Publications

2006

• Visualisation tridimensionnelle de la superposition linéaire de 6 états propres de l'atome d'Hydrogène (calcul tridimensionnel)
  
  • Colonna Jean-François
  , 2006. Tridimensional display of a linear superposition of 6 eigenstates of the Hydrogen atom (tridimensional computation) (Visualisation tridimensionnelle de la superposition linéaire de 6 états propres de l'atome d'Hydrogène (calcul tridimensionnel))
• Visualisation tridimensionnelle de la superposition linéaire de 6 états propres de l'atome d'Hydrogène (calcul tridimensionnel)
  
  • Colonna Jean-François
  , 2006. Tridimensional display of a linear superposition of 6 eigenstates of the Hydrogen atom (tridimensional computation) (Visualisation tridimensionnelle de la superposition linéaire de 6 états propres de l'atome d'Hydrogène (calcul tridimensionnel))
• Sequential control variates for functionals of Markov processes
  
  • Gobet Emmanuel
  • Maire Sylvain
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2006, 43 (3), pp.1256-1275. Using a sequential control variates algorithm, we compute Monte Carlo approximations of solutions of linear partial differential equations connected to linear Markov processes by the Feynman--Kac formula. It includes diffusion processes with or without absorbing/reflecting boundary and jump processes. We prove that the bias and the variance decrease geometrically with the number of steps of our algorithm. Numerical examples show the efficiency of the method on elliptic and parabolic problems. (10.1137/040609124)
  DOI : 10.1137/040609124
• Probability of fixation under weak selection: a branching process unifying approach.
  
  • Lambert Amaury
  Theoretical Population Biology, Elsevier, 2006, 69 (4), pp.419-41. We link two-allele population models by Haldane and Fisher with Kimura's diffusion approximations of the Wright-Fisher model, by considering continuous-state branching (CB) processes which are either independent (model I) or conditioned to have constant sum (model II). Recent works by the author allow us to further include logistic density-dependence (model III), which is ubiquitous in ecology. In all models, each allele (mutant or resident) is then characterized by a triple demographic trait: intrinsic growth rate r, reproduction variance sigma and competition sensitivity c. Generally, the fixation probability u of the mutant depends on its initial proportion p, the total initial population size z, and the six demographic traits. Under weak selection, we can linearize u in all models thanks to the same master formula u = p + p(1 - p)[g(r)s(r) + g(sigma)s(sigma) + g(c)s(c)] + o(s(r),s(sigma),s(c), where s(r) = r' - r, s(sigma) = sigma-sigma' and s(c) = c - c' are selection coefficients, and g(r), g(sigma), g(c) are invasibility coefficients (' refers to the mutant traits), which are positive and do not depend on p. In particular, increased reproduction variance is always deleterious. We prove that in all three models g(sigma) = 1/sigma and g(r) = z/sigma for small initial population sizes z. In model II, g(r) = z/sigma for all z, and we display invasion isoclines of the 'mean vs variance' type. A slight departure from the isocline is shown to be more beneficial to alleles with low sigma than with high r. In model III, g(c) increases with z like ln(z)/c, and g(r)(z) converges to a finite limit L > K/sigma, where K = r/c is the carrying capacity. For r > 0 the growth invasibility is above z/sigma when z < K, and below z/sigma when z > K, showing that classical models I and II underestimate the fixation probabilities in growing populations, and overestimate them in declining populations. (10.1016/j.tpb.2006.01.002)
  DOI : 10.1016/j.tpb.2006.01.002
• Texture géométrique bidimensionnelle
  
  • Colonna Jean-François
  , 2006. texture géométrique bidimensionnelle
• 3-D electromagnetics, asymptotic models and MUSIC-type imaging of a collection of small scatterers
  
  • Iakovleva Ekaterina
  • Lesselier Dominique
  • Perrusson Gaële
  • Ammari Habib
  , 2006, pp.13-18.
• Texture fractale
  
  • Colonna Jean-François
  , 2006. Texture fractale
• La guerre
  
  • Colonna Jean-François
  , 2006. The war (La guerre)
• A class of financial products and models where super-replication prices are explicit
  
  • Carassus Laurence
  • Gobet Emmanuel
  • Temam Emmanuel
  , 2006, pp.67-84. We consider a multidimensional financial model with mild conditions on the underlying asset price process. The trading is only allowed at some fixed discrete times and the strategy is constrained to lie in a closed convex cone. We show how the minimal cost of a super hedging strategy can be easily computed by a backward recursive scheme. As an application, when the underlying follows a stochastic differential equation including stochastic volatility or Poisson jumps, we compute those super-replication prices for a range of European and American style options, including Asian, Lookback or Barrier Options. We also perform some multidimensional computations. (10.1142/9789812770448_0004)
  DOI : 10.1142/9789812770448_0004
• Improved interface condition for 2D domain decomposition with corner : a theoretical determination.
  
  • Chniti Chokri
  • Nataf Frédéric
  • Nier Francis
  , 2006. This article deals with a local improvement of domain decomposition methods for 2-dimensional elliptic problems for which either the geometry or the domain decomposition presents conical singularities. The problems amounts to determining the coefficients of some interface boundary conditions so that the domain decomposition algorithm converges rapidly. Specific problems occur in the presence of conical singularities. Starting from the method used for regular interfaces, we derive a local improvement by matching the singularities, that is the first terms of the asymptotic expansion around the corner, provided by Kondratiev theory. This theoretical approach leads to the explicit computation of some coefficients in the interface boundary conditions, to be tested numerically. This final numerical step is presented in a companion article. This part focuses on the method used to compute these coefficients and provides detailed examples on a model problem. A MODIFIED VERSION OF THIS PREPRINT HAS BEEN PUBLISHED IN CALCOLO
• Le champ des normales d'une surface fractale definie a l'aide de trois champs bidimensionnels
  
  • Colonna Jean-François
  , 2006. The normal field of a fractal surface defined by means of three bidimensional fields (Le champ des normales d'une surface fractale definie a l'aide de trois champs bidimensionnels)
• Le champ des normales d'une surface fractale definie a l'aide de trois champs bidimensionnels
  
  • Colonna Jean-François
  , 2006. The normal field of a fractal surface defined by means of three bidimensional fields (Le champ des normales d'une surface fractale definie a l'aide de trois champs bidimensionnels)
• Le champ des normales d'une représentation tridimensionnelle d'une variété quadridimensionnelle de Calabi-Yau
  
  • Colonna Jean-François
  , 2006. The normal field of a tridimensional representation of a quadridimensional Calabi-Yau manifold (Le champ des normales d'une représentation tridimensionnelle d'une variété quadridimensionnelle de Calabi-Yau)
• Bounds on Regeneration Times and Limit Theorems for Subgeometric Markov Chains
  
  • Douc Randal
  • Guillin Arnaud
  • Moulines Éric
  , 2006. This paper studies limit theorems for Markov Chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster-Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof.
• Convex Sobolev inequalities and spectral gap
  
  • Dolbeault Jean
  • Bartier Jean-Philippe
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2006, 342 (5), pp.307-312. This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.
• Homogenization of the Schrodinger equation with a time oscillating potential
  
  • Allaire Grégoire
  • Vanninathan Muthusamy
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2006, 6, pp.1-16. We study the homogenization of a Schrodinger equation in a periodic medium with a time dependent potential. This is a model for semiconductors excited by an external electromagnetic wave. We prove that, for a suitable choice of oscillating (both in time and space) potential, one can partially transfer electrons from one Bloch band to another. This justifies the famous "Fermi golden rule" for the transition probability between two such states which is at the basis of various optical properties of semiconductors. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves theory.
• Discretization and simulation of the Zakai equation
  
  • Gobet Emmanuel
  • Pagès Gilles
  • Pham Huyên
  • Printems Jacques
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2006, 44 (6), pp.2505-2538. This paper is concerned with numerical approximations for the stochastic partial differential Zakai equation of nonlinear filtering problems. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyze the error caused by an Euler-type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order √δ (δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. Formally, this consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number N of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from a correlated Kalman–Bucy filter. (10.1137/050623140)
  DOI : 10.1137/050623140
• Modification of UCT with Patterns in Monte-Carlo Go
  
  • Gelly Sylvain
  • Wang Yizao
  • Munos Rémi
  • Teytaud Olivier
  , 2006. Algorithm UCB1 for multi-armed bandit problem has already been extended to Algorithm UCT (Upper bound Confidence for Tree) which works for minimax tree search. We have developed a Monte-Carlo Go program, MoGo, which is the first computer Go program using UCT. We explain our modification of UCT for Go application and also the intelligent random simulation with patterns which has improved significantly the performance of MoGo. UCT combined with pruning techniques for large Go board is discussed, as well as parallelization of UCT. MoGo is now a top level Go program on $9\times9$ and $13\times13$ Go boards.
• A Forward-Backward Stochastic Algorithm For Quasi-Linear PDEs
  
  • Delarue François
  • Menozzi Stéphane
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2006, 16, pp.140-184. We propose a time-space discretization scheme for quasi-linear PDEs. The algorithm relies on the theory of fully coupled Forward-Backward SDEs, which provides an efficient probabilistic representation of this type of equations. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE.
• Geometric Variance Reduction in Markov Chains: Application to Value Function and Gradient Estimation
  
  • Munos Rémi
  Journal of Machine Learning Research, Microtome Publishing, 2006, 7, pp.413-427. We study a variance reduction technique for Monte Carlo estimation of functionals in Markov chains. The method is based on designing sequential control variates using successive approximations of the function of interest V. Regular Monte Carlo estimates have a variance of O(1/N), where N is the number of sample trajectories of the Markov chain. Here, we obtain a geometric variance reduction O(ρ^N) (with ρ<1) up to a threshold that depends on the approximation error V-AV, where A is an approximation operator linear in the values. Thus, if V belongs to the right approximation space (i.e. AV=V), the variance decreases geometrically to zero. An immediate application is value function estimation in Markov chains, which may be used for policy evaluation in a policy iteration algorithm for solving Markov Decision Processes. Another important domain, for which variance reduction is highly needed, is gradient estimation, that is computing the sensitivity ∂αV of the performance measure V with respect to some parameter α of the transition probabilities. For example, in policy parametric optimization, computing an estimate of the policy gradient is required to perform a gradient optimization method. We show that, using two approximations for the value function and the gradient, a geometric variance reduction is also achieved, up to a threshold that depends on the approximation errors of both of those representations.
• Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects
  
  • Cakoni Fioralba
  • Fares M'Barek
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2006, 22 (3), pp.845--867. (10.1088/0266-5611/22/3/007)
  DOI : 10.1088/0266-5611/22/3/007
• An anti-diffusive scheme for viability problems
  
  • Bokanowski Olivier
  • Martin Sophie
  • Munos Rémi
  • Zidani Hasnaa
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2006, 56 (9), pp.1147-1162. This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and then use a specially relevant numerical scheme for its approximation. Since this value function is discontinuous, usual discretization schemes (such as finite differences) would provide a poor approximation quality because of numerical diffusion. Hence, we investigate the Ultra-Bee scheme, particularly interesting here for its anti-diffusive property in the transport of discontinuous functions. Although currently there is no available convergence proof for this scheme, we observed that numerically, the experiments done on several benchmark problems for computing viability kernels and capture basins are very encouraging compared to the viability algorithm, which fully illustrates the relevance of this scheme for numerical approximation of viability problems.
• Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models
  
  • Champagnat Nicolas
  • Ferrière Régis
  • Méléard Sylvie
  Theoretical Population Biology, Elsevier, 2006, 69 (3), pp.297-321. A distinctive signature of living systems is Darwinian evolution, that is, a propensity to generate as well as self-select individual diversity. To capture this essential feature of life while describing the dynamics of populations, mathematical models must be rooted in the microscopic, stochastic description of discrete individuals characterized by one or several adaptive traits and interacting with each other. The simplest models assume asexual reproduction and haploid genetics: an offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take a mutation step to new trait values; selection follows from ecological interactions among individuals. Here we present a rigorous construction of the microscopic population process that captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by the trait values of each individual, and interactions between individuals. A by-product of this formal construction is a general algorithm for effcient numerical simulation of the individual-level model. Once the microscopic process is in place, we derive different macroscopic models of adaptive evolution. These models differ in the renormalization they assume, i.e. in the limits taken, in specific orders, on population size, mutation rate, mutation step, while rescaling time accordingly. The macroscopic models also differ in their mathematical nature: deterministic, in the form of ordinary, integro-, or partial differential equations, or probabilistic, like stochastic partial differential equations or superprocesses. These models include extensions of Kimura's equation (and of its approximation for small mutation effects) to frequency- and density-dependent selection. A novel class of macroscopic models obtains when assuming that individual birth and death occur on a short timescale compared with the timescale of typical population growth. On a timescale of very rare mutations, we establish rigorously the models of "trait substitution sequences" and their approximation known as the "canonical equation of adaptive dynamics". We extend these models to account for mutation bias and random drift between multiple evolutionary attractors. The renormalization approach used in this study also opens promising avenues to study and predict patterns of life-history allometries, thereby bridging individual physiology, genetic variation, and ecological interactions in a common evolutionary framework. (10.1016/j.tpb.2005.10.004)
  DOI : 10.1016/j.tpb.2005.10.004
• A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics
  
  • Baudouin Lucie
  Portugaliae Mathematica, European Mathematical Society Publishing House, 2006, 63 (3), pp.293-325. We study a problem of bilinear optimal control for the electronic wave function of an Helium atom by an external time dependent electric field. The behavior of the atom is modeled by the Hartree-Fock equation, whose solution is the wave function of the electrons, coupled with the classical Newtonian dynamics, corresponding to the motion of the nucleus. We prove the existence of a bilinear optimal control in the case when the position of the nucleus is known and also prove the corresponding optimality condition. Then, we detail the proof of the existence of an optimal control for the coupled system and complete the study giving a formal optimality condition to define the electric control.
• 3D Charge Distributions Along Edges and Corners of Electrodes in SAW Transducers
  
  • Jerez-Hanckes Carlos F.
  • Laude Vincent
  • Lardat Raphael
  • Nédélec Jean-Claude
  , 2006, pp.92-95.