.2283297
• XLiFE++, an eXtended Library of Finite Elements in C++
  
  • Lunéville Éric
  • Kielbasiewicz Nicolas
  , 2014. XLiFE++ is an FEM-BEM C++ library that can solve 1D / 2D / 3D, scalar / vector, transient / stationnary / harmonic problems. It is autonomous, providing everything required for solving PDE problems : mesh tools, a wide range of finite elements on every mesh cell (nodal at any order, edge at any order and H_2 elements), a wide range of essential conditions, including periodic and quasi-periodic conditions, absorbing conditions (DtN, PML), direct / iterative / eigen solvers.
• Variance Optimal Hedging for discrete time processes with independent increments. Application to Electricity Markets
  
  • Goutte Stéphane
  • Oudjane Nadia
  • Russo Francesco
  The Journal of Computational Finance, Incisive Media, 2014, 17 (2), pp.71-111. We consider the discretized version of a (continuous-time) two-factor model introduced by Benth and coauthors for the electricity markets. For this model, the underlying is the exponent of a sum of independent random variables. We provide and test an algorithm, which is based on the celebrated Foellmer-Schweizer decomposition for solving the mean-variance hedging problem. In particular, we establish that decomposition explicitely, for a large class of vanilla contingent claims. Interest is devoted in the choice of rebalancing dates and its impact on the hedging error, regarding the payoff regularity and the non stationarity of the log-price process. (10.21314/JCF.2013.261)
  DOI : 10.21314/JCF.2013.261