Publications

2008

• Probabilistic approach for granular media equations in the non uniformly convex case.
  
  • Cattiaux Patrick
  • Guillin Arnaud
  • Malrieu Florent
  Probability Theory and Related Fields, Springer Verlag, 2008, 140 (1-2), pp.19-40. We use here a particle system to prove a convergence result as well as a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. Proof is straightforward, simplifying deeply proofs of Carrillo-McCann-Villani \cite{CMV,CMV2} and completing results of Malrieu \cite{malrieu03} in the uniformly convex case. It relies on an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a $T_1$ transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.
• Singular arcs in the generalized Goddard's Problem
  
  • Bonnans J. Frederic
  • Martinon Pierre
  • Trélat Emmanuel
  Journal of Optimization Theory and Applications, Springer Verlag, 2008, 139 (2), pp.439--461. We investigate variants of Goddard's problems for nonvertical tra- jectories. The control is the thrust force, and the ob jective is to max imize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal tra jectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control.
• Seismic motion in urban sites consisting of blocks in welded contact with a soft layer overlying a hard half space: II. large and infinite number of identical equispaced blocks
  
  • Wirgin Armand
  • Groby Jean-Philippe
  Geophysical Journal International, Oxford University Press (OUP), 2008, 172, pp.725-758. We address the problem of the response to a seismic wave of an urban site consisting of a large and infinite number ($N_{b}$) of identical, equispaced blocks overlying a soft layer underlain by a hard substratum. The results of the theoretical analysis, appealing to a space-frequency mode-matching (MM) technique, are compared to those obtained by a space-time finite element (FE) method. The two methods are shown to give rise to much the same prediction of seismic response for $N_{b}=10$. The MM technique is also applied to the case $N_{b}=\infty$, notably to reveal the structure and natural frequencies of the vibration modes of the urban site. The mechanism of the interaction between blocks and the ground, as well as that of the collective effects of the blocks, are studied. It is shown that the presence of a large number of blocks modifies the seismic disturbance in a manner which evokes, and may partially account for, what was observed during many earthquakes in Mexico City. Disturbances at a much smaller level, induced by a small number of blocks are studied in the companion paper.
• Shoot-1.1 Package - User Guide
  
  • Martinon Pierre
  • Gergaud Joseph
  , 2008, pp.41. This package implements a shooting method for solving boundary value problems, for instance resulting of the application of Pontryagin's Minimum Principle to an optimal control problem. The software is mostly Fortran90, with some third party Fortran77 codes for the numerical integration and non-linear equations system. Its features include the handling of right hand side discontinuities (such as caused by a bang-bang control) for the integration of the trajectory and the computation of Jacobians for the shooting method. The particular case of singular arcs for optimal control problems is also addressed.
• Identification of generalized impedance boundary conditions in inverse scattering problems
  
  • Bourgeois Laurent
  • Haddar Houssem
  , 2008, pp.27. In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact set. We also extend local stability results to the case of back-scattering data.
• Numerical study of optimal trajectories with singular arcs for space launcher problems
  
  • Martinon Pierre
  • Bonnans Frédéric J.
  • Laurent-Varin Julien
  • Trélat Emmanuel
  , 2008, pp.26. The subject of this paper is the study of singular arcs (i.e. with a non maximal thrust) for a space launcher problem. We consider a flight to the GTO orbit for a heavy multi-stage launcher (Ariane 5 class), and use a realistic physical model for the drag force and rocket thrust. As a preliminary result, we first solve the complete flight with stage separations, at full thrust. Then we focus on the first atmospheric climbing phase, to investigate the possible existence of optimal trajectories with singular arcs. We primarily use an indirect shooting method (based on Pontryagin's Minimum Principle), coupled to a continuation (homotopy) approach. Some additional experiments are made with a basic direct method, and confirm the solutions obtained by the shooting. We study two slightly different launcher models, and observe that modifying parameters such as the aerodynamic reference area and specific impulsion can indeed lead to optimal trajectories with either full thrust or singular arcs.
• Discounted and finitely repeated minority games with public signals
  
  • Scarsini Marco
  • Scarlatti Sergio
  • Renault Jérôme
  Mathematical Social Sciences, Elsevier, 2008, 56 (1), pp.44-74. We consider a repeated game where at each stage players simultaneously choose one of two rooms. The players who choose the less crowded room are rewarded with one euro. The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced, hence the game has imperfect public monitoring. An undiscounted version of this game was considered by Renault et al. (2005), who proved a folk theorem. Here we consider a discounted version and a finitely repeated version of the game, and we strengthen our previous result by showing that the set of equilibrium payos Hausdor-converges to the feasible set as either the discount factor goes to one or the number of repetition goes to infinity. We show that the set of public equilibria for this game is strictly smaller than the set of private equilibria. (10.1016/j.mathsocsci.2007.12.004)
  DOI : 10.1016/j.mathsocsci.2007.12.004
• Adaptive Importance Sampling in General Mixture Classes
  
  • Cappé Olivier
  • Douc Randal
  • Guillin Arnaud
  • Marin Jean-Michel
  • Robert Christian P.
  Statistics and Computing, Springer Verlag (Germany), 2008, 18 (4), pp.447-459. In this paper, we propose an adaptive algorithm that iteratively updates both the weights and component parameters of a mixture importance sampling density so as to optimise the importance sampling performances, as measured by an entropy criterion. The method is shown to be applicable to a wide class of importance sampling densities, which includes in particular mixtures of multivariate Student t distributions. The performances of the proposed scheme are studied on both artificial and real examples, highlighting in particular the benefit of a novel Rao-Blackwellisation device which can be easily incorporated in the updating scheme. (10.1007/s11222-008-9059-x)
  DOI : 10.1007/s11222-008-9059-x
• High-order angles in almost-Riemannian geometry
  
  • Boscain Ugo
  • Sigalotti Mario
  Séminaire de Théorie Spectrale et Géométrie, Grenoble : Université de Grenoble 1, Institut Fourier, 1983-, 2008, 24, pp.41-54.
• Minimum stress optimal design with the level set method
  
  • Allaire Grégoire
  • Jouve François
  Engineering Analysis with Boundary Elements, Elsevier, 2008, 32, pp.909-918. This paper is devoted to minimum stress design instructural optimization. We propose a simple andefficient numerical algorithm for shape and topologyoptimization based on the level set method coupledwith the topological derivative. We compute ashape derivative, as well as a topological derivative,for a stress-based objective function. Using anadjoint equation we implement a gradient algorithmfor the minimization of the objective function.Several numerical examples in 2-d and 3-d are discussed.
• Fluid-Structure Interaction and multi-body contact. Application to the aortic valves
  
  • Astorino Matteo
  • Gerbeau Jean-Frédéric
  • Pantz Olivier
  • Traore Karim-Frédéric
  , 2008, pp.23. We present a partitioned procedure for fluid-structure interaction problems in which contacts among different deformable bodies can occur. A typical situation is the movement of a thin valve (e.g. the aortic valve) immersed in an incompressible viscous fluid (e.g. the blood). In the proposed strategy the fluid and structure solvers are considered as independent ``black-boxes'' that exchange forces and displacements; the structure solvers are moreover not supposed to manage contact by themselves. The hypothesis of non-penetration among solid objects defines a non-convex optimization problem. To solve the latter, we use an internal approximation algorithm that is able to directly handle the cases of thin structures and self-contacts. A numerical simulation on an idealized aortic valve is finally realized with the aim of illustrating the proposed scheme.
• Analyse théorique et numérique du modèle de Webster Lokshin
  
  • Haddar Houssem
  • Matignon Denis
  , 2008. Les ondes acoustiques qui se propagent dans un pavillon dont la paroi est le siège de pertes visco-thermiques et dont les deux extremités sont sujettes à des conditions de rayonnement obéissent à un modèle de Webster-Lokshin, lequel fait intervenir des dérivées fractionnaires en temps dans le milieu et des conditions aux limites dynamiques. Ce système peut s'interpréter comme le couplage de trois sous-systèmes : une équation des ondes, une réalisation diffusive de l'opérateur pseudo-différentiel en temps, et une réalisation dissipative de l'impédance par le lemme de Kalman-Yakubovich-Popov. En utilisant le théorème de Hille-Yosida, l'existence et l'unicité des solutions fortes de ce système sont établies. De plus, des schémas numériques sont proposés et leur stabilité est analysée en utilisant des techniques d'énergie ; de nombreuses simulations numériques viennent illustrer le comportement du modèle pour diverses valeurs des paramètres.
• Particle filter-based policy gradient for pomdps
  
  • Coquelin Pierre-Arnaud
  • Deguest Romain
  • Munos Rémi
  , 2008. Our setting is a Partially Observable Markov Decision Process with continuous state, observation and action spaces. Decisions are based on a Particle Filter for estimating the belief state given past observations. We consider a policy gradient approach for parameterized policy optimization. For that purpose, we investigate sensitivity analysis of the performance measure with respect to the parameters of the policy, focusing on Finite Difference (FD) techniques. We show that the naive FD is subject to variance explosion because of the non-smoothness of the resampling procedure. We propose a more sophisticated FD method which overcomes this problem and establish its consistency.
• Generalized Fast Marching Method: Applications to Image Segmentation
  
  • Forcadel Nicolas
  • Le Guyader Carole
  • Gout Christian
  Numerical Algorithms, Springer Verlag, 2008, 48 (1-3), pp.189-211. In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan-Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266-277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm. (10.1007/s11075-008-9183-x)
  DOI : 10.1007/s11075-008-9183-x
• Implicit time discretization of the mean curvature flow with a discontinuous forcing term.
  
  • Chambolle Antonin
  • Novaga Matteo
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2008, 10 (3), pp.283--300. We consider an implicit time discretization for the motion of a hypersurface driven by its anisotropic mean curvature. We prove some convergence results of the scheme under very general assumptions on the forcing term, which include in particular the case of a typical path of the Brownian motion. We compare this limit with other available solutions, whenever they are defined. As a by-product of the analysis, we also provide a simple proof of the coincidence of the limit flow with the regular evolutions, defined for small times, in the case of a regular forcing term.
• Numerical Linear Algebra
  
  • Allaire Grégoire
  • Kaber Sidi-Mahmoud
  , 2008, 55.
• On the existence of transmission eigenvalues in an inhomogeneous medium
  
  • Cakoni Fioralba
  • Haddar Houssem
  , 2008, pp.24. We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both scalar Helmholtz equation and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that this contrast is large enough.