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
• 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
• 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.
• 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
• 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
• 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
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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.