Publications

2009

• A distribution framework for the generalized Fourier transform associated with a Sturm--Liouville operator
  
  • Hazard Christophe
  , 2009, pp.18. The generalized Fourier transform associated with a selfadjoint Sturm--Liouville operator is a unitary transformation which converts the action of this operator into a simple product by a spectral variable. For a particular operator defined on the half-line and which involves a step function, we show how to extend such a transformation to generalized functions, or distributions, with a suitable definition of such distributions. This extension is based essentially on the fact that, as the usual Fourier transform, this transformation has the property to exchange regularity and decay between the physical and spectral variables.
• Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition
  
  • Bécache Eliane
  • Rodríguez Jerónimo
  • Tsogka Chrysoula
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2009, 43 (2), pp.377-398. The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly taken into account when the first family of mixed finite elements is used. We, therefore, introduce the second family of mixed finite elements for which a theoretical convergence analysis is presented and error estimates are obtained. A numerical study of the convergence is also considered for a particular object geometry which shows that our theoretical error estimates are optimal. © 2009 EDP Sciences SMAI. (10.1051/m2an:2008047)
  DOI : 10.1051/m2an:2008047
• Stability Analysis of Optimal Control Problems with a Second-order State Constraint
  
  • Hermant Audrey
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 20 (1), pp.104-129. This paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second-order. The strengthened Legendre-Clebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak second-order sufficient condition (taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the $L^2$ norm, and Hölder continuous in the $L^\infty$ norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager [SIAM J. Control Optim., 1998]. The difficulty is that multipliers associated with second-order state constraints have a low regularity (they are only bounded measures). We obtain Lipschitz stability of a ``primitive'' of the state constraint multiplier. (10.1137/070707993)
  DOI : 10.1137/070707993
• Direct computation of stresses in planar linearized elasticity
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (7), pp.1043-1064. Given a simply-connected domain Ω in ℝ2, consider a linearly elastic body with Ω as its reference configuration, and define the Hilbert space E(Ω)={e(eαβ) ∈ L2s (Ω) ∂11e22- 2∂12e12}+∂22e11 = 0 in H-2(Ω)}. Then we recently showed that the associated pure traction problem is equivalent to finding a 2 × 2 matrix field = (∈αβ) ∈E(Ω) that satisfies j(∈)= inf e∈E(Ω) j(e), where j(e) = 1/2 ∫Ω Aαβστ eστ eαβ dx - l(e), where (A αβστ ) is the elasticity tensor, and l is a continuous linear form over E(Ω) that takes into account the applied forces. Since the unknown stresses (σαβ) inside the elastic body are then given by σαβ = Aαβστ eστ, this minimization problem thus directly provides the stresses. We show here how the above Saint Venant compatibility condition ∂11 e22 - 2∂12e12 + ∂22e11 = 0 in H-2(Ω) can be exactly implemented in a finite element space h, which uses "edge" finite elements in the sense of J. C. Nédélec. We then establish that the unique solution h of the associated discrete problem, viz., find ∈h ∈ Eh such that j(∈h)=inf eh∈Eh j(eh) converges to in the space L2 s(Ω). We emphasize that, by contrast with a mixed method, only the approximate stresses are computed in this approach. © 2009 World Scientific Publishing Company. (10.1142/s0218202509003711)
  DOI : 10.1142/s0218202509003711
• Application of Discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic
  
  • Castel Nicolas
  • Cohen Gary
  • Duruflé Marc
  Journal of Computational Acoustics, World Scientific Publishing, 2009, 17 (2), pp.175-196. A discontinuous Galerkin method is developed for linear hyperbolic systems on general hexahedral meshes. The use of hexahedral elements and tensorized quadrature formulas to evaluate the integrals leads to an efficient matrix-vector product. It is shown for high order approximations, the reduction in computational time can be very important, compared to tetrahedral elements. Two choices of quadrature points are considered, the Gauss points or Gauss-Lobatto points. The method is applied to the aeroacoustic system (simplified Linearized Euler Equations). Some 3-D numericals experiments show the importance of penalization, and the advantage of using high order.
• Resonances of an elastic plate coupled with a compressible confined flow
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2009, 62 (2), pp.105-129. A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid-structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations λ(k) = k2, where λ is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically. © The author 2009. Published by Oxford University Press; all rights reserved. (10.1093/qjmam/hbp004)
  DOI : 10.1093/qjmam/hbp004
• Control problems with mixed constraints and application to an optimal investment problem
  
  • Bonnans J. Frederic
  • Tiba Dan
  Mathematical Reports, Romanian Academy of Sciences, 2009, 4, pp.293-306.
• Comparison principle for a Generalized Fast Marching Method
  
  • Forcadel Nicolas
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.pp. 1923-1951. In \cite{CFFM06}, the authors have proposed a generalization of the classical Fast Marching Method of Sethian for the eikonal equation in the case where the normal velocity depends on space and time and can change sign. The goal of this paper is to propose a modified version of the Generalized Fast Marching Method proposed in \cite{CFFM06} for which we state a general comparison principle. We also prove the convergence of the new algorithm.
• Numerical study of optimal trajectories with singular arcs for an Ariane 5 launcher
  
  • Martinon Pierre
  • Bonnans J. Frederic
  • Laurent-Varin Julien
  • Trélat Emmanuel
  Journal of Guidance, Control, and Dynamics, American Institute of Aeronautics and Astronautics, 2009, 32 (1), pp.51--55. We consider a flight mission to the geostationary transfer orbit (GTO) for an Ariane 5 launcher, while maximizing the payload or, as a variant, minimizing the fuel consumption. We first solve the complete flight sequence up to the final orbit, assuming a maximal thrust for all propulsion systems. Then we focus on the tmospheric ascent phase, which has been studied for instance in [1, 2, 3]. We are more specifically interested in optimal tra jectories with singular arcs (flight phases with a non maximal thrust) for the boosters. Due to the presence of tabulated data in the physical model, the exact expression of the singular control cannot be obtained from the time derivatives of the switching function. An alternate way to compute the singular control is provided, and numerical experiments are carried out for for several launcher variants.
• Blockers and Transversals
  
  • Zenklusen Rico
  • Ries Bernard
  • Picouleau Christophe
  • de Werra Dominique
  • Costa Marie-Christine
  • Bentz Cédric
  Discrete Mathematics, Elsevier, 2009, 13, pp.4306--4314. We explore connections between d-blockers B in a graph G = (V;E) (i.e. subsets of edges whose removal decreases by at least d the cardinality of maximum matchings) and d-transversals T (i.e. subsets of edges such that every maximum matching M has at least d edges in T. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, grid graphs, chains and cycles. We also study the complexity status of finding minimum transversals and blockers. Algorithms for d-transversals and d- blockers based on dynamic programming are given for trees. (10.1016/j.disc.2009.01.006)
  DOI : 10.1016/j.disc.2009.01.006
• Diffraction by a defect in an open waveguide: A Mathematical analysis based on a modal radiation condition
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Dakhia Ghania
  • Hazard Christophe
  • Chorfi Lahcène
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2009, 70 (3), pp.677-693. We consider the scattering of a time-harmonic acoustic wave by a defect in a twodimensional open waveguide. The scattered wave satisfies the Helmholtz equation in a perturbed layered half-plane. We introduce a modal radiation condition based on a generalized Fourier transform which diagonalizes the transverse contribution of the Helmholtz operator. The uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized Fourier variable. The existence is then deduced classically from Fredholm's alternative by reformulating the scattering problem as a Lippmann-Schwinger equation by means of the Green's function for the layered half-plane. © 2009 Society for Industrial and Applied Mathematics. (10.1137/080740155)
  DOI : 10.1137/080740155
• Solving electromagnetic eigenvalue problems in polyhedral domains with nodal finite elements
  
  • Buffa Annalisa
  • Ciarlet Patrick
  • Jamelot Erell
  Numerische Mathematik, Springer Verlag, 2009, 113 (4), pp.497-518. A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with the help of a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, their method required a parameterization of the variational formulation. In order to avoid this difficulty, we use a mixed variational setting instead of the parameterization, which allows us to handle the divergence-free constraint on the field in a straightforward manner. The numerical analysis of the method is carried out, and numerical examples are provided to show the efficiency of our approach. © Springer-Verlag 2009. (10.1007/s00211-009-0246-2)
  DOI : 10.1007/s00211-009-0246-2
• Reactive transport in porous media
  
  • Apoung-Kamga Jean-Baptiste
  • Have Pascal
  • Houot Jean
  • Kern Michel
  • Semin Adrien
  ESAIM: Proceedings, EDP Sciences, 2009, 28, pp.227 - 245. We present a numerical method for coupling transport with chemistry in porous media. Our method is based on a fixed-point algorithm that enables us to coupled different transport and chemistry modules. We present the methods for solving the sub-problems, detail the formulation for the coupled problem and show numerical examples to validate the method. Résumé. Nous présentons une méthode numérique pour le couplage du transport et de la chimie en milieu poreux. Notre méthode utilise un algorithme de point fixe, qui nous permet de coupler des modules de transport et de chimie différents. Nous présentons les méthodes numériques utilisées pour chacun des sous-problèmes, ainsi qu'une formulation pour le problème couplé, et nous validons la méthode sur quelques exemples numériques. (10.1051/proc/2009049)
  DOI : 10.1051/proc/2009049
• Planning reinforcement on gas transportation networks with optimization methods
  
  • Bonnans Joseph Frederic
  • André Jean
  • Cornibert Laurent
  European Journal of Operational Research, Elsevier, 2009, 197 (3), pp.1019-1027.
• La méthode des éléments finis : de la théorie à la pratique. Tome 1 : Concepts généraux
  
  • Ciarlet Patrick
  • Lunéville Éric
  , 2009, pp.194. La méthode des éléments finis, apparue dans les années 50 pour traiter des problèmes de mécanique des structures, a connu depuis lors un développement continu et est utilisée, aujourd’hui, dans tous les domaines d’applications : mécanique, physique, chimie, économie, finance et biologie. Elle est maintenant utilisée dans la plupart des logiciels de calcul scientifique, et de nombreux ingénieurs y sont confrontés dans le cadre de leur activité de modélisation et de simulation numérique. Il est donc important d’en maîtriser les divers aspects. Cet ouvrage recouvre un cours enseigné à l'ENSTA depuis plusieurs années. On y présente tous les éléments essentiels de la méthode : les fondements théoriques (formulations variationnelles d’équations aux dérivées partielles, principes généraux et analyse numérique de la méthode), les considérations pratiques de mise en œuvre (structure creuse des matrices, principe d’assemblage), les algorithmes (en particulier ceux relatifs à la résolution des systèmes linéaires) et enfin des illustrations numériques.
• Approximations of Stochastic Optimization Problems Subject to Measurability Constraints
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2009, 19 (4), pp.1719-1734. Motivated by the numerical resolution of stochastic optimization problems subject to measurability constraints, we focus upon the issue of discretization. There exist indeed two components to be discretized for such problems, namely, the random variable modelling uncertainties (noise) and the $\sigma$-field modelling the knowledge (information) according to which decisions are taken. There is no reason to bind these two discretizations, which are a priori unrelated. In this setting, we present conditions under which the discretized problems converge to the original one. The focus is put on the convergence notions ensuring the quality of the approximation; we illustrate their importance by means of a counterexample based on the Monte Carlo approximation. Copyright © 2009 Society for Industrial and Applied Mathematics (10.1137/070692376)
  DOI : 10.1137/070692376
• The fictitious domain method and applications in wave propagation
  
  • Bécache Eliane
  • Rodríguez Garcia Jerónimo
  • Tsogka Chrysoula
  , 2009. This paper deals with the convergence analysis of the fictitious domain method used for taking into account the Neumann boundary condition on the surface of a crack (or more generally an object) in the context of acoustic and elastic wave propagation. For both types of waves we consider the first order in time formulation of the problem known as mixed velocity-pressure formulation for acoustics and velocity-stress formulation for elastodynamics. The convergence analysis for the discrete problem depends on the mixed finite elements used. We consider here two families of mixed finite elements that are compatible with mass lumping. When using the first one which is less expensive and corresponds to the choice made in a previous paper, it is shown that the fictitious domain method does not always converge. For the second one a theoretical convergence analysis was carried out in [7] for the acoustic case. Here we present numerical results that illustrate the convergence of the method both for acoustic and elastic waves.
• Some convergence results for Howard's algorithm
  
  • Bokanowski Olivier
  • Maroso Stefania
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (4), pp.3001--3026. This paper deals with convergence results of Howard's algorithm for the resolution of $\min_{a\in \cA} (B^a x - b^a)=0$ where $B^a$ is a matrix, $b^a$ is a vector (possibly of infinite dimension), and $\cA$ is a compact set. We show a global super-linear convergence result, under a monotonicity assumption on the matrices $B^a$. In the particular case of an obstacle problem of the form $\min(A x - b,\, x-g)=0$ where $A$ is an $N\times N$ matrix satisfying a monotonicity assumption, we show the convergence of Howard's algorithm in no more than $N$ iterations, instead of the usual $2^N$ bound. Still in the case of obstacle problem, we establish the equivalence between Howard's algorithm and a primal-dual active set algorithm (M. Hintermüller et al., {\em SIAM J. Optim.}, Vol 13, 2002, pp. 865-888). We also propose an Howard-type algorithm for a "double-obstacle" problem of the form $\max(\min(Ax-b,x-g),x-h)=0$. We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the double-obstacle problem. (10.1007/s00245-006-0865-2)
  DOI : 10.1007/s00245-006-0865-2
• Model reduction for a class of linear descriptor systems
  
  • Hechme Grace
  • Nechepurenko Yu.M.
  • Sadkane Miloud
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 229 (1), pp.54-60. For linear descriptor systems of the form Bẋ=Ax+Cu, this paper constructs reduced order systems associated with a given part of the finite spectrum of the pencil P(λ)=A−λB. It is known that the reduction can be obtained by a block diagonalization of the generalized Schur decomposition of P(λ). In this paper we consider the special case when B = [(H, 0; 0, 0)]and A = [(J, G; - F*, 0)]. This case is suited, in particular, for linearized hydrodynamic problems. We derive a sufficient condition under which the reduced system can approximate the initial one and show that it can be obtained in significantly cheap and efficient approaches. We consider first in detail the case when F = G and H is the identity matrix and then treat the general case. © 2008 Elsevier B.V. All rights reserved. (10.1016/j.cam.2008.10.001)
  DOI : 10.1016/j.cam.2008.10.001
• Numerical approximation for a superreplication problem under gamma constraints
  
  • Bruder Benjamin
  • Bokanowski Olivier
  • Maroso Stefania
  • Zidani Hasnaa
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2009, 47 (3), pp.2289-2320. We study a superreplication problem of European options with gamma constraints, in mathematical finance. The initially unbounded control problem is set back to a problem involving a viscosity PDE solution with a set of bounded controls. Then a numerical approach is introduced, inconditionnally stable with respect to the mesh steps. A generalized finite difference scheme is used since basic finite differences cannot work in our case. Numerical tests illustrate the validity of our approach. (10.1137/080725222)
  DOI : 10.1137/080725222
• Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain.
  
  • Duruflé Marc
  • Grob Pascal
  • Joly Patrick
  Numerical Methods for Partial Differential Equations, Wiley, 2009, 25 (3), pp.526-551. In this paper, we examine the infl uence of numerical integration on finite element methods using quadrilateral or hexahedral meshes in the time domain. We pay special attention to the use of Gauss-Lobatto points to perform mass lumping for any element order. We provide some theoretical results through several error estimates that are completed by various numerical experiments. (10.1002/num.20353)
  DOI : 10.1002/num.20353
• Fast and accurate computation of layer heat potentials
  
  • Li Jing-Rebecca
  • Greengard Leslie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2009. We discuss the numerical evaluation of single and double layer heat potentials in two dimensions on stationary and moving boundaries. One of the principal difficulties in designing high order methods concerns the local behavior of the heat kernel, which is both weakly singular in time and rapidly decaying in space. We show that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which we refer to as geometrically-induced stiffness, but that rules based on product integration of the full heat kernel in time are robust. When combined with previously developed fast algorithms for the evolution of the history part of layer potentials, diffusion processes in complex, moving geometries can be computed accurately and in nearly optimal time.
• Existence of solutions for a model describing the dynamics of junctions between dislocations
  
  • Forcadel Nicolas
  • Monneau Régis
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2009, 40 (6), pp.pp. 2517-2535. We study a dynamical version of a multi-phase field model of Koslowski and Ortiz for planar dislocation networks. We consider a two-dimensional vector field which describes phase transitions between constant phases. Each phase transition corresponds to a dislocation line, and the vectorial field description allows the formation of junctions between dislocations. This vector field is assumed to satisfy a non-local vectorial Hamilton-Jacobi equation with non-zero viscosity. For this model, we prove the existence for all time of a weak solution. (10.1137/070710925)
  DOI : 10.1137/070710925
• Revisiting the Analysis of Optimal Control Problems with Several State Constraints
  
  • Bonnans Joseph Frederic
  • Hermant Audrey
  Control and Cybernetics, Polish Academy of Sciences, 2009, 38 (4), pp.1021--1052.
• Space-time mesh refinement for discontunuous Galerkin methods for symmetric hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1886-1895. We present a new non-conforming space-time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method. (10.1016/j.cam.2009.08.094)
  DOI : 10.1016/j.cam.2009.08.094