Publications

2009

• Le système solaire avec une planète virtuelle verte -point de vue de la planète virtuelle
  
  • Colonna Jean-François
  , 2009. The Solar System with a green virtual planet -virtual planet point of view- (Le système solaire avec une planète virtuelle verte -point de vue de la planète virtuelle-)
• Le systeme solaire avec une planete virtuelle verte -point de vue de la planete virtuelle
  
  • Colonna Jean-François
  , 2009. The Solar System with a green virtual planet -virtual planet point of view- (Le systeme solaire avec une planete virtuelle verte -point de vue de la planete virtuelle-)
• Le système solaire avec une planète virtuelle verte -point de vue de la planète virtuelle
  
  • Colonna Jean-François
  , 2009. The Solar System with a green virtual planet -virtual planet point of view- (Le système solaire avec une planète virtuelle verte -point de vue de la planète virtuelle-)
• Le système solaire avec une planète virtuelle verte -point de vue de la planète virtuelle
  
  • Colonna Jean-François
  , 2009. The Solar System with a green virtual planet -virtual planet point of view- (Le système solaire avec une planète virtuelle verte -point de vue de la planète virtuelle-)
• Sparse Super-Resolution with Space Matching Pursuits
  
  • Yu Guoshen
  • Mallat Stéphane
  , 2009. Super-resolution image zooming is possible when the image has some geometric regularity. Directional interpolation algorithms are used in industry, with ad-hoc regularity measurements. Sparse signal decompositions in dictionaries of curvelets or bandlets find indirectly the directions of regularity by optimizing the sparsity. However, super-resolution interpolations in such dictionaries do not outperform cubic spline interpolations. It is necessary to further constraint the sparse representation, which is done through projections over structured vector spaces. A space matching pursuit algorithm is introduced to compute image decompositions over spaces of bandlets, from which a super-resolution image zooming is derived. Numerical experiments illustrate the efficiency of this super-resolution procedure compared to cubic spline interpolations.
• A direct approach to the duality of grand and small Lebesgue spaces
  
  • Di Fratta Giovanni
  • Fiorenza Alberto
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2009, 70 (7), pp.2582-2592. In this paper we show, by elementary methods, that the quasinorms of the grand and small Lebesgue spaces \[ \nnorm{f}_{L^{p)}} \approx \sup_{0<t<1} (1-\log t)^{-{\frac 1p}} \left(\int_{t}^1 [f^{\ast}(s)]^p ds\right)^{{\frac 1p}}\, , \,\, \,\, 1<p<\infty \] \[ \nnorm{f}_{L^{(q}} \approx \int_0^1 (1-\log t)^{-{\frac 1q}}\left(\int_0^t f^{\ast}(s)^ {q}ds\right)^{{\frac 1q}} \frac{dt}t\,\, ,\,\, \,\, q=\frac p{p-1} \] found by Fiorenza and Karadzhov in \cite{FK}, by using deeply extrapolation-interpolation techniques, are associate each other. In other terms, the sharp H\"older's type inequality: \[ \int_0^1fgdx\le c(p) \nnorm{f}_{L^{(q}}\nnorm{g}_{L^{p)}} \] is proven, where the sharpness means that $ \nnorm{\cdot}_{L^{(q}}$ is the smallest quasinorm (up to equivalences) such that the inequality holds. The method is based entirely on integral estimates, makes use of asymptotic properties of the Euler's Gamma function, and gives an explicit estimate of the constant $c(p)$. All the results are expressed in terms of the more general spaces $L^{p),\theta}$ and $L^{(q,\theta}$, $\theta>0$. (10.1016/j.na.2008.03.044)
  DOI : 10.1016/j.na.2008.03.044
• Une interpolation 'double' entre deux entrelacs
  
  • Colonna Jean-François
  , 2009. A 'double' interpolation between two intertwinings (Une interpolation 'double' entre deux entrelacs)
• Quasi-stationary distributions for structured birth and death processes with mutations
  
  • Collet Pierre
  • Martinez Servet
  • Méléard Sylvie
  • San Martin Jaime
  , 2009. We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of quasi-stationary distributions when the process is conditioned on non-extinction. We firstly show in this general setting, the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.
• Trait Substitution Sequence process and Canonical Equation for age-structured populations
  
  • Méléard Sylvie
  • Tran Viet Chi
  Journal of Mathematical Biology, Springer, 2009, 58 (6), pp.881-921. We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin-McCamy equation in Demography. Another particularity is that they involve a fitness function, describing the probability of invasion of the resident population by the mutant one, that can not always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried. (10.1007/s00285-008-0202-2)
  DOI : 10.1007/s00285-008-0202-2
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs
  
  • Fahim Arash
  • Touzi Nizar
  • Warin Xavier
  , 2009. We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations a! rising in the theory of portfolio optimization in financial mathematics.
• Echiquier
  
  • Colonna Jean-François
  , 2009. Chessboard (Echiquier)
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• Entrelacs
  
  • Colonna Jean-François
  , 2009. Intertwining (Entrelacs)
• Echiquier
  
  • Colonna Jean-François
  , 2009. Chessboard (Echiquier)
• l'oeil etait dans l'ordinateur
  
  • Colonna Jean-François
  , 2009. L'oeil etait dans l'ordinateur. Image obtenue a l'aide d'un generateur entrelacs a partir d'unes structure non periodique
• Subgeometric rates of convergence of f-ergodic strong Markov processes
  
  • Douc Randal
  • Fort Gersende
  • Guillin Arnaud
  Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.897-923. We provide a condition for f-ergodicity of strong Markov processes at a subgeometric rate. This condition is couched in terms of a supermartingale property for a functional of the Markov process. Equivalent formulations in terms of a drift inequality on the extended generator and on the resolvent kernel are given. Results related to (f,r)-regularity and to moderate deviation principle for integral (bounded) functional are also derived. Applications to specific processes are considered, including elliptic stochastic differential equation, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian system and storage models. (10.1016/j.spa.2008.03.007)
  DOI : 10.1016/j.spa.2008.03.007
• Subgeometric rates of convergence of f-ergodic strong Markov processes
  
  • Douc Randal
  • Fort Gersende
  • Guillin Arnaud
  Stochastic Processes and their Applications, Elsevier, 2009, 119 (3), pp.897 - 923. We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f-ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of a drift inequality on the extended generator is also given. Results related to (f,r)-regularity of the process, of some skeleton chains and of the resolvent chain are also derived. Applications to specific processes are considered, including elliptic stochastic differential equations, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian systems and storage models (10.1016/j.spa.2008.03.007)
  DOI : 10.1016/j.spa.2008.03.007
• Some Variations on Total Variation--Based Image Smoothing
  
  • Chambolle Antonin
  • Levine Stacey
  • Lucier Bradley
  , 2009. In this paper we study finite-difference approximations to the variational problem using the BV smoothness penalty that was introduced in an image smoothing context by Rudin, Osher, and Fatemi. We give a dual formulation for an ``upwind'' finite-difference approximation for the BV seminorm; this formulation is in the same spirit as one popularized by Chambolle for a simpler, more anisotropic, finite-difference approximation to the BV seminorm. We introduce a multiscale method for speeding the approximation of both Chambolle's original method and of the new formulation of the upwind scheme. We demonstrate numerically that the multiscale method is effective, and we provide numerical examples that illustrate both the qualitative and quantitative behavior of the solutions of the numerical formulations.
• Échiquier -Échecs et Maths
  
  • Colonna Jean-François
  , 2009. Chessboard (Echiquier -Echecs et Maths-)
• Échiquier -Échecs et Maths
  
  • Colonna Jean-François
  , 2009. Chessboard (Echiquier -Echecs et Maths-)