Publications

2010

• Asymptotic modelling of conductive thin sheets
  
  • Schmidt Kersten
  • Tordeux Sébastien
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2010, 61 (4), pp.603-626. We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H1-modelling error for an expansion with N terms is bounded by O(ε^{N+1}) in the exterior of the sheet and by O(ε^{N+1/2}) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results. (10.1007/s00033-009-0043-x)
  DOI : 10.1007/s00033-009-0043-x
• About stability and regularization of ill-posed elliptic Cauchy problems: The case of C1,1 domains
  
  • Bourgeois Laurent
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (4), pp.715-735. This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with C 1,1 boundary. It is an extension of an earlier result of [Phung, ESAIM: COCV 9 (2003) 621-635] for domains of class C∞. Our estimate is established by using a Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility introduced in [Lattès and Lions, Dunod (1967)] to solve the ill-posed Cauchy problems. © EDP Sciences, SMAI, 2010. (10.1051/m2an/2010016)
  DOI : 10.1051/m2an/2010016
• Lipschitz solutions of optimal control problems with state constraints of arbitrary order
  
  • Bonnans J. Frederic
  Mathematics and its Applications: Annals of the Academy of Romanian Scientists, Academy of Romanian Scientists Publishing House, 2010, 2 (1), pp.78-98. In this paper we generalize to an arbitrary order, under minimal hypotheses, some sufficient conditions for Lipschitz continuity of the solution of a state constrained optimal control problems. The proof combines the approach by Hager in 1979 for dealing with first-order state constraints, and the high-order alternative formulation of the optimality conditions.
• Initialization of the shooting method via the Hamilton-Jacobi-Bellman approach
  
  • Cristiani Emiliano
  • Martinon Pierre
  Journal of Optimization Theory and Applications, Springer Verlag, 2010, 146 (2), pp.321-346. The aim of this paper is to investigate from the numerical point of view the possibility of coupling the Hamilton-Jacobi-Bellman (HJB) approach and the Pontryagin's Minimum Principle (PMP) to solve some control problems. We show that an approximation of the value function computed by the HJB method on rough grids can be used to obtain a good initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered. The CPU time for the proposed method is less than four minutes up to dimension four, without code parallelization. (10.1007/s10957-010-9649-6)
  DOI : 10.1007/s10957-010-9649-6
• A kinetic mechanism inducing oscillations in simple chemical reactions networks
  
  • Coatléven Julien
  • Altafini Claudio
  Mathematical Biosciences and Engineering, AIMS Press, 2010, 7 (2), pp.301-312. It is known that a kinetic reaction network in which one or more secondary substrates are acting as cofactors may exhibit an oscillatory behavior. The aim of this work is to provide a description of the functional form of such a cofactor action guaranteeing the onset of oscillations in sufficiently simple reaction networks. (10.3934/mbe.2010.7.301)
  DOI : 10.3934/mbe.2010.7.301
• Decomposition of large-scale stochastic optimal control problems
  
  • Carpentier Pierre
  • Barty Kengy
  • Girardeau Pierre
  RAIRO - Operations Research, EDP Sciences, 2010, 44, pp.167-183. In this paper, we present an Uzawa-based heuristic that is adapted to certain type of stochastic optimal control problems. More precisely, we consider dynamical systems that can be divided into smallscale subsystems linked through a static almost sure coupling constraint at each time step. This type of problem is common in production/ portfolio management where subsystems are, for instance, power units, and one has to supply a stochastic power demand at each time step. We outline the framework of our approach and present promising numerical results on a simplified power management problem. (10.1051/ro/2010013)
  DOI : 10.1051/ro/2010013
• Analysis of Acoustic Wave Propagation in a Thin Moving Fluid
  
  • Joly Patrick
  • Weder Ricardo
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2010, 70, pp.2449-2472. We study the propagation of acoustic waves in a fluid that is contained in a thin two-dimensional tube and that it is moving with a velocity profile that depends only on the transversal coordinate of the tube. The governing equations are the Galbrun equations or, equivalently, the linearized Euler equations. We analyze the approximate model that was recently derived by Bonnet-Bendhia, Durufle, and Joly to describe the propagation of the acoustic waves in the limit when the width of the tube goes to zero. We study this model for strictly monotonic stable velocity profiles. We prove that the equations of the model of Bonnet-Bendhia, Durufle, and Joly are well posed, i.e., that there is a unique global solution, and that the solution depends continuously on the initial data. Moreover, we prove that for smooth profiles the solution grows at most as t(3) as t -> infinity, and that for piecewise linear profiles it grows at most as t(4). This establishes the stability of the model in a weak sense. These results are obtained by constructing a quasi-explicit representation of the solution. Our quasi-explicit representation gives a physical interpretation of the propagation of acoustic waves in the fluid and provides an efficient way to compute the solution numerically. (10.1137/09077237X)
  DOI : 10.1137/09077237X
• Generation of Higher-Order Polynomial Bases of Nédélec H(curl) Finite Elements for Maxwell's Equations
  
  • Bergot Morgane
  • Lacoste Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (6). The goal of this study is the automatic construction of a vectorial polynomial basis for Nédélec mixed finite elements, particular, the generation of finite elements without the expression of the polynomial basis functions, using symbolic calculus: the exhibition of basis functions has no practical interest.
• Blockers and Transversals in some subclasses of bipartite graphs: when caterpillars are dancing on a grid
  
  • Ries Bernard
  • Bentz Cédric
  • de Werra Dominique
  • Costa Marie-Christine
  • Zenklusen Rico
  • Picouleau Christophe
  Discrete Mathematics, Elsevier, 2010, 310, pp.132--146. (10.1016/j.disc.2009.08.009)
  DOI : 10.1016/j.disc.2009.08.009
• Time harmonic wave diffraction problems in materials with sign-shifting coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  • Zwölf Carlo Maria
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (6), pp.1912-1919. Some electromagnetic materials present, in a given frequency range, an effective dielectric permittivity and/or magnetic permeability which are negative. We are interested in the reunion of such a "negative" material and a classical one. More precisely, we consider here a scalar model problem for the simulation of a wave transmission between two such materials. This model is governed by a Helmholtz equation with a weight function in the ΔΔ principal part which takes positive and negative real values. Introducing additional unknowns, we have already proposed in Bonnet-Ben Dhia et al. (2006) [1] some new variational formulations of this problem, which are of Fredholm type provided the absolute value of the contrast of permittivities is large enough, and therefore suitable for a finite element discretization. We prove here that, under similar conditions on the contrast, the natural variational formulation of the problem, although not "coercive plus compact", is nonetheless suitable for a finite element discretization. This leads to a numerical approach which is straightforward, less costly than the previous ones, and very accurate. (10.1016/j.cam.2009.08.041)
  DOI : 10.1016/j.cam.2009.08.041
• Quadratic growth conditions in optimal control problems
  
  • Bonnans Joseph Frederic
  • Osmolovskii Nikolai P.
  Contemporary mathematics, American Mathematical Society, 2010, 514, pp.85--98.
• A biomechanical inactivation principle
  
  • Berret Bastien
  • Jean Frédéric
  • Gauthier Jean-Paul
  Proceedings of the Steklov Institute of Mathematics, MAIK Nauka/Interperiodica, 2010, 268, pp.93--116. (10.1134/S0081543810010098)
  DOI : 10.1134/S0081543810010098
• Comparison of High-Order Absorbing Boundary Conditions and Perfectly Matched Layers in the Frequency Domain
  
  • Rabinovich Daniel
  • Givoli Dan
  • Bécache Eliane
  International Journal for Numerical Methods in Biomedical Engineering, John Wiley and Sons, 2010, 26, pp.1351-1369.
• Security Analysis of Word Problem-Based Cryptosystems
  
  • Levy-Dit-Vehel Françoise
  • Perret Ludovic
  Designs, Codes and Cryptography, Springer Verlag, 2010, 54 (1), pp.29-41. We investigate two schemes based on the word problem on groups. From a complexity-theoretic point of view, we show that the problems underlying those schemes are equivalent. We then present a reaction attack on one of the schemes, thus easily transposed to the other. The attack, besides its efficiency, permits to recover an equivalent secret key. (10.1007/s10623-009-9307-x)
  DOI : 10.1007/s10623-009-9307-x
• A Numerical Study of Variable Depth KdV Equations and Generalizations of Camassa-Holm-like Equations
  
  • Duruflé Marc
  • Israwi Samer
  , 2010. In this paper we study numerically the KdV-top equation and compare it with the Boussinesq equations over uneven bottom. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa-Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method
• Efficient computation of photonic crystal waveguide modes with dispersive material
  
  • Schmidt Kersten
  • Kappeler Roman
  Optics Express, Optical Society of America - OSA Publishing, 2010, 18 (7), pp.7307-7322. The optimization of PhC waveguides is a key issue for successfully designing PhC devices. Since this design task is computationally expensive, efficient methods are demanded. The available codes for computing photonic bands are also applied to PhC waveguides. They are reliable but not very efficient, which is even more pronounced for dispersive material. We present a method based on higher order finite elements with curved cells, which allows to solve for the band structure taking directly into account the dispersiveness of the materials. This is accomplished by reformulating the wave equations as a linear eigenproblem in the complex wave-vectors k. For this method, we demonstrate the high efficiency for the computation of guided PhC waveguide modes by a convergence analysis. © 2010 Optical Society of America. (10.1364/oe.18.007307)
  DOI : 10.1364/oe.18.007307
• Second-order analysis of optimal control problems with control and initial-final state constraints
  
  • Bonnans J. Frederic
  • Osmolovskii Nikolai P.
  Journal of Convex Analysis, Heldermann, 2010, 17 (3), pp.885-913. This paper provides an analysis of Pontryagine minima satisfying a quadratic growth condition, for optimal control problems of ordinary differential equations with constraints on initial-final state, as well as control constraints satisfying the uniform positive linear independence condition.
• A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Inverse Problems, IOP Publishing, 2010, 26 (9), pp.095016. In this paper, we introduce a new version of the method of quasi-reversibility to solve the ill-posed Cauchy problems for the Laplace's equation in the presence of noisy data. It enables one to regularize the noisy Cauchy data and to select a relevant value of the regularization parameter in order to use the standard method of quasi-reversibility. Our method is based on duality in optimization and is inspired by the Morozov's discrepancy principle. Its efficiency is shown with the help of some numerical experiments in two dimensions. © 2010 IOP Publishing Ltd. (10.1088/0266-5611/26/9/095016)
  DOI : 10.1088/0266-5611/26/9/095016
• Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string
  
  • Chabassier Juliette
  • Joly Patrick
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2010, 199 (45-48), pp.2779-2795. This paper considers a general class of nonlinear systems, "nonlinear Hamiltonian systems of wave equations". The first part of our work focuses on the mathematical study of these systems, showing central properties (energy preservation, stability, hyperbolicity, finite propagation velocity, etc.). Space discretization is made in a classical way (variational formulation) and time discretization aims at numerical stability using an energy technique. A definition of "preserving schemes" is introduced, and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is trivial. A general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear Hamiltonian systems of wave equations class. The problem of the vibration of a piano string is taken as an example. Nonlinear coupling between longitudinal and transversal modes is modeled in the "geometrically exact model", or approximations of this model. Numerical results are presented. (10.1016/j.cma.2010.04.013)
  DOI : 10.1016/j.cma.2010.04.013
• Finite Element Modeling of Airflow During Phonation
  
  • Sidlof Petr
  • Lunéville Éric
  • Chambeyron Colin
  • Doaré Olivier
  • Chaigne A
  • Horáček J
  Journal of Computational and Applied Mechanics, Miskolci Egyetemi Kiadó, 2010, 4, pp.121-132. In the paper a mathematical model of airflow in human vocal folds is presented. The geometry of the glottal channel is based on measurements of excised human larynges. The airflow is modeled by nonstationary incompressible Navier-Stokes equations in a 2D computational domain, which is deformed in time due to vocal fold vibration. The paper presents numerical results and focuses on flow separation in glottis. Quantitative data from numerical simulations are compared to results of measurements by Particle Image Velocimetry (PIV), performed on a scaled self-oscillating physical model of vocal folds.
• Weak vector and scalar potentials. Applications to Poincaré's theorem and Korn's inequality in Sobolev spaces with negative exponents.
  
  • Amrouche Chérif
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Analysis and Applications, World Scientific Publishing, 2010, 8 (1), pp.1-17. In this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré's theorem and to Korn's inequality. (10.1142/s0219530510001497)
  DOI : 10.1142/s0219530510001497
• Electrowetting of a 3D drop: Numerical modelling with electrostatic vector fields
  
  • Ciarlet Patrick
  • Scheid Claire
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (4), pp.647-670. The electrowetting process is commonly used to handle very small amounts of liquid on a solid surface. This process can be modelled mathematically with the help of the shape optimization theory. However, solving numerically the resulting shape optimization problem is a very complex issue, even for reduced models that occur in simplified geometries. Recently, the second author obtained convincing results in the 2D axisymmetric case. In this paper, we propose and analyze a method that is suitable for the full 3D case. © EDP Sciences, SMAI, 2010. (10.1051/m2an/2010014)
  DOI : 10.1051/m2an/2010014
• Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous data
  
  • Bokanowski Olivier
  • Megdich Nadia
  • Zidani Hasnaa
  Numerische Mathematik, Springer Verlag, 2010, 115 (1), pp.1--44. On étudie un schéma non monotone pour l'équation Hamilton Jacobi Bellman du premier ordre, en dimension 1. Le schéma considèré est lié au schéma anti-diffusif, appellé UltraBee, proposé par Bokanowski-Zidani (publié en 2007 dans J. Sci. Compt.). Ici, on prouve la convergence, en norme $L^1$, à l'ordre 1, pour une condition initiale discontinue. Le caractère anti-diffusif du schéma est illustré par quelques exemples numériques. (10.1007/s00211-009-0271-1)
  DOI : 10.1007/s00211-009-0271-1