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
• 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
• 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.
• Les fameux points de Lagrange -- Fameux, pour qui les connaît !
  
  • Perez Jérôme
  Images des mathématiques, CNRS, 2014. Parmi tous les domaines abordés par Joseph-Louis Lagrange la mécanique céleste tient une place de choix. C'est pendant sa période berlinoise, de 1766 à 1788 qu'il découvre une famille de points d'équilibre de certaines extensions du problème des deux corps. Les points de Lagrange étaient nés ! Depuis cette époque, nombreuses sont les extensions de cette théorie à différentes configurations. Et nombreuses sont aussi les observations astronomiques en relation directe avec ces théories.
• La clé du mystère de la lettre H ?
  
  • Perez Jérôme
  Images des mathématiques, CNRS, 2014. En physique théorique, en mécanique quantique, en optimisation, et dans bien d'autres domaines la lettre $H$ est traditionnellement rattachée à $H\!$amilton à travers le terme hamiltonien. Lorsque l'on fait l'exégèse de cette notation on constate pourtant que la notation lui est antérieure et a été introduite par Lagrange dans un contexte où $H\!$uygens semble être mis en avant... Un manuscrit redécouvert récemment dans l'un des ouvrages de la seconde édition de la mécanique analytique, publié par Lagrange en 1815 alors qu'Hamilton n'avait pas 10 ans, pourrait bien être la clé de ce mystère.
• 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.
• 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
• 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
• 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
• 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
• 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
• Propagation in waveguides with varying cross-section and curvature: A new light on the role of supplementary modes in multimodal methods
  
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 470 (2166), pp.20130743. We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems. (10.1098/rspa.2014.0008)
  DOI : 10.1098/rspa.2014.0008
• Level-set approach for Reachability Analysis of Hybrid Systems under Lag Constraints
  
  • Granato Giovanni
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (1), pp.606--628. This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite differences schemes for solving the system of inequalities is proposed and the convergence of the numerical solution towards the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application, namely, that of range extender electric vehicles. This class of electric vehicles uses an additional module -- the range extender -- as an extra source of energy in addition to its main source -- a high voltage battery. The reachability study of this system is used to establish the maximum range of a simple vehicle model. (10.1137/120874205)
  DOI : 10.1137/120874205
• BSDEs under partial information and financial applications.
  
  • Ceci Claudia
  • Cretarola Alessandra
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2014. In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Föllmer-Schweizer decomposition of a square integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model. (10.1016/j.spa.2014.03.003)
  DOI : 10.1016/j.spa.2014.03.003
• Generalized covariation for Banach space valued processes, Itô formula and applications
  
  • Di Girolami Cristina
  • Russo Francesco
  Osaka Journal of Mathematics, Osaka University, 2014, 51 (3). This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
• Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional
  
  • Warner James E.
  • Diaz Manuel I.
  • Aquino Wilkins
  • Bonnet Marc
  Computational Mechanics, Springer Verlag, 2014, 54, pp.645-659. This work focuses on the identification of heterogeneous linear elastic moduli in the context of frequency-domain, coupled acoustic-structure interaction (ASI), using either solid displacement or fluid pressure measurement data. The approach postulates the inverse problem as an optimization problem where the solution is obtained by minimizing a modified error in constitutive equation (MECE) functional. The latter measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, while incorporating the measurement data as additional quadratic error terms. We demonstrate two strategies for selecting the MECE weighting coefficient to produce regularized solutions to the ill-posed identification problem: 1) the discrepancy principle of Morozov, and 2) an error-balance approach that selects the weight parameter as the minimizer of another functional involving the ECE and the data misfit. Numerical results demonstrate that the proposed methodology can successfully recover elastic parameters in 2D and 3D ASI systems from response measurements taken in either the solid or fluid subdomains. Furthermore, both regularization strategies are shown to produce accurate reconstructions when the measurement data is polluted with noise. The discrepancy principle is shown to produce nearly optimal solutions, while the error-balance approach, although not optimal, remains effective and does not need a priori information on the noise level. (10.1007/s00466-014-1018-0)
  DOI : 10.1007/s00466-014-1018-0
• Variance optimal hedging for continuous time additive processes and applications
  
  • Goutte Stéphane
  • Oudjane Nadia
  • Russo Francesco
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2014, 81 (1), pp.147--185. For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is an exponential of an additive process.
This allows to provide an efficient algorithm for solving the
mean variance hedging problem.
Applications to models derived from the electricity market are performed. (10.1080/17442508.2013.774402)
  DOI : 10.1080/17442508.2013.774402
• Second-order necessary conditions in Pontryagin form for optimal control problems
  
  • Bonnans J. Frederic
  • Dupuis Xavier
  • Pfeiffer Laurent
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (6), pp.3887-3916. In this report, we state and prove first- and second-order necessary conditions in Pontryagin form for optimal control problems with pure state and mixed control-state constraints. We say that a Lagrange multiplier of an optimal control problem is a Pontryagin multiplier if it is such that Pontryagin's minimum principle holds, and we call optimality conditions in Pontryagin form those which only involve Pontryagin multipliers. Our conditions rely on a technique of partial relaxation, and apply to Pontryagin local minima. (10.1137/130923452)
  DOI : 10.1137/130923452
• 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
• 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
• 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.