Publications

2009

• Asymptotic analysis of an optimal control problem connected to the human locomotion
  
  • Bayen Terence
  • Chitour Yacine
  • Jean Frédéric
  • Mason Paolo
  , 2009, pp.2248-2253. The article is devoted to the analysis of two optimal control problems. We first consider a model proposed by Arechavaleta et al. (see [3]) describing the goal-oriented locomotion, for which the control on the derivative of the curvature script K̇ along the trajectory is supposed bounded. Necessary conditions on optimal trajectories are given. We then investigate an extension of this model obtained by removing the boundedness assumption on script K. In this framework several properties of the optimal trajectories are detected and in particular we determine an asymptotic behavior of the initial value of the associated covector with respect to the final point. (10.1109/CDC.2009.5400873)
  DOI : 10.1109/CDC.2009.5400873
• A global steering method for general dynamical nonholonomic systems
  
  • Chitour Yacine
  • Jean Frédéric
  • Long Ruixing
  , 2009, pp.27-32. In this paper, we extend the globally convergent steering algorithm for regular nonholonomic systems presented in [7] to a much larger class of systems which contain singularities. This extension is based on the construction of a continuous first order approximation of the control system. We also propose an exact motion planning method for nilpotent systems. The method makes use of sinusoidal control laws and generalizes the algorithm presented in [12] for steering chained-form systems. It gives rise to C1 trajectories, then makes easy dynamical extension. ©2009 IEEE. (10.1109/CDC.2009.5400197)
  DOI : 10.1109/CDC.2009.5400197
• Couplage des méthodes modale et éléments finis pour la diffraction des ondes élastiques guidées : Application au Contrôle Non Destructif
  
  • Baronian Vahan
  , 2009. En vue de simuler une expérience de contrôle non destructif par ondes ultrasonores guidées, on considère un guide élastique 2D (une plaque) ou 3D (une barre) qui comporte un défaut (fissure, hétérogénéité locale due à une soudure etc...). L'objectif est de résoudre numériquement le problème de la diffraction d'un mode du guide par le défaut. Nous nous sommes attachés à mettre au point une méthode couplant des éléments finis dans une portion (aussi petite que possible) du guide, contenant le défaut, avec des décompositions modales de part et d'autre du défaut. La difficulté consiste à écrire la bonne condition de raccord entre ces deux représentations. Le point important est d'avoir à sa disposition une relation d'orthogonalité permettant de projeter la solution éléments finis sur les modes. Ceci conduit à formuler le problème à l'aide de vecteurs hybrides déplacement/contrainte pour lesquels il existe une relation de bi-orthogonalité : la relation dite de Fraser. On peut alors écrire une condition exacte (ou transparente) à la troncature modale près, sur les frontières artificielles du domaine de calcul. Il faut enfin intégrer cette condition aux limites dans une approche variationnelle (en déplacements) en vue de développer une méthode d'éléments finis. Du fait du caractère hybride de la condition, on doit pour cela introduire comme inconnue supplémentaire la composante normale de la contrainte normale définie sur la frontière artificielle et écrire une formulation mixte. Nous avons traité numériquement les cas bidimensionnel et tridimensionnel d'un guide isotrope à bords libres. Les modes du guide sont calculés numériquement par une approche originale utilisant à nouveau les vecteurs hybrides déplacement/contrainte, qui permet de conserver au niveau discret la relation de biorthogonalité. Le code développé permet de calculer très rapidement la "matrice de scattering
• Graph coloring with cardinality constraints on the neighborhoods
  
  • Costa Marie-Christine
  • de Werra Dominique
  • Picouleau Christophe
  • Ries Bernard
  Discrete Optimization, Elsevier, 2009, 6 (4), pp.362--369. Extensions and variations of the basic problem of graph coloring are introduced. It consists essentially in finding in a graph G a k-coloring, i.e., a partition V 1, ..., V k of the vertex set of G such that for some specified neighborhood ˜N (v) of each vertex v, the number of vertices in ˜N (v) \ V i is (at most) a given integer hi v. The complexity of some variations is discussed according to ˜N (v) which may be the usual neighbors, or the vertices at distance at most 2 or the closed neighborhood of v (v and its neighbors). Polynomially solvable cases are exhibited (in particular when G is a special tree). (10.1016/j.disopt.2009.04.005)
  DOI : 10.1016/j.disopt.2009.04.005
• Characterisation of the value function of final state constrained control problems with BV trajectories
  
  • Briani Ariela
  • Zidani Hasnaa
  , 2009. This paper aims to investigate a control problem governed by differential equations with Random measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo), we obtain that the value function can be characterized by means of an auxiliary control problem involving absolutely continuous trajectories. We study the characterization of the value function of this auxiliary problem and discuss its discrete approximations.
• Schémas numériques pour la résolution de l’équation des ondes
  
  • Bécache Eliane
  , 2009. L’objectif de ce cours est d’appréhender les problèmes de propagation d’ondes, de les étudier sur le plan mathématique et de proposer et d’analyser des méthodes numériques pour les résoudre. Cette partie du cours est plus spécifiquement consacrée aux méthodes numériques et nous renvoyons au cours de P. Joly pour ce qui concerne l’étude mathématique des équations continues.
• Optimized higher order time discretization of second order hyperbolic problems: Construction and numerical study
  
  • Joly Patrick
  • Rodríguez Jerónimo
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1953-1961. We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail. The optimal schemes are validated through various numerical results. (10.1016/j.cam.2009.08.046)
  DOI : 10.1016/j.cam.2009.08.046
• Improved Successive Constraint Method Based A Posteriori Error Estimate for Reduced Basis Approximation of 2D Maxwells Problem
  
  • Chen Yanlai
  • Hesthaven Jan Sickmann
  • Maday Yvon
  • Rodríguez Jerónimo
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2009, 43 (6), pp.1099--1116. In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup stability constants is essential. In [7], the authors presented an efficient method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient due to a nice property, namely, that the computed lower bound is monotonically increasing with respect to the size of the nested sets. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.
• On the theoretical justification of Pocklington's equation
  
  • Claeys Xavier
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (8), pp.1325-1355. Pocklington's model consists in a one-dimensional integral equation relating the current at the surface of a straight finite wire to the tangential trace of an incident electromagnetic field. It is a simplification of the more usual single layer potential equation posed on a two-dimensional surface. We are interested in estimating the error between the solution of the exact integral equation and the solution of Pocklington's model. We address this problem for the model case of acoustics in a smooth geometry using results of asymptotic analysis. (10.1142/S0218202509003802)
  DOI : 10.1142/S0218202509003802
• Energy preserving scheme for non linear systems of wave equations. Application to piano strings.
  
  • Chabassier Juliette
  • Joly Patrick
  , 2009, pp.00. The linear wave equation does not describe the com- plexity of the piano strings vibration enough for physics based sound synthesis. The nonlinear cou- pling between transversal and longitudinal modes has to be taken into account, as does the "geometrically exact" model. This system of equations can be clas- sified among a general energy preserving class of sys- tems. We present an implicit, centered, second order accurate, numerical scheme that preserves a discrete energy, leading to unconditional stability of the nu- merical scheme. The complete model takes into ac- count the bridge coupling the strings, and the ham- mer non linear attack on the strings.
• Non Spurious Mixed Spectral Element Methods for Maxwell's Equations
  
  • Cohen Gary
  • Sinding Alexandre
  , 2009. In this paper, we describe a new continuous approximation of Maxwell’s equations well-suited to mass-lumping and which ensures low storage. Then, we introduce a dissipative jump derived from Discontinuous Galerkin Methods (DGM) to get rid of spurious waves for both edge and continuous elements. This new approach leads to efficient spectral elements for Maxwell’s equations which are cheaper than DGM. On the other hand, this approach provides a good approximation of singularities generated by reentrant corners.
• Spectral controllability for 2D and 3D linear Schrödinger equations
  
  • Beauchard Karine
  • Chitour Yacine
  • Kateb Djalil
  • Long Ruixing
  Journal of Functional Analysis, Elsevier, 2009, 256 (12), pp.3916-3976. We consider a quantum particle in an infinite square potential well of RnRn, n=2,3n=2,3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: (Kal ) if Ω is the bottom of the well, then for every eigenvalue λ of View the MathML source−ΔΩD, the projections of the dipolar moment onto every (normalized) eigenvector associated to λ are linearly independent in RnRn. In 3D, our main result states that spectral controllability in finite time never holds for one-directional dipolar moment. The proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions. In 2D, we first prove the existence of a minimal time Tmin(Ω)>0Tmin(Ω)>0 for spectral controllability, i.e., if T>Tmin(Ω)T>Tmin(Ω), one has spectral controllability in time T if condition (Kal ) holds true for (Ω ) and, if TTmin(Ω) holds generically with respect to the domain. The proof relies on shape differentiation and a careful study of Dirichlet-to-Neumann operators associated to certain Helmholtz equations. We also show that one can recover exact controllability in abstract spaces from this 2D spectral controllability, by adapting a classical variational argument from control theory. (10.1016/j.jfa.2009.02.009)
  DOI : 10.1016/j.jfa.2009.02.009
• Analyse mathématique et numérique de problèmes de propagation des ondes dans des milieux périodiques infinis localement perturbés
  
  • Fliss Sonia
  , 2009. Les milieux périodiques présentent des propriétés intéressantes dans un grand nombre d'applications (les cristaux photoniques en optique, les matériaux composites en mécanique,...). Dans ces applications, on rencontre souvent ces milieux présentant des défauts localisés, c'est-à-dire des milieux qui diffèrent de milieux périodiques dans des régions bornées. Il nous semble intéressant de proposer des méthodes mathématiques et numériques nouvelles spécifiques au traitement des structures périodiques de grande taille, pouvant présenter des défauts localisés. Les caractéristiques du problème rendant très souvent les méthodes d'homogénéisation inapplicables, l'idée est d'exploiter la structure particulière des milieux périodiques pour restreindre les calculs au voisinage du défaut. Nous avons donc approfondi la question de trouver des conditions aux bords parfaitement transparentes. C'est pourquoi nous avons cherché à généraliser les techniques de conditions transparentes non locales, de type Neumann-to-Dirichlet, bien établies pour les milieux homogènes à l'extérieur de la perturbation. La difficulté est que lorsque le milieu extérieur est homogène, on ne dispose plus d'une représentation explicite de la solution. Nous traitons successivement trois situations de difficulté croissante : le cas mono-dimensionnel qui est un cas classique mais dont l'étude a des vertus pédagogiques, le problème du guide périodique localement perturbé et le problème plus complexe du milieu périodique dans les deux dimensions. Pour chaque situation, la démarche est la même : elle consiste tout d'abord à résoudre le problème pour un milieu absorbant puis pour un milieu non absorbant par absorption limite. Nous pouvons alors montrer que les opérateurs DtN peuvent être caractérisés en utilisant la solution de problèmes de cellule locaux, l'utilisation d'outils mathématiques tels que la Transformée de Floquet-Bloch et la solution d'équations quadratiques et linéaires à valeurs et inconnus opérateurs.
• Time-homogenization of a first order system arising in the modelling of the dynamics of dislocation densities
  
  • Briani Ariela
  • Monneau Régis
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2009, 347, pp.231-236. In this Note we are interested in the dynamics of dislocation densities in a material submitted to a time periodic stress. The dislocation densities solve a set of two coupled first order equations of Burgers' type. Our main aim is to give a description of the long time behaviour of those densities. By an homogenization procedure in the framework of viscosity solutions, we obtain that at the limit, the dislocation densities fulfills a single diffusion equation. (10.1016/j.crma.2009.01.006)
  DOI : 10.1016/j.crma.2009.01.006
• Computations of lossy bloch waves in two-dimensional photonic crystals
  
  • Engström Christian
  • Hafner Christian
  • Schmidt Kersten
  Journal of computational and theoretical nanoscience, American Scientific Publishers, 2009, 6, pp.775-783. In this article we compute lossy Bloch waves in two-dimensional photonic crystals with dispersion and material loss. For given frequencies these waves are determined from non-linear eigenvalue problems in the wave vector. We applied two numerical methods to a demanding test case, a photonic crystal with embedded quantum dots that exhibits very strong and anamolous dispersion. The first method is based on the formulation with periodic boundary conditions leading to a quadratic eigenvalue problem. We discretize this problem by the finite element method (FEM), first of quadratic order and, second, of higher orders using curved cells (p-FEM). Second, we use the multiple-multipole method (MMP) with artificial sources and compute extrema in the field response determining the eigenvalues. Both MMP and FEM provide robust solutions for the investigated dispersive and lossy photonic crystal, and can approximate the Bloch waves to a high accuracy. Moreover, the MMP method and p-FEM show low computational effort for very accurate solutions.
• Computation of the band structure of two-dimensional photonic crystals with hp finite elements
  
  • Schmidt Kersten
  • Kauf Peter
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2009, 198, pp.1249-1259. The band structure of 2D photonic crystals -- a periodic material with discontinuous dielectrical properties -- and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using {\em p}-FEM for smooth and {\em hp}-FEM for polygonal interfaces and boundaries. In this article we discretise the variational eigenvalue problems for photonic crystals with smooth and polygonal interfaces in scalar variables with quasi-periodic boundary conditions by means of {\em p}- and {\em hp}-FEM -- this for the transverse electric (TE) and transverse magnetic (TM) modes. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.
• Advances on 3D geoelectric forward solver techniques
  
  • Blome Mark
  • Maurer Hansruedi
  • Schmidt Kersten
  Geophysical Journal International, Oxford University Press (OUP), 2009, 176, pp.740-752. Modern geoelectrical data acquisition systems allow large amounts of data to be collected in a short time. Inversions of such data sets require powerful forward solvers for predicting the electrical potentials. State-of-the-art solvers are typically based on finite elements. Recent developments in numerical mathematics led to direct matrix solvers that allow the equation systems arising from such finite element problems to be solved very efficiently. They are particularly useful for 3D geoelectrical problems, where many electrodes are involved. Although modern direct matrix solvers include optimized memory saving strategies, their application to realistic, large-scale 3D problems is still somewhat limited. Therefore, we present two novel techniques that allow the number of grid points to be reduced considerably, while maintaining a high solution accuracy. In the areas surrounding an electrode array we attach infinite elements that continue the electrical potentials to infinity. This does not only reduces the number of grid points, but also avoid the artificial Dirichlet or mixed boundary conditions that are well known to be the cause of numerical inaccuracies. Our second development concerns the singularity removal in the presence of significant surface topography. We employ a fast multipole boundary element method for computing the singular potentials. This renders unnecessary mesh refinements near the electrodes, which results in substantial savings of grid points of up to more than 50%. By means of extensive numerical tests we demonstrate that combined application of infinite elements and singularity removal allows the number of grid points to be reduced by a factor of $\approx$ 6 -- 10 compared with traditional finite element methods. This will be key for applying finite elements and direct matrix solver techniques to realistic 3D inversion problems.
• Decomposition of large-scale stochastic optimal control problems
  
  • Barty Kengy
  • Carpentier Pierre
  • Girardeau Pierre
  , 2009. In this paper, we present an Uzawa-based heuristic that is adapted to some type of stochastic optimal control problems. More precisely, we consider dynamical systems that can be divided into small-scale independent subsystems, though 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.
• Dynamique gravitationnelle multi-échelle : formation et évolution des systèmes auto-gravitants non isolés
  
  • Kielbasiewicz Nicolas
  , 2009. L'objet de cette thèse est l'étude de la formation et des propriétés des systèmes auto-gravitants isolés et non isolés à l'aide de simulations numériques d'effondrements gravitationnels de systèmes à N corps. Dans une première partie, nous synthétisons les principaux résultats analytiques concernant le système d'équations couplées Boltzmann sans collisions - Poisson, qui modélise les systèmes auto-gravitants non collisionnels, et leur extension en présence d'un champ extérieur, ainsi que certaines solutions analytiques. Dans une seconde partie, nous présentons les codes que nous avons utilisés et nous introduisons des notions de calcul parallèle et réparti, certains éléments ayant été parallélisés à l'aide de la bibliothèque d'échange de messages M.P.I. Enfin, dans une dernière partie, nous exposons les différents résultats de nos simulations et leurs analyses. La première partie d'entre eux concerne l'approfondissement de l'étude des conditions initiales nécessaires au déclenchement des instabilités d'Antonov, ainsi que l'étude de la dynamique interne des systèmes auto-gravitants isolés. La seconde partie concerne l'étude de ces systèmes mis en orbites dans un potentiel extérieur, visant à étudier son influence sur le processus de formation des systèmes auto-gravitants par effondrement gravitationnels.
• 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