Publications

2014

• 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
• 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.
• 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
• 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
• Complexity in control-affine systems
  
  • Jean Frédéric
  • Prandi Dario
  , 2014. We will consider affine-control systems, i.e., systems in the form _ q(t) = f0(q(t)) + Xm i=1 ui (t)fi (q(t)) Here, the point q belongs to a smooth manifold M the fi 's are smooth vector fields on M u 2 L1([0;T];Rm) This type of system appears in many applications Mechanical systems Quantum control Microswimmers (Tucsnak, Alouges) Neuro-geometry of vision (Mumfor, Petitot)
• Wood's anomalies for arrays of dielectric scatterers
  
  • Maurel Agnès
  • Félix Simon
  • Mercier Jean-François
  • Ourir Abdelwaheb
  • Djeffal Zine Eddine
  Journal of the European Optical Society : Rapid publications, European Optical Society, 2014, 9, pp.14001. The Rayleigh Wood anomalies refer to an unexpected repartition of the electromagnetic energy between the several interference orders of the light emerging from a grating. Since Hessel and Oliner (Appl. Opt. 4, 1275-1297 (1965)), several studies have been dedicated to this problem, focusing mainly on the case of metallic gratings. In this paper, we derive explicit expressions of the reflection coefficients in the case of dielectric gratings using a perturbative approach. This is done in a multimodal description of the field combined with the use of the admittance matrix, analog to the so-called electromagnetic impedance. Comparisons with direct numerical calculations show a good agreement with our analytical prediction. (10.2971/jeos.2014.14001)
  DOI : 10.2971/jeos.2014.14001
• Mathematical modeling of a discontinuous Myers condition
  
  • Lunéville Éric
  • Mercier Jean-François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2014, 48 (5), pp.1529-1555. (10.1051/m2an/2014008)
  DOI : 10.1051/m2an/2014008
• Generalized method for retrieving effective parameters of anisotropic metamaterials
  
  • Mercier Jean-François
  • Castanié Aurore
  • Félix Simon
  • Maurel Agnes
  Optics Express, Optical Society of America - OSA Publishing, 2014, 22 (24), pp.29977-29953. Electromagnetic or acoustic metamaterials can be described in terms of equivalent effective, in general anisotropic, media and several techniques exist to determine the effective permeability and permittivity (or effective mass density and bulk modulus in the context of acoustics). Among these techniques, retrieval methods use the measured reflection and transmission coefficients (or scattering coefficients) for waves incident on a metamaterial slab containing few unit cells. Until now, anisotropic effective slabs have been considered in the literature but they are limited to the case where one of the axes of anisotropy is aligned with the slab interface. We propose an extension to arbitrary orientations of the principal axes of anisotropy and oblique incidence. The retrieval method is illustrated in the electromagnetic case for layered media, and in the acoustic case for array of tilted elliptical particles. (10.1364/OE.22.029937)
  DOI : 10.1364/OE.22.029937
• Local transformation leading to an efficient Fourier modal method for perfectly conducting gratings
  
  • Félix Simon
  • Maurel Agnes
  • Mercier Jean-François
  Journal of the Optical Society of America, Optical Society of America, 2014, 31 (10), pp.2249-2255. We present an efficient Fourier modal method for wave scattering by perfectly conducting gratings (in the two polarizations). The method uses a geometrical transformation, similar to the one used in the C-method, that transforms the grating surface into a flat surface, thus avoiding to question the Rayleigh hypothesis; also, the transformation only affects a bounded inner region that naturally matches the outer region; this allows applying a simple criterion to select the ingoing and outgoing waves. The method is shown to satisfy reciprocity and energy conservation, and it has an exponential rate of convergence for regular groove shapes. Besides, it is shown that the size of the inner region, where the solution is computed, can be reduced to the groove depth, that is, to the minimal computation domain. (10.1364/JOSAA.31.002249)
  DOI : 10.1364/JOSAA.31.002249
• Edge Element Methods for Maxwell's Equations with Strong Convergence for Gauss' Laws
  
  • Ciarlet Patrick
  • Wu Haijun
  • Zou Jun
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2014, 52 (2), pp.779-807. In this paper we propose and investigate some edge element approximations for three Maxwell systems in three dimensions: the stationary Maxwell equations, the time-harmonic Maxwell equations and the time-dependent Maxwell equations. These approximations have three novel features. First, the resulting discrete edge element systems can be solved by some existing preconditioned solvers with optimal convergence rate independent of finite element meshes, including the stationary Maxwell equations. Second, they ensure the optimal strong convergence of the Gauss' laws in some appropriate norm, in addition to the standard optimal convergence in energy-norm, under the general weak regularity assumptions that hold for both convex and non-convex polyhedral domains and for the discontinuous coefficients that may have large jumps across the interfaces between different media. Finally, no saddle-point discrete systems are needed to solve for the stationary Maxwell equations, unlike most existing edge element schemes. (10.1137/120899856)
  DOI : 10.1137/120899856
• A new Fast Multipole formulation for the elastodynamic half-space Green's tensor
  
  • Chaillat Stéphanie
  • Bonnet Marc
  Journal of Computational Physics, Elsevier, 2014, 258, pp.787-808. In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green's tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green's tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the full-space Green's tensor, the elastodynamic half-space Green's tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about $N=6 10^5$ degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. (10.1016/j.jcp.2013.11.010)
  DOI : 10.1016/j.jcp.2013.11.010
• 2-Stage Robust MILP with continuous recourse variables
  
  • Billionnet Alain
  • Costa Marie-Christine
  • Poirion Pierre-Louis
  Discrete Applied Mathematics, Elsevier, 2014, 170, pp.21-32. We solve a linear robust problem with mixed-integer first-stage variables and continuous second stage variables. We consider column wise uncertainty. We first focus on a problem with right hand-side uncertainty which satisfies a "full recourse property" and a specific definition of the uncertainty. We propose a solution based on a generation constraint algorithm. Then we give some generalizations of the approach: for left-hand side uncertainty and for uncertainty sets defined by a polytope. Finally we solve the problem when the "full recourse property" is not satisfied. (10.1016/j.dam.2014.01.017)
  DOI : 10.1016/j.dam.2014.01.017
• Optimal control of leukemic cell population dynamics
  
  • Dupuis Xavier
  Mathematical Modelling of Natural Phenomena, EDP Sciences, 2014, 9 (1), pp.4-26. We are interested in optimizing the co-administration of two drugs for some acute myeloid leukemias (AML), and we are looking for in vitro protocols as a first step. This issue can be formulated as an optimal control problem. The dynamics of leukemic cell populations in culture is given by age-structured partial differential equations, which can be reduced to a system of delay differential equations, and where the controls represent the action of the drugs. The objective function relies on eigenelements of the uncontrolled model and on general relative entropy, with the idea to maximize the efficiency of the protocols. The constraints take into account the toxicity of the drugs. We present in this paper the modeling aspects, as well as theoretical and numerical results on the optimal control problem that we get. (10.1051/mmnp/20149102)
  DOI : 10.1051/mmnp/20149102
• Optimal feedback control of undamped wave equations by solving a HJB equation
  
  • Kröner Axel
  • Kunisch Karl
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 21 (2), pp.442 - 464. An optimal fi nite-time horizon feedback control problem for (semi linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on nite elements lead to approximated problems governed by ODEs in high dimensional space which makes infeasible the numerical resolution by HJB approach. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The e ffect of noise is considered and numerical simulations are presented to show the relevance of the approach. (10.1051/cocv/2014033)
  DOI : 10.1051/cocv/2014033
• The covariation for Banach space valued processes and applications
  
  • Di Girolami Cristina
  • Fabbri Giorgio
  • Russo Francesco
  Metrika, Springer Verlag, 2014, 77 (1), pp.51–104. This article focuses on a new concept of quadratic variation for processes taking values in a Banach space $B$ and a corresponding covariation. This is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace $\chi$ of the dual of the projective tensor product of $B$ with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of $\bar \nu_0$-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark-Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type. (10.1007/s00184-013-0472-6)
  DOI : 10.1007/s00184-013-0472-6
• Study of a Model Equation in Detonation Theory
  
  • Faria Luiz
  • Kasimov Aslan
  • Rosales Rodolfo
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2014, 74 (2), pp.547-570. (10.1137/130938232)
  DOI : 10.1137/130938232