Publications

2012

• Optimizing the deployment of a multilevel optical FTTH network
  
  • Chardy Matthieu
  • Costa Marie-Christine
  • Faye Alain
  • Trampont Mathieu
  European Journal of Operational Research, Elsevier, 2012, 222 (3), pp.430--440. Due to the emergence of bandwidth-requiring services, telecommunication operators are being compelled to renew their fix access network, most of them favoring the Fiber To The Home (FTTH) technology. This paper focuses on the optimization of FTTH deployment, which is of prime importance due to the economic stakes. The key design issue here is locating splitters and routing fi bers in an existing network infrastructure to which is associated a graph with given capacities on the edges. No assumption is made on the structure of the graph. First we propose a mixed integer formulation for this decision problem. Then, valid inequalities and problem size reduction schemes are presented. Finally efficiency of solving approaches is assessed through extensive numerical tests performed on Orange real-life data (10.1016/j.ejor.2012.05.024)
  DOI : 10.1016/j.ejor.2012.05.024
• Helmholtz equation in periodic media with a line defect
  
  • Coatléven Julien
  Journal of Computational Physics, Elsevier, 2012, 231 (4), pp.1675-1704. We consider the Helmholtz equation in an unbounded periodic media perturbed by an unbounded defect whose structure is compatible with the periodicity of the underlying media. We exhibit a method coupling Dirichlet-to-Neumann maps with the Lippmann-Schwinger equation approach to solve this problem, where the Floquet-Bloch transform in the direction of the defect plays a central role. We establish full convergence estimates that makes the link between the rate of decay of a function and the good behavior of a quadrature rule to approximate the inverse Floquet-Bloch transform. Finally we exhibit a few numerical results to illustrate the efficiency of the method. © 2011 Elsevier Inc. (10.1016/j.jcp.2011.10.022)
  DOI : 10.1016/j.jcp.2011.10.022
• Error estimates for the logarithmic barrier method in stochastic linear quadratic optimal control problems
  
  • Bonnans Joseph Frédéric
  • Silva Francisco J.
  Systems and Control Letters, Elsevier, 2012, 61 (1), pp.143-147. We consider a linear quadratic stochastic optimal control problem whith non-negativity control constraints. The latter are penalized with the classical logarithmic barrier. Using a duality argument and the stochastic minimum principle, we provide an error estimate for the solution of the penalized problem which is the natural extension of the well known estimate in the deterministic framework.
• On the use of sampling methods to identify cracks in acoustic waveguides
  
  • Bourgeois Laurent
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2012, 28 (10), pp.105011.1-105011.18. We consider the identification of cracks in an acoustic 2D/3D waveguide with the help of sampling methods such as the linear sampling method or the factorization method. A modal version of these sampling methods is used. Our paper emphasizes the fact that if one a priori knows the type of boundary condition which actually applies on the crack, then we shall adapt the formulation of our sampling method to such boundary conditions in order to improve the efficiency of the method. The need for such adaptation is proved theoretically and illustrated numerically with the help of 2D examples. We also show by using our modal formulation that the factorization method is applicable in a waveguide with the same data as the linear sampling method. © 2012 IOP Publishing Ltd. (10.1088/0266-5611/28/10/105011)
  DOI : 10.1088/0266-5611/28/10/105011
• Propagation of guided waves through weak penetrable scatterers
  
  • Maurel Agnes
  • Mercier Jean-François
  Journal of the Acoustical Society of America, Acoustical Society of America, 2012, 131 (3), pp.1874-1889. The scattering of a scalar wave propagating in a waveguide containing weak penetrable scatterers is inspected in the Born approximation. The scatterers are of arbitrary shape and present a contrast both in density and in wavespeed (or bulk modulus), a situation that can be translated in the context of SH waves, water waves, or transverse electric/transverse magnetic polarized electromagnetic waves. For small size inclusions compared to the waveguide height, analytical expressions of the transmission and reflection coefficients are derived, and compared to results of direct numerical simulations. The cases of periodically and randomly distributed inclusions are considered in more detail, and compared with unbounded propagation through inclusions. Comparisons with previous results valid in the low frequency regime are proposed. © 2012 Acoustical Society of America. (10.1121/1.3682037)
  DOI : 10.1121/1.3682037
• T-coercivity for scalar interface problems between dielectrics and metamaterials
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2012, 46, pp.1363-1387. Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of R^d, with d=2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria. (10.1051/m2an/2012006)
  DOI : 10.1051/m2an/2012006
• An elementary introduction to the construction and the analysis of Perfectly Matched Layers for time domain wave propagation
  
  • Joly Patrick
  SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, Springer, 2012, 57, pp.5-48.
• Operator Factorization for Multiple-Scattering Problems and an Application to Periodic Media
  
  • Coatléven Julien
  • Joly Patrick
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.303-318. This work concerns multiple-scattering problems for time-harmonic equations in a reference generic media. We consider scatterers that can be sources, obstacles or compact perturbations of the reference media. Our aim is to restrict the computational domain to small compact domains containing the scatterers. We use Robin-to-Robin (RtR) operators (in the most general case) to express boundary conditions for the interior problem. We show that one can always factorize the RtR map using only operators defined using single-scatterer problems. This factorization is based on a decomposition of the diffracted field, on the whole domain where it is defined. Assuming that there exists a good method for solving single-scatterer problems, it then gives a convenient way to compute RtR maps for a random number of scatterers.
• Solving multi-stage stochastic mixed integer linear programs by the dual dynamic programming approach
  
  • Cen Zhihao
  , 2012. We consider a model of medium-term commodity contracts management. Randomness takes place only in the prices on which the commodities are exchanged, whilst state variable is multi-dimensional, and decision variable is integer. In our previous article, we proposed an algorithm based on the quantization of random process and a dual dynamic programming type approach to solve the continuous relaxation problem. In this paper, we study the multi-stage stochastic mixed integer linear program (SMILP) and show the difficulty when using dual programming type algorithm. We propose an approach based on the cutting plane method combined with the algorithm in our previous article, which gives an upper and a lower bound of the optimal value and a sub-optimal integer solution. Finally, a numerical test on a real problem in energy market is provided.
• A preconditioned 3-D multi-region fast multipole solver for seismic wave propagation in complex geometries
  
  • Chaillat Stéphanie
  • Semblat Jean-François
  • Bonnet Marc
  Communications in Computational Physics, Global Science Press, 2012, 11, pp.594-609. The analysis of seismic wave propagation and amplification in complex geological structures requires efficient numerical methods. In this article, following up on recent studies devoted to the formulation, implementation and evaluation of 3-D single- and multi-region elastodynamic fast multipole boundary element methods (FM-BEMs), a simple preconditioning strategy is proposed. Its efficiency is demonstrated on both the single- and multi-region versions using benchmark examples (scattering of plane waves by canyons and basins). Finally, the preconditioned FM-BEM is applied to the scattering of plane seismic waves in an actual configuration (alpine basin of Grenoble, France), for which the high velocity contrast is seen to significantly affect the overall efficiency of the multi-region FM-BEM. (10.4208/cicp.231209.030111s)
  DOI : 10.4208/cicp.231209.030111s
• Solving the Homogeneous Isotropic Linear Elastodynamics Equations Using Potentials and Finite Elements. The Case of the Rigid Boundary Condition
  
  • Burel Aliénor
  • Imperiale Sébastien
  • Joly Patrick
  Numerical Analysis and Applications, Springer, 2012, 5 (2), pp.136-143. In this article, elastic wave propagation in a homogeneous isotropic elastic medium with rigid boundary is considered. A method based on the decoupling of pressure and shear waves via the use of scalar potentials is proposed. This method is adapted to a finite elements discretization, which is discussed. A stable, energy preserving numerical scheme is presented, as well as 2D numerical results. (10.1134/S1995423912020061)
  DOI : 10.1134/S1995423912020061
• About Fokker-Planck equation with measurable coefficients and applications to the fast diffusion equation
  
  • Belaribi Nadia
  • Russo Francesco
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2012, 17 (84), pp.1-28. The object of this paper is the uniqueness for a $d$-dimensional Fokker-Planck type equation with non-homogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so called Barenblatt solution of the fast diffusion equation which is the partial differential equation $\partial_t u = \partial^2_{xx} u^m$ with $m\in(0,1)$. Together with the mentioned Fokker-Planck equation, we make use of small time density estimates uniformly with respect to the initial condition
• Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section
  
  • Imperiale Sébastien
  • Joly Patrick
  Advances in Applied Mechanics, New York ; London ; Paris [etc] : Academic Press, 2012, xx. This paper is the first contribution towards the rigorous justification of asymptotic 1D models for the time-domain simulation of the propagation of electromagnetic waves in coaxial cables. Our general objective is to derive error estimates between the "exact" solution of the full 3D model and the "approximate" solution of the 1D model known as the Telegraphist's equation. (10.4208/aamm.12-12S06)
  DOI : 10.4208/aamm.12-12S06
• Perfectly Matched Layer with Mixed Spectral Elements for the Propagation of Linearized Water Waves
  
  • Cohen Gary
  • Imperiale Sébastien
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.285-302. After setting a mixed formulation for the propagation of linearized water waves problem, we define its spectral element approximation. Then, in order to take into account unbounded domains, we construct absorbing perfectly matched layer for the problem. We approximate these perfectly matched layer by mixed spectral elements and show their stability using the 'frozen coefficient' technique. Finally, numerical results will prove the efficiency of the perfectly matched layer compared to classical absorbing boundary conditions. (10.4208/cicp.201109.261110s)
  DOI : 10.4208/cicp.201109.261110s
• Approximate Models for Wave Propagation Across Thin Periodic Interfaces
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2012, 98 (1), pp.28-71. This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy. (10.1016/j.matpur.2012.01.003)
  DOI : 10.1016/j.matpur.2012.01.003
• Complete Radiation Boundary Conditions for Convective Waves
  
  • Hagstrom Thomas
  • Bécache Eliane
  • Givoli Dan
  • Stein Kurt
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.610-628. Local approximate radiation boundary conditions of optimal efficiency for the convective wave equation and the linearized Euler equations in waveguide geometry are formulated, analyzed, and tested. The results extend and improve for the convective case the general formulation of high-order local radiation boundary condition sequences for anisotropic scalar equations developed in [4]. (10.4208/cicp.231209.060111s)
  DOI : 10.4208/cicp.231209.060111s
• Source point discovery through high frequency asymptotic time reversal
  
  • Benamou Jean-David
  • Collino Francis
  • Marmorat Simon
  Journal of Computational Physics, Elsevier, 2012, 231, pp.4643-4661. (10.1016/j.jcp.2012.03.012)
  DOI : 10.1016/j.jcp.2012.03.012
• d-Transversals of Stable Sets and Vertex Covers in Weighted Bipartite Graphs
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Picouleau Christophe
  • Ries Bernard
  • de Werra Dominique
  Journal of Discrete Algorithms, Elsevier, 2012, 17, pp.95-102. (10.1016/j.jda.2012.06.002)
  DOI : 10.1016/j.jda.2012.06.002
• Characterization of a local quadratic growth of the Hamiltonian for control constrained optimal control problems
  
  • Bonnans J. Frédéric
  • Osmolovskii Nikolai P.
  Dynamics of Continuous, Discrete and Impulsive Systems, University of Waterloo, Ontario, Canada, 2012, 19 (1-2), pp.1-16. We consider an optimal control problem with inequality control constraints given by smooth functions satisfying the hypothesis of linear independence of gradients of active constraints. For this problem, we formulate a generalization of strengthened Legendre condition and prove that this generalization is equivalent to the condition of a local quadratic growth of the Hamiltonian subject to control constraints.
• Passive Optical Network design under Operations Administration and Maintenance considerations
  
  • Hervet Cedric
  • Chardy Matthieu
  Journal of Applied Operational Research, 2012, 4 (3), pp.152-172. Due to the emergence of bandwidth-requiring services, telecommunication operators are being compelled to renew their fixed access network, most of them favouring the Fiber To The Home (FTTH) technology. This paper focuses on the decision problem of the optimization FTTH networks under Operations, Administration and Maintenance (OA&M) considerations. Mixed integer formulations, based on integer generalized flow models, are proposed for the modelling of these decision problems for which we design a specific branch and bound-based solution algorithm. Extensive numerical tests performed on real-life instances prove the efficiency of branch and bound solving approaches for such models. An empirical assessment of the impact of OA&M considerations is also made, leading to the design of operational recommendations.
• Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations: Convergence Result and Error Estimate
  
  • Briani Ariela
  • Camilli Fabio
  • Zidani Hasnaa
  Differential Equations and Applications, Element, 2012, 4, pp.297-317. We consider approximation schemes for monotone systems of fully nonlinear second order partial di erential equations. We rst prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of Hamilton-Jacobi-Bellman equations. Examples including nite di erence schemes and Semi-Lagrangian schemes are discussed. (10.7153/dea-04-18)
  DOI : 10.7153/dea-04-18
• Uniform controllability of scalar conservation laws in the vanishing viscosity limit
  
  • Léautaud Matthieu
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2012, 50 (3), pp.1661-1699. We deal with viscous perturbations of scalar conservation laws on a bounded interval with a general flux function f and a small dissipation coefficient Ɛ. Acting on this system on both endpoints of the interval, we prove global exact controllability to constant states with nonzero speed. More precisely, we construct boundary controls so that the solution is driven to the targeted constant state, and we moreover require these controls to be uniformly bounded as Ɛ → 0+ in an appropriate space. For general (nonconvex) flux functions this can be done for sufficiently large time, and for convex fluxes f, we have a precise estimate on the minimal time needed to control. © 2012 Society for Industrial and Applied Mathematics. (10.1137/100803043)
  DOI : 10.1137/100803043
• Giens 2011
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2012.
• La RO au cœur du déploiement du Fiber To The Home à France-Télécom Orange: La RO récompensée par le prix Orange de l'Innovation 2012 (catégorie Réseau)
  
  • Chardy Matthieu
  • Costa Marie-Christine
  • Faye Alain
  • Francfort Stanislas
  • Hervet Cedric
  • Trampont Mathieu
  Bulletin de la ROADEF, Société Française de Recherche Opérationnelle et d'Aide à la Décision, 2012, 29, pp.8-11. L'objectif actuel des opérateurs télécoms est de proposer et fournir des services à valeur ajoutée, toujours plus consommateurs en ressources réseaux. Longtemps situé dans le réseau dit " cœur ", le goulot d'étrangle- ment en termes de capacité s'est aujourd'hui dé- placé vers le réseau dit d'" accès ". En ce qui concerne le ré- seau d'accès fixe, cette nécessaire montée en dé- bit s'est effectuée sans changer l'infrastructure phy- sique cuivre, profitant d'une " simple " modula- tion de la fréquence des signaux transportés (bas- débit versus haut-débit). Mais aujourd'hui, le ré- seau cuivre montre ses limites à accompagner la bande passante toujours plus importante attendue par les utilisateurs. Et cette limite ne ferait que s'accentuer dans le futur. Face à ce constat, les opérateurs télécoms sont contraints d'engager un renouvellement de leur ré- seau d'accès fixe. La technologie fibre retenue par France-Télécom Orange est la Fiber To The Home (FTTH), et plus précisément l'architecture Passive Optical Network (PON). Il s'agit d'une archi- tecture dite Point à Multipoints, où chaque fibre optique partant du cœur de réseau est démultipliée lors de son passage dans des équipements passifs ap- pelés coupleurs optiques : une fibre dessert jusqu'à 64 clients qui ont, en bas de chez eux, une fibre optique individuelle. Ce renouvellement complet de l'infrastructure du réseau d'accès fixe s'effectue de manière progres- sive et nécessitera un investissement échelonné es- timé à plusieurs dizaines de milliards d'euros d'ici sa fin prévue en 2020. Face à de tels enjeux financiers et à une telle complexité dans le choix du déploiement d'un réseau FTTH, un outil d'optimisation du design des réseaux FTTH PON apparaissait donc crucial.
• A low frequency model for acoustic propagation in a 2D flow duct: numerical computation
  
  • Joubert Lauris
  • Joly Patrick
  Communications in Computational Physics, Global Science Press, 2012, 11 (2), pp.508-524. In this paper we study a low frequency model for acoustic propagation in a 2D flow duct. For some Mach profile flow, we are able to give a well-posedness theorem. Its proof relies on a quasi-explicit expression of the solution which provides us an efficient numerical method. We give and comment numerical results for particular linear, tangent and quadratic profiles. Finally, we give a numerical validation of our asymptotic model.