Publications

2008

• Localization and characterization of simple defects in finite-size photonic crystals
  
  • Groby Jean-Philippe
  • Lesselier Dominique
  Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2008, 25 (1), pp.146-152. Structured materials like photonic crystals require for optimal use a high degree of precision with respect to both position and optical characteristics of their components. Here we present a simple tomographic algorithm, based on a specific Green's function together with a first-order Born approximation, which enables us to localize and characterize identical defects in finite-sized photonic crystals. This algorithm is proposed as a first step to the monitoring of such materials. Illustrative numerical results show in particular the possibility of focalization beyond the Rayleigh criterion. (10.1364/JOSAA.25.000146)
  DOI : 10.1364/JOSAA.25.000146
• 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.
• 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
• Infinitely many-armed bandits
  
  • Wang Yizao
  • Audibert Jean-Yves
  • Munos Rémi
  , 2008. We consider multi-armed bandit problems where the number of arms is larger than the possible number of experiments. We make a stochastic assumption on the mean-reward of a new selected arm which characterizes its probability of being a near-optimal arm. Our assumption is weaker than in previous works. We describe algorithms based on upper-confidence-bounds applied to a restricted set of randomly selected arms and provide upper-bounds on the resulting expected regret. We also derive a lower-bound which matches (up to a logarithmic factor) the upper-bound in some cases.
• From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution
  
  • Champagnat Nicolas
  • Ferrière Régis
  • Méléard Sylvie
  Stochastic Models, Taylor & Francis, 2008, 24 (S1), pp.2-44. We are interested in modelling Darwinian evolution, resulting from the interplay of phenotypic variation and natural selection through ecological interactions. Our models are rooted in the microscopic, stochastic description of a population of discrete individuals characterized by one or several adaptive traits. The population is modelled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. We look for tractable large population approximations. By combining various scalings on population size, birth and death rates, mutation rate, mutation step, or time, a single microscopic model is shown to lead to contrasting macroscopic limits, of different nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. In the limit of rare mutations, we show that a possible approximation is a jump process, justifying rigorously the so-called trait substitution sequence. We thus unify different points of view concerning mutation-selection evolutionary models. (10.1080/15326340802437710)
  DOI : 10.1080/15326340802437710
• 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
• 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.
• 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.
• 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
• 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.
• 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
• 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.