Publications

2008

• The value of Repeated Games with an informed controller
  
  • Renault Jérôme
  , 2008. We consider the general model of zero-sum repeated games (or stochastic games with signals), and assume that one of the players is fully informed and controls the transitions of the state variable. We prove the existence of the uniform value, generalizing several results of the literature. A preliminary existence result is obtained for a certain class of stochastic games played with pure strategies.
• Localisation et caractérisation de défauts simples dans un cristal photonique 2D de dimension finie
  
  • Groby Jean-Philippe
  • Lesselier Dominique
  , 2008.
• Uniform value in Dynamic Programming
  
  • Renault Jérôme
  , 2008. We consider dynamic programming problems with a large time horizon, and give sufficient conditions for the existence of the uniform value. As a consequence, we obtain an existence result when the state space is precompact, payoffs are uniformly continuous and the transition correspondence is non expansive. In the same spirit, we give an existence result for the limit value. We also apply our results to Markov decision processes and obtain a few generalizations of existing results.
• Placement 'harmonieux' de 26 points sur une sphère par recuit simulé
  
  • Colonna Jean-François
  , 2008. Harmonious' arrangement of 26 points on a sphere by means of simulated annealing (Placement 'harmonieux' de 26 points sur une sphère par recuit simulé)
• Placement 'harmonieux' de 26 points sur une sphere par recuit simule
  
  • Colonna Jean-François
  , 2008. Harmonious' arrangement of 26 points on a sphere by means of simulated annealing (Placement 'harmonieux' de 26 points sur une sphere par recuit simule)
• Multi-static response of spherical scatterers and the back-propagation of singular fields
  
  • Iakovleva Ekaterina
  • Lesselier Dominique
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2008, 56 (3), pp.825-833. In view of the development of multiple signal clas- sification-type, non-iterative imaging procedures based on the singular value decomposition of the multistatic response (MSR) matrix of a collection of inclusions, the full 3-D electromagnetic case with arbitrary contrasts of permeability and permittivity (including perfectly electric conducting or perfectly magnetic con- ducting limit cases) is studied. The structure of the MSR matrix of a single inclusion (or a set of well-separated ones) is analyzed. Emphasis is on a far-field situation, in which one considers an electric dipole array operated in the transmit/receive mode at a single frequency. Back-propagated electric and magnetic fields associated to the singular vectors of the MSR matrix for a single spherical inclusion (or again for well-separated ones) are given in closed form and their leading-order values are exhibited. Numerical illustrations (cross-sectional maps of back-propagated fields computed from a singular value decomposition of the MSR matrix) are presented, for one and two inclusions, to illustrate this behavior as a function of the geometric and electromagnetic parameters of the configuration in a possibly noisy case. (10.1109/TAP.2008.916913)
  DOI : 10.1109/TAP.2008.916913
• Texture tridimensionnelle
  
  • Colonna Jean-François
  , 2008. texture tridimensionnelle obtenue dans le corps des quaternions avec une arithmétique "non standard
• Proof(s) of the Lamperti representation of Continuous-State Branching Processes
  
  • Caballero Maria-Emilia
  • Lambert Amaury
  • Uribe Bravo Geronimo
  , 2008. The representation of continuous-state branching processes (CSBPs) as time-changed Lévy processes with no negative jumps was discovered by John Lamperti in 1967 but was never proved. The goal of this paper is to provide a proof, and we actually provide two. The first one relies on studying the time-change, using martingales and the Lévy-Itô representation of Lévy processes. It gives insight into a stochastic differential equation satisfied by CSBPs and on its relevance to the branching property. The other method studies the time-change in a discrete model, where an analogous Lamperti representation is evident, and provides functional approximations to Lamperti transforms by introducing a new topology on Skorohod space. Some classical arguments used to study CSBPs are reconsidered and simplified.
• Vibrations 0003
  
  • Colonna Jean-François
  , 2008. Vibrations 0003 (Vibrations 0003)
• Sequential Monte Carlo smoothing with application to parameter estimation in non-linear state space models
  
  • Olsson Jimmy
  • Cappé Olivier
  • Douc Randal
  • Moulines Eric
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2008, 14 (1), pp.155-179. This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algorithm relies on forgetting properties of the filtering dynamics and the quality of the estimates produced is investigated, both theoretically and via simulations. (10.3150/07-BEJ6150)
  DOI : 10.3150/07-BEJ6150
• Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations
  
  • Haddar Houssem
  • Joly Patrick
  • Nguyen Hoai-Minh
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2008, 18 (10), pp.1787-1827. (10.1142/S0218202508003194)
  DOI : 10.1142/S0218202508003194
• Shape and topology optimization of the robust compliance via the level set method
  
  • Allaire Grégoire
  • de Gournay Frédéric
  • Jouve François
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2008, 14 (14), pp.43--70. The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it. Then, in the framework of Hadamard method, we compute the directional shape derivative of this criterion which is used in a numerical algorithm, based on the level set method, to find optimal shapes that minimize the worst-case compliance. Since this criterion is usually merely directionally differentiable, we introduce a semidefinite programming approach to select the best descent direction at each step of a gradient method. Numerical examples are given in 2-d and 3-d. Mathematics Subject Classification. 49Q10, 74P10, 74P15, 74P20. (10.1051/cocv:2007048)
  DOI : 10.1051/cocv:2007048
• Global weak solutions to the equations of motion for magnetic fluids
  
  • Amirat Youcef
  • Hamdache Kamel
  • Murat François
  Journal of Mathematical Fluid Mechanics, Springer Verlag, 2008, 10 (8), pp.326-351.
• 3D Electrostatic Hybrid Elements Model for SAW transducers modeling
  
  • Jerez-Hanckes Carlos F.
  • Laude Vincent
  • Lardat Raphael
  • Nédélec Jean-Claude
  IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Institute of Electrical and Electronics Engineers, 2008, 55 (3), pp.686-695. In this work, the singular behavior of charges at corners and edges on the metallized areas in SAW transducers are investigated. In particular, it is demonstrated that a tensor product of the commonly used Tchebychev bases overestimates the singularities at corners, and, hence, it cannot be used in a proper boundary element method formulation. On the other hand, it is shown that a simple finite element method-like approach is impractical due to the enormous number of unknowns required to model the electrode's large length-to-width ratio. These considerations are then used for defining a hybrid element model, which combines Tchebychev and linear polynomials over differently meshed domains. Such an approach is shown to suitably account for charge singularities while greatly reducing the number of unknowns. Results are obtained for isotropic and anisotropic substrates for non-periodic configurations.
• Asymptotics for steady state voltage potentials in a bidimensional highly contrasted medium with thin layer
  
  • Poignard Clair
  Mathematical Methods in the Applied Sciences, Wiley, 2008, 31 (4), pp.443-479. We study the behavior of steady state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviors of the potentials. (10.1002/mma.923)
  DOI : 10.1002/mma.923
• Sensitivity Analysis in Particle Filters. Application to Policy Optimization in 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.
• A Post-Treatment of the Homogenization Method for Shape Optimization
  
  • Pantz O.
  • Trabelsi K.
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2008, 47 (3), pp.1380-1398. We propose an alternative to the classical post-treatment of the homogenization method for shape optimization. Rather than penalize the material density once the optimal composite shape is obtained (by the homogenization method) in order to produce a workable shape close to the optimal one, we macroscopically project the microstructure of the former through an appropriate procedure that roughly consists in laying the material along the directions of lamination of the composite. We have tested our approach in the framework of compliance minimization in two-dimensional elasticity. Numerical results are provided. (10.1137/070688900)
  DOI : 10.1137/070688900
• Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems
  
  • Benner Peter
  • Li Jing-Rebecca
  • Penzl Thilo
  Numerical Linear Algebra with Applications, Wiley, 2008, 15 (9), pp.755-777. We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. (10.1002/nla.622)
  DOI : 10.1002/nla.622
• 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.
• Playing off-line games with bounded rationality
  
  • Scarsini Marco
  • Tomala Tristan
  • Renault Jérôme
  Mathematical Social Sciences, Elsevier, 2008, 56 (2), pp.2078-223. We study a two-person zero-sum game where players simultaneously choose sequences of actions, and the overall payoff is the average of a one-shot payoff over the joint sequence. We consider the maxmin value of the game played in pure strategies by boundedly rational players and model bounded rationality by introducing complexity limitations. First we define the complexity of a sequence by its smallest period (a non-periodic sequence being of infinite complexity) and study the maxmin of the game where player~1 is restricted to strategies with complexity at most $n$ and player~2 is restricted to strategies with complexity at most $m$. We study the asymptotics of this value and a complete characterization in the matching pennies case. We extend the analysis of matching pennies to strategies with bounded recall. (10.1016/j.mathsocsci.2008.01.005)
  DOI : 10.1016/j.mathsocsci.2008.01.005
• Continuous cascade models for asset returns
  
  • Bacry Emmanuel
  • Kozhemyak Alexey
  • Muzy J.-F.
  Journal of Economic Dynamics and Control, Elsevier, 2008, pp.156-199. In this paper, we make a short overview of continuous cascade models recently introduced to model asset return fluctuations. We show that these models account in a very parcimonious manner for most of 'stylized facts' of financial time-series. We review in more details the simplest continuous cascade namely the log-normal multifractal random walk (MRW). It can simply be considered as a stochastic volatility model where the (log-) volatility memory has a peculiar 'logarithmic' shape. This model possesses some appealing stability properties with respect to time aggregation. We describe how one can estimate it using a GMM method and we present some applications to volatility and (VaR) Value at Risk forecasting. (10.1016/j.jedc.2007.01.024)
  DOI : 10.1016/j.jedc.2007.01.024
• Non-iterative MUSIC-type algorithm for reconstructing two-dimensional thin dielectric inclusions
  
  • Park Won-Kwang
  • Ammari Habib
  • Lesselier Dominique
  , 2008, 135 (2), pp.297-305. We consider a non-iterative MUSIC-type imaging algorithm for reconstructing thin, curve-like penetrable inclusions in a two-dimensional homogeneous space. It is based on an appropriate asymptotic formula of the scattering amplitude. Operating at fixed nonzero frequency, it yields the shape of the inclusion from scattered fields in addition to estimates of the length of the supporting curve. Numerical implementation shows that it is a fast and efficient algorithm.