Publications

2010

• High-order Absorbing Boundary Conditions for anisotropic and convective wave equations
  
  • Bécache Eliane
  • Givoli Dan
  • Hagstrom Thomas
  Journal of Computational Physics, Elsevier, 2010, 229 (4), pp.1099-1129. High-order Absorbing Boundary Conditions (ABCs), applied on a rectangular artificial computational boundary that truncates an unbounded domain, are constructed for a general two-dimensional linear scalar time-dependent wave equation which represents acoustic wave propagation in anisotropic and subsonically convective media. They are extensions of the construction of Hagstrom, Givoli and Warburton for the isotropic stationary case. These ABCs are local, and involve only low-order derivatives owing to the use of auxiliary variables on the artificial boundary. The accuracy and well-posedness of these ABCs is analyzed. Special attention is given to the issue of mismatch between the directions of phase and group velocities, which is a potential source of concern. Numerical examples for the anisotropic case are presented, using a finite element scheme. © 2009 Elsevier Inc. All rights reserved. (10.1016/j.jcp.2009.10.012)
  DOI : 10.1016/j.jcp.2009.10.012
• An efficient data structure to solve front propagation problems
  
  • Bokanowski Olivier
  • Cristiani Emiliano
  • Zidani Hasnaa
  Journal of Scientific Computing, Springer Verlag, 2010, 42 (2), pp.251--273. In this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d>= 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an Hamilton-Jacobi-Bellman equation and use an adapted anti-diffusive Ultra-Bee scheme. We obtain a general method which is faster than a full storage technique. We show that we can compute problems that are out of reach by full storage techniques (because of memory). Numerical experiments are provided in dimension d=2,3,4. (10.1007/s10915-009-9329-6)
  DOI : 10.1007/s10915-009-9329-6
• About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Applicable Analysis, Taylor & Francis, 2010, 89 (11), pp.1745-1768. This article is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for Laplace's equation in domains with Lipschitz boundary. It completes the results obtained by Bourgeois [Conditional stability for ill-posed elliptic Cauchy problems: The case of C1,1 domains (part I), Rapport INRIA 6585, 2008] for domains of class C1,1. This estimate is established by using an interior Carleman estimate and a technique based on a sequence of balls which approach the boundary. This technique is inspired by Alessandrini et al. [Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Annali della Scuola Normale Superiore di Pisa 29 (2000), pp. 755-806]. We obtain a logarithmic stability estimate, the exponent of which is specified as a function of the boundary's singularity. Such stability estimate induces a convergence rate for the method of quasi-reversibility introduced by Lattés and Lions [Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967] to solve the Cauchy problems. The optimality of this convergence rate is tested numerically, precisely a discretized method of quasi-reversibility is performed by using a nonconforming finite element. The obtained results show very good agreement between theoretical and numerical convergence rates. © 2010 Taylor & Francis. (10.1080/00036810903393809)
  DOI : 10.1080/00036810903393809
• 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
• 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
• Reachability and minimal times for state constrained nonlinear problems without any controllability assumption
  
  • Bokanowski Olivier
  • Forcadel Nicolas
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2010, 48 (7), pp.pp. 4292-4316. We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent state constraints, as well as a target problem for a two-player game with state constraints. Our method gives a good framework for numerical approximations, and some numerical illustrations are included in the paper.
• Weighted regularization for composite materials in electromagnetism
  
  • Ciarlet Patrick
  • Lefèvre François
  • Lohrengel Stéphanie
  • Nicaise Serge
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (1), pp.75-108. In this paper, a weighted regularization method for the time-harmonic Maxwell equations with perfect conducting or impedance boundary condition in composite materials is presented. The computational domain Ω is the union of polygonal or polyhedral subdomains made of different materials. As a result, the electromagnetic field presents singularities near geometric singularities, which are the interior and exterior edges and corners. The variational formulation of the weighted regularized problem is given on the subspace of H(curl;Ω) whose fields u satisfy wα div(εu) ∈ L 2(Ω) and have vanishing tangential trace or tangential trace in L2(δΩ). The weight function w(x) is equivalent to the distance of x to the geometric singularities and the minimal weight parameter α is given in terms of the singular exponents of a scalar transmission problem. A density result is proven that guarantees the approximability of the solution field by piecewise regular fields. Numerical results for the discretization of the source problem by means of Lagrange Finite Elements of type P1 and P2 are given on uniform and appropriately refined two-dimensional meshes. The performance of the method in the case of eigenvalue problems is addressed. © EDP Sciences, SMAI 2009. (10.1051/m2an/2009041)
  DOI : 10.1051/m2an/2009041
• 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
• L^1-error estimate for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1.
  
  • Bokanowski Olivier
  • Forcadel Nicolas
  • Zidani Hasnaa
  Mathematics of Computation, American Mathematical Society, 2010, 79 (271), pp.1395--1426. The goal of this paper is to study some numerical approximations of particular Hamilton-Jacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes, the first one is based on the Ultra-Bee scheme and the second one is based on the Fast Marching Method. We prove the convergence and derive $L^1$-error estimates for both schemes. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions.
• 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.
• 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
• 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
• 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.
• 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
• 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
• 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
• 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.
• Quadratic growth conditions in optimal control problems
  
  • Bonnans Joseph Frederic
  • Osmolovskii Nikolai P.
  Contemporary mathematics, American Mathematical Society, 2010, 514, pp.85--98.
• 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
• 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