Publications

2014

• 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
• 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
• 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 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
• 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
• 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.
• 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
• 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
• 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
• 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
• 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)
• 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
• 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
• 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 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
• The finite element method in solid mechanics
  
  • Bonnet Marc
  • Frangi Attilio
  • Rey Christian
  , 2014, pp.365. The book focuses on topics that are at the core of the Finite Element Method (FEM) for the mechanics of deformable solids and structures.Its main objective is to provide the reader, who is assumed to be familiar with standard continuum solid mechanics, with a clear grasp of the essentials, sufficient background for reading and exploiting the research literature on computational solid mechanics, and a working knowledge of the main implementational issues of the FEM.This book arises from a course taught since 2004 to last-year students of Ecole Polytechnique (France). It is intended for Master and PhD students, as well as scientists and engineers looking for a rigorous introduction to FEM theory and programming for linear and non-linear analyses in solid mechanics.As a distinguishing feature, in addition to sections devoted to theory and concepts presented in general terms, each chapter also features other sections (interspersed with the former) devoted to detailed description of specific features (e.g. the construction of a specific finite element), annotated Matlab code and/or numerical examples produced with it, or worked-out analytical examples.
• 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
• Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning
  
  • Jean Frédéric
  , 2014. Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems. The notes are divided into three chapters and two appendices. In Chapter 1 we introduce the basic definitions on nonholonomic systems and sub-Riemannian geometry, and give the main result on controllability, namely the Chow-Rashevsky Theorem. Chapter 2 provides a detailed exposition of the notions of first-order approximation, including nonholonomic orders, privileged coordinates, nilpotent approximations, and distance estimates such as the Ball-Box Theorem. As an application we show how these notions allow us to describe the tangent structure to a Carnot-Carathéodory space (the metric space defined by a sub-Riemannian distance). The chapter ends with the presentation of desingularization procedures, that are necessary to recover uniformity in approximations and distance estimates. Chapter 3 is devoted to the motion planning problem for nonholonomic systems. We show in particular how to apply the tools from sub-Riemannian geometry in order to give solutions to this problem, first in the case where the system is nilpotent, and then in the general case. An overview of the existing methods for nonholonomic motion planning concludes this chapter. Finally, we present some results on composition of flows in connection with the Campbell-Hausdorff formula in Appendix A, and some complements on the different systems of privileged coordinates in Appendix B. (10.1007/978-3-319-08690-3)
  DOI : 10.1007/978-3-319-08690-3
• The "exterior approach" to solve the inverse obstacle problem for the Stokes system
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2014, pp.Pages: 23 - 51. We apply an "exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements. Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming infite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand. Some numerical experiments in 2D show the effciency of the two mixed formulations and of the exterior approach based on one of them. (10.3934/ipi.2014.8.23)
  DOI : 10.3934/ipi.2014.8.23
• Space-time focusing of acoustic waves on unknown scatterers
  
  • Cassier Maxence
  • Hazard Christophe
  Wave Motion, Elsevier, 2014, pp.19. Consider a propagative medium, possibly inhomogeneous, containing some scatterers whose positions are unknown. Using an array of transmit-receive transducers, how can one generate a wave that would focus in space and time near one of the scatterers, that is, a wave whose energy would confine near the scatterer during a short time? The answer proposed in the present paper is based on the so-called DORT method (French acronym for: decomposition of the time reversal operator) which has led to numerous applications owing to the related space-focusing properties in the frequency domain, i.e., for time-harmonic waves. This method essentially consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. By introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space-time focusing. We consider the case of the scalar wave equation and we make use of an asymptotic model for small sound-soft scatterers, usually called the Foldy-Lax model. In this context, several mathematical and numerical arguments that support our idea are explored. (10.1016/j.wavemoti.2014.07.009)
  DOI : 10.1016/j.wavemoti.2014.07.009
• Wave propagation through penetrable scatterers in a waveguide and through a penetrable gratings
  
  • Maurel Agnès
  • Mercier Jean-François
  • Félix Simon
  Journal of the Acoustical Society of America, Acoustical Society of America, 2014, 135 (1), pp.165-174. A multimodal method based on the admittance matrix is used to analyze wave propagation through scatterers of arbitrary shape. Two cases are considered: a waveguide containing scatterers, and the scattering of a plane wave at oblique incidence to an infinite periodic row of scatterers. In both cases, the problem reduces to a system of two sets of first-order differential equations for the modal components of the wavefield, similar to the system obtained in the rigorous coupled wave analysis. The system can be solved numerically using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed (convergence, reciprocity, energy conservation). Alternatively, the admittance matrix can be used to get analytical results in the weak scattering approximation. This is done using the plane wave approximation, leading to a generalized version of the Webster equation and using a perturbative method to analyze the Wood anomalies and Fano resonances. (10.1121/1.4836075)
  DOI : 10.1121/1.4836075
• High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls
  
  • Schmidt Kersten
  • Anastasia Thöns-Zueva
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014, pp.1823. (10.1142/S0218202514500080)
  DOI : 10.1142/S0218202514500080
• Quasi-local transmission conditions for non-overlapping domain decomposition methods for the Helmholtz equation
  
  • Collino Francis
  • Joly Patrick
  • Lecouvez Matthieu
  • Stupfel Bruno
  Comptes Rendus. Physique, Académie des sciences (Paris), 2014, 15 (5), pp.403-414. In this article, we present new transmission conditions for a domain decomposition method, applied to a scattering problem. Unlike other conditions used in the literature, the conditions developed here are non-local, but can be written as an integral operator (as a Riesz potential) on the interface between two domains. This operator, of order View the MathML source12, leads to an exponential convergence of the domain decomposition algorithm. A spectral analysis of the influence of the operator on simple cases is presented, as well as some numerical results and comparisons. (10.1016/j.crhy.2014.04.005)
  DOI : 10.1016/j.crhy.2014.04.005