Publications

2012

• Courbe de Hilbert tridimensionnelle -itération 4
  
  • Colonna Jean-François
  , 2012. Tridimensional Hilbert Curve -iteration 4- (Courbe de Hilbert tridimensionnelle -itération 4-)
• Courbe de Hilbert tridimensionnelle -itération 3
  
  • Colonna Jean-François
  , 2012. Tridimensional Hilbert Curve -iteration 3- (Courbe de Hilbert tridimensionnelle -itération 3-)
• Concentration Bounds for Stochastic Approximations
  
  • Frikha Noufel
  • Menozzi Stephane
  , 2012. We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of the Euler scheme of some diffusion process at a fixed deterministic time and its empirical mean obtained by the Monte-Carlo procedure. We then give some estimates concerning the deviation between the value at a given time-step of a stochastic approximation algorithm and its target. Under suitable assumptions both concentration bounds turn out to be Gaussian. The key tool consists in exploiting accurately the concentration properties of the increments of the schemes. For the first case, as opposed to the previous work of Lemaire and Menozzi (EJP, 2010), we do not have any systematic bias in our estimates. Also, no specific non-degeneracy conditions are assumed.
• Moderate Deviations for Mean Field Particle Models
  
  • del Moral Pierre
  • Hu Shulan
  • Wu Liming
  , 2012, pp.37. This article is concerned with moderate deviation principles of a general class of mean eld type interacting particle models. We discuss functional moderate deviations of the occupation measures for both the strong -topology on the space of fi nite and bounded measures as well as for the corresponding stochastic processes on some class of functions equipped with the uniform topology. Our approach is based on an original semigroup analysis combined with stochastic perturbation techniques and projective limit large deviation methods.
• A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule
  
  • Boscain Ugo
  • Caponigro Marco
  • Chambrion Thomas
  • Sigalotti Mario
  Communications in Mathematical Physics, Springer Verlag, 2012, 311 (2), pp.423-455. In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields. (10.1007/s00220-012-1441-z)
  DOI : 10.1007/s00220-012-1441-z
• Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer
  
  • Allaire Grégoire
  • Pankratova Iryna
  • Piatnitski Andrey
  SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, Springer, 2012, 58 (1), pp.53-95. The paper deals with the homogenization of a non-stationary convection-diffusion equation defined in a thin rod or in a layer with Dirichlet boundary condition. Under the assumption that the convection term is large, we describe the evolution of the solution’s profile and determine the rate of its decay. The main feature of our analysis is that we make no assumption on the support of the initial data which may touch the domain’s boundary. This requires the construction of boundary layer correctors in the homogenization process which, surprisingly, play a crucial role in the definition of the leading order term at the limit. Therefore we have to restrict our attention to simple geometries like a rod or a layer for which the definition of boundary layers is easy and explicit. (10.1007/BF03322605)
  DOI : 10.1007/BF03322605
• Monument Valley sous la neige
  
  • Colonna Jean-François
  , 2012. Snowy Monument Valley (Monument Valley sous la neige)
• Monument Valley sous la neige
  
  • Colonna Jean-François
  , 2012. Snowy Monument Valley (Monument Valley sous la neige)
• Monument Valley sous la neige
  
  • Colonna Jean-François
  , 2012. Snowy Monument Valley (Monument Valley sous la neige)
• Monument Valley sous la neige
  
  • Colonna Jean-François
  , 2012. Snowy Monument Valley (Monument Valley sous la neige)
• Small time heat kernel asymptotics at the sub-Riemannian cut locus
  
  • Barilari Davide
  • Boscain Ugo
  • Neel Robert W.
  , 2012. For a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\ all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane we get the expansion $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.
• A Remark on the Anisotropic Outer Minkowski content
  
  • Chambolle Antonin
  • Lisini Stefano
  • Lussardi Luca
  , 2012. We study an anisotropic version of the outer Minkowski content of a closed set in Rn. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') transformed using the tridimensional life game (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel)
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') transformed using the tridimensional life game (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel)
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') transformed using the tridimensional life game (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel)
• Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.4% de cellules occupées
  
  • Colonna Jean-François
  , 2012. The tridimensional John Conway's life game with random initial conditions -1.4% of occupied cells- (Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.4% de cellules occupées-)
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') transformed using the tridimensional life game (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb') transformé en utilisant le jeu de la vie tridimensionnel)
• Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.6% de cellules occupées
  
  • Colonna Jean-François
  , 2012. The tridimensional John Conway's life game with random initial conditions -1.6% of occupied cells- (Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.6% de cellules occupées-)
• Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.2% de cellules occupées
  
  • Colonna Jean-François
  , 2012. The tridimensional John Conway's life game with random initial conditions -1.2% of occupied cells- (Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.2% de cellules occupées-)
• Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -2.0% de cellules occupées
  
  • Colonna Jean-François
  , 2012. The tridimensional John Conway's life game with random initial conditions -2.0% of occupied cells- (Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -2.0% de cellules occupées-)
• Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.8% de cellules occupées
  
  • Colonna Jean-François
  , 2012. The tridimensional John Conway's life game with random initial conditions -1.8% of occupied cells- (Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.8% de cellules occupées-)
• Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.0% de cellules occupées
  
  • Colonna Jean-François
  , 2012. The tridimensional John Conway's life game with random initial conditions -1.0% of occupied cells- (Le jeu de la vie tridimensionnel de John Conway avec des conditions initiales aléatoires -1.0% de cellules occupées-)
• Regularization of inverse problems in image processing
  
  • Jalalzai Khalid
  , 2012. Inverse problems are to recover the data that has been processed or corrupted. Since they are ill-posed they require a regularization. In image processing, the total variation as a regularization tool has the advantage of preserving the discontinuities while creating smooth regions. These results are established in this thesis in a continuous setting for general energies. In addition, we propose and examine a variant of the total variation. We establish a dual formulation that allows us to prove that this variant coincides with the total variation for sets of finite perimeter. Nowadays, non-local methods exploiting the self-similarities of images is particularly successful. We adapt this approach to the problem of spectrum completion, which has applications for general inverse problems. The final part is devoted to the algorithmic aspects inherent to the optimization of the convex energies we considered. We study the convergence and the complexity of the recently developed Primal-Dual algorithms.
• On the gradient flow of a one-homogeneous functional
  
  • Briani Ariela
  • Chambolle Antonin
  • Novaga Matteo
  • Orlandi Giandomenico
  Confluentes Mathematici, Université de Lyon, 2012, 03 (04), pp.617-635. We consider the gradient flow of a one-homogeneous functional, whose dual involves the derivative of a constrained scalar function. We show in this case that the gradient flow is related to a weak, generalized formulation of the Hele-Shaw flow. The equivalence follows from a variational representation, which is a variant of well-known variational representations for the Hele-Shaw problem. As a consequence we get existence and uniqueness of a weak solution to the Hele-Shaw flow. We also obtain an explicit representation for the Total Variation flow in one dimension and easily deduce basic qualitative properties, concerning in particular the ''staircasing effect''. (10.1142/S1793744211000461)
  DOI : 10.1142/S1793744211000461
• A maximin characterization of the escape rate of non-expansive mappings in metrically convex spaces
  
  • Gaubert Stéphane
  • Vigeral Guillaume
  Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press (CUP), 2012, 152 (2), pp.341-363. We establish a maximin characterisation of the linear escape rate of the orbits of a non-expansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's non-positive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterisation, which involves horofunctions, generalises the Collatz-Wielandt characterisation of the spectral radius of a non-negative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning non-expansive maps in Banach spaces, a variant of a Denjoy-Wolff type theorem of Karlsson (2001), together with a refinement of a theorem of Gunawardena and Walsh (2003), concerning order-preserving positively homogeneous self-maps of symmetric cones. An application to zero-sum stochastic games is also given. (10.1017/S0305004111000673)
  DOI : 10.1017/S0305004111000673