Publications

2012

• Evaluation of 3-D Singular and Nearly Singular Integrals in Galerkin BEM for Thin Layers
  
  • Lenoir Marc
  • Salles Nicolas
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 36, pp.3057-3078. An explicit method for the evaluation of singular and near-singular integrals arising in three-dimensional Galerkin BEM is presented. It is based on a recursive reduction of the dimension of the integration domain leading to a linear combination of one-dimensional regular integrals, which can be exactly evaluated. This method has appealing properties in terms of reliability, precision, and flexibility. The results we present here are devoted to the case of thin layers for the Helmholtz equation, a situation where the panels are close and parallel, known to be difficult in terms of accuracy. Nevertheless, the method applies as well to two-dimensional BEM, secant planes, or even volume integral equations. A MATLAB implementation of the formulas presented here is available online. (10.1137/120866567)
  DOI : 10.1137/120866567
• The Variational Theory of Complex Rays for three-dimensional Helmholtz problems
  
  • Kovalevsky Louis
  • Ladevèze Pierre
  • Riou Hervé
  • Bonnet Marc
  Journal of Computational Acoustics, World Scientific Publishing, 2012, 20, pp.125021 (25 pages). This article proposes an extension of the Variational Theory of Complex Rays (VTCR) to three-dimensional linear acoustics, The VTCR is a Trefftz-type approach designed for mid-frequency range problems and has been previously investigated for structural dynamics and 2D acoustics. The proposed 3D formulation is based on a discretization of the amplitude portrait using spherical harmonics expansions. This choice of discretization allows to substantially reduce the numerical integration work by taking advantage of well-known analytical properties of the spherical harmonics. It also permits (like with the previous 2D Fourier version) an effective \emph{a priori} selection method for the discretization parameter in each sub-region, and allows to estimate the directivity of the pressure field by means of a natural definition of rescaled amplitude portraits. The accuracy and performance of the proposed formulation are demonstrated on a set of numerical examples that include results on an actual case study from the automotive industry. (10.1142/S0218396X1250021X)
  DOI : 10.1142/S0218396X1250021X
• A Patchy Dynamic Programming Scheme for a Class of Hamilton-Jacobi-Bellman Equations
  
  • Cacace Simone
  • Cristiani Emiliano
  • Falcone Maurizio
  • Picarelli Athena
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (5), pp.A2625-A2649. In this paper we present a new parallel algorithm for the solution of Hamilton-Jacobi- Bellman equations related to optimal control problems. The main idea is to divide the domain of computation into subdomains following the dynamics of the control problem. This results in a rather complex geometrical subdivision, but has the advantage that every subdomain is invariant with respect to the optimal controlled vector field, so that we can compute the value function in each subdomain assigning the task to a processor and avoiding the classical transmission condition on the boundaries of the subdomains. For this specific feature the subdomains are patches in the sense introduced by Ancona and Bressan in [1]. Several examples in dimension two and three illustrate the properties of the new method. (10.1137/110841576)
  DOI : 10.1137/110841576
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (3), pp.A1824-A1848. We consider the inverse obstacle scattering problem of determining both the shape and the "equiva- lent impedance" from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface differential operator (refer to as generalized impedance boundary condition) as opposed to a scalar function. The generalized impedance boundary condition can be seen as a more accurate model for effective impedances and is widely used in the scattering problem for thin coatings. Our approach is based on a least square optimization technique. A major part of our analysis is to characterize the derivative of the cost function with respect to the boundary and this complex surface impedance configuration. In particular, we provide an extension of the notion of shape derivative to the case where the involved impedance parameters do not need to be surface traces of given functions, which leads (in general) to a non-vanishing tangential boundary perturbation. The efficiency of considering this type of derivative is illustrated by several 2D numerical experiments based on a (classical) steepest descent method. The feasibility of retrieving both the shape and the impedance parameters is also discussed in our numerical experiments. (10.1137/110850347)
  DOI : 10.1137/110850347
• Usual Anderson localization restored in bilayered left- and right-handed structures
  
  • Maurel Agnès
  • Ourir Abdelwaheb
  • Mercier Jean-François
  • Pagneux Vincent
  Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2012, 85 (20). We present a study of the attenuation length in a one-dimensional array of alternating left- and right-handed materials in which both the permittivities and the permeabilities are disordered. This type of structure has been shown to present an anomaly in the attenuation length when only permeabilities are disordered. We derive a simple analytical expression of the attenuation length, when the disorder in the refraction index is due to perturbations in both the permeability and the permittivity. Our expression is able to explain the transition to the anomalous behavior when perturbation only in the permeability or only in the permittivity is considered. Besides, we show that the anomaly is dramatically affected when considering perturbations in permeability and permittivity. The coupling effects are able to restore the ordinary localization length. © 2012 American Physical Society. (10.1103/physrevb.85.205138)
  DOI : 10.1103/physrevb.85.205138
• Interior transmission eigenvalue problem for Maxwell's equations: The T-coercivity as an alternative approach
  
  • Chesnel Lucas
  Inverse Problems, IOP Publishing, 2012, 28 (6). In this paper, we examine the interior transmission problem for Maxwells equations in the case where both and , the physical parameters of the scattering medium, differ from 0 and 0 modelling the background medium. Using the T-coercivity method, we propose an alternative approach to the classical techniques to prove that this problem is of Fredholm type and that the so-called transmission eigenvalues form at most a discrete set. The T-coercivity approach allows us to deal with cases where 0 and 0 can change sign. We also provide results of localization and FaberKrahn-type inequalities for the transmission eigenvalues. © 2012 IOP Publishing Ltd. (10.1088/0266-5611/28/6/065005)
  DOI : 10.1088/0266-5611/28/6/065005
• Transparent boundary conditions for evolution equations in infinite periodic strips
  
  • Coatléven Julien
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (3), pp.1563-1583. We consider the solution of a generic equation $\gamma\rho(\mathbf{x})\partial^p_tu(\mathbf{x},t)-\Delta u(\mathbf{x},t) +V(\mathbf{x})u(\mathbf{x},t) = f(\mathbf{x},t)$, $\mathbf{x} = (x,y)$, for $t>0$, $p=1,2$ in a domain $\Omega$ which is infinite in $x$ and bounded in $y$. We assume that $f(\cdot,t)$ is supported for all $t>0$ in $\Omega_0 = \{\mathbf{x} \in \Omega \; | \; -a_- < x < a_+\}$ and that $\rho(\mathbf{x})$ and $V(\mathbf{x})$ are x-periodic in $\Omega \setminus \Omega_0$. We consider the associated $\theta$-scheme in time, to obtain a semidiscretized problem. We then show how to obtain for each time step exact boundary conditions on the vertical segments, $\Gamma_0^- = \{\mathbf{x}\in \Omega\; | \; x=-a_-\}$ and $\Gamma_0^+ = \{\mathbf{x}\in \Omega \;| \; x=a_+\}$, that will enable us to find the solution on $\Omega_0 \cup \Gamma_0^+ \cup \Gamma_0^-$. Then the solution can be extended in $\Omega$ in a straightforward manner from the values on $\Gamma_0^-$ and $\Gamma_0^+$. The method is based on the solution of local problems on a single periodicity cell, solved during an initialization step. The exact boundary conditions as well as the extension operators can be obtained for each time step through elementary computations using the solution of these local cell problems. (10.1137/110838030)
  DOI : 10.1137/110838030
• Sensitivity analysis of energy contracts management problem by stochastic programming techniques
  
  • Cen Zhihao
  • Bonnans J. Frederic
  • Christel Thibault
  , 2012, 12 (2012), pp.447-471.. 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. In our previous article, we proposed an algorithm to deal with such problem, based on quantization of random process and a dual dynamic programming type approach. We obtained accurate estimates of the optimal value and a suboptimal strategy from this algorithm. In this paper, we analyse the sensitivity with respect to parameters driving the price model. We discuss the estimate of marginal price based on the Danskin's theorem. Finally, some numerical results applied to realistic energy market problems have been performed. Comparisons between results obtained by our algorithm and other classical methods are provided and evidence the accuracy of the estimate of marginal prices.
• 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
• 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.
• 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
• 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.
• 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.
• 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.
• Giens 2011
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2012.