Publications

2013

• 24 points répartis équitablement sur une sphère à l'aide de la spirale de Fibonacci
  
  • Colonna Jean-François
  , 2013. 24 evenly distributed points on a sphere by means of the Fibonacci spiral (24 points répartis équitablement sur une sphère à l'aide de la spirale de Fibonacci)
• 12 points répartis équitablement sur une sphère -un Icosaèdre- par recuit simulé
  
  • Colonna Jean-François
  , 2013. 12 evenly distributed points on a sphere -an Icosahedron- by means of simulated annealing (12 points répartis équitablement sur une sphère -un Icosaèdre- par recuit simulé)
• 6 points répartis équitablement sur une sphère -un Octaèdre- par recuit simulé
  
  • Colonna Jean-Francois
  , 2013. 6 evenly distributed points on a sphere -an Octahedron- by means of simulated annealing (6 points répartis équitablement sur une sphère -un Octaèdre- par recuit simulé)
• 24 points répartis équitablement sur une sphère par recuit simulé
  
  • Colonna Jean-François
  , 2013. 24 evenly distributed points on a sphere by means of simulated annealing (24 points répartis équitablement sur une sphère par recuit simulé)
• Simulation numérique d'un système d'EDP hyperboliques avec des flux discontinus
  
  • Aymard Benjamin
  • Clément Frédérique
  • Coquel Frédéric
  • Monniaux Danielle
  • Postel Marie
  , 2013.
• 2000 points répartis 'équitablement' sur une sphère à l'aide de la spirale de Fibonacci -zoom sur les 10 premiers points de la spirale
  
  • Colonna Jean-Francois
  , 2013. 2000 evenly distributed points on a sphere by means of the Fibonacci spiral -zoom in on the first 10 points of the spiral- (2000 points répartis 'équitablement' sur une sphère à l'aide de la spirale de Fibonacci -zoom sur les 10 premiers points de la spirale-)
• Geometry and analysis of control-affine systems: motion planning, heat and Schrödinger evolution
  
  • Prandi Dario
  , 2013. This thesis is dedicated to two problems arising from geometric control theory, regarding control-affine systems $\dot q= f_0(q)+\sum_{j=1}^m u_j f_j(q)$, where $f_0$ is called the drift. In the first part we extend the concept of complexity of non-admissible trajectories, well understood for sub-Riemannian systems, to this more general case, and find asymptotic estimates. Then, in the second part of the thesis, we consider a family of 2-dimensional driftless control systems. For these, we study how the set where the control vector fields become collinear affects the evolution of the heat and of a quantum particle with respect to the associated Laplace-Beltrami operator.
• 2000 points répartis 'équitablement' sur une sphère à l'aide de la spirale de Fibonacci
  
  • Colonna Jean-Francois
  , 2013. 2000 evenly distributed points on a sphere by means of the Fibonacci spiral (2000 points répartis 'équitablement' sur une sphère à l'aide de la spirale de Fibonacci)
• 2000 points répartis 'équitablement' sur une sphère à l'aide de la spirale de Fibonacci -avec visualisation de la spirale
  
  • Colonna Jean-Francois
  , 2013. 2000 evenly distributed points on a sphere by means of the Fibonacci spiral -with display of the spiral- (2000 points répartis 'équitablement' sur une sphère à l'aide de la spirale de Fibonacci -avec visualisation de la spirale-)
• Nonlinear Perron-Frobenius theory and max-plus numerical methods for Hamilton-Jacobi equations
  
  • Qu Zheng
  , 2013. Dynamic programming is one of the main approaches to solve optimal control problems. It reduces the latter problems to Hamilton-Jacobi partial differential equations (PDE). Several techniques have been proposed in the literature to solve these PDE. We mention, for example, finite difference schemes, the so-called discrete dynamic programming method or semi-Lagrangian method, or the antidiffusive schemes. All these methods are grid-based, i.e., they require a discretization of the state space, and thus suffer from the so-called curse of dimensionality. The present thesis focuses on max-plus numerical solutions and convergence analysis for medium to high dimensional deterministic optimal control problems. We develop here max-plus based numerical algorithms for which we establish theoretical complexity estimates. The proof of these estimates is based on results of nonlinear Perron-Frobenius theory. In particular, we study the contraction properties of monotone or non-expansive nonlinear operators, with respect to several classical metrics on cones (Thompson's metric, Hilbert's projective metric), and obtain nonlinear or non-commutative generalizations of the "ergodicity coefficients" arising in the theory of Markov chains. These results have applications in consensus theory and also to the generalized Riccati equations arising in stochastic optimal control.
• Policy iteration for perfect information stochastic mean payoff games with bounded first return times is strongly polynomial
  
  • Akian Marianne
  • Gaubert Stéphane
  , 2013. Recent results of Ye and Hansen, Miltersen and Zwick show that policy iteration for one or two player (perfect information) zero-sum stochastic games, restricted to instances with a fixed discount rate, is strongly polynomial. We show that policy iteration for mean-payoff zero-sum stochastic games is also strongly polynomial when restricted to instances with bounded first mean return time to a given state. The proof is based on methods of nonlinear Perron-Frobenius theory, allowing us to reduce the mean-payoff problem to a discounted problem with state dependent discount rate. Our analysis also shows that policy iteration remains strongly polynomial for discounted problems in which the discount rate can be state dependent (and even negative) at certain states, provided that the spectral radii of the nonnegative matrices associated to all strategies are bounded from above by a fixed constant strictly less than 1.
• Un réseau cubique tridimensionnel défini à l'aide de trois champs tridimensionnels
  
  • Colonna Jean-François
  , 2013. A tridimensional distorted cubic mesh defined by means of three tridimensional fields (Un réseau cubique tridimensionnel défini à l'aide de trois champs tridimensionnels)
• Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian
  
  • Graber Philip Jameson
  , 2013. We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed. Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE, viscosity solutions, front propagation, mean field games
• Un plan distordu défini à l'aide de trois champs bidimensionnels
  
  • Colonna Jean-François
  , 2013. A distorted plane defined by means of three bidimensional fields (Un plan distordu défini à l'aide de trois champs bidimensionnels)
• Convergence to equilibrium for discrete gradient-like flows and An accurate method for the motion of suspended particles in a Stokes fluid
  
  • Nguyen Thanh Nhan
  , 2013. The thesis has two independent parts. The first part concerns the convergence toward equilibrium of discrete gradient flows or, with more generality, of some discretizations of autonomous systems which admit a Lyapunov function. The study is performed assuming sufficient conditions for the solutions of the continuous problem to converge toward a stationary state as time goes to infinity. It is shown that under mild hypotheses, the discrete system has the same property. This leads to new results on the large time asymptotic behavior of some known non-linear schemes. The second part concerns the numerical simulation of the motion of particles suspended in a viscous fluid. It is shown that the most widely used methods for computing the hydrodynamic interactions between particles lose their accuracy in the presence of large non-hydrodynamic forces and when at least two particles are close from each other. This case arises in the context of medical engineering for the design of nano-robots that can swim. This loss of accuracy is due to the singular character of the Stokes flow in areas of almost contact. A new method is introduced here. Numerical experiments are realized to illustrate its better accuracy.
• Diffusion Microscopist Simulator: A General Monte Carlo Simulation System for Diffusion Magnetic Resonance Imaging
  
  • Yeh Chun-Hung
  • Schmitt Benoit
  • Le Bihan Denis
  • Li Jing-Rebecca
  • Lin Ching-Po
  • Poupon Cyril
  PLoS ONE, Public Library of Science, 2013. This article describes the development and application of an integrated, generalized, and efficient Monte Carlo simulation system for diffusion magnetic resonance imaging (dMRI), named Diffusion Microscopist Simulator (DMS). DMS comprises a random walk Monte Carlo simulator and an MR image synthesizer. The former has the capacity to perform large-scale simulations of Brownian dynamics in the virtual environments of neural tissues at various levels of complexity, and the latter is flexible enough to synthesize dMRI datasets from a variety of simulated MRI pulse sequences. The aims of DMS are to give insights into the link between the fundamental diffusion process in biological tissues and the features observed in dMRI, as well as to provide appropriate ground-truth information for the development, optimization, and validation of dMRI acquisition schemes for different applications. The validity, efficiency, and potential applications of DMS are evaluated through four benchmark experiments, including the simulated dMRI of white matter fibers, the multiple scattering diffusion imaging, the biophysical modeling of polar cell membranes, and the high angular resolution diffusion imaging and fiber tractography of complex fiber configurations. We expect that this novel software tool would be substantially advantageous to clarify the interrelationship between dMRI and the microscopic characteristics of brain tissues, and to advance the biophysical modeling and the dMRI methodologies. (10.1371/journal.pone.0076626)
  DOI : 10.1371/journal.pone.0076626
• Rough wall effect on micro-swimmers
  
  • Gérard-Varet David
  • Giraldi Laetitia
  , 2013. We study the effect of a rough wall on the controllability of micro-swimmers made of several balls linked by thin jacks: the so-called 3-sphere and 4-sphere swimmers. We show that a controllable swimmer (the 4-sphere swimmer) is not impacted by the roughness. On the contrary, we show that the roughness changes the dynamics of the 3-sphere swimmer, so that it can reach any direction almost everywhere.
• Mathematical methods for analysis swimming at low Reynolds number
  
  • Giraldi Laëtitia
  , 2013. This thesis is devoted to the mathematical study of the swimming at low Reynolds number. The controllability and the optimal problems associated with the displacement of micro- swimmers are the main points developed in this work. In the first part, we study the controllability and the optimal control problem in time associated with a reduced model of swimmers, called the"N-link swimmer". In the second part, we study the boundary effect on the controllability of particular micro-swimmers made by several balls linked each others by thin jacks. Firstly, we analyze the effect of a plane wall on the mobility of these swimmers. Then, we generalize these results where the wall is rough. We demonstrate that a controllable swimmer remains controllable in a half space delimited by a wall (plane or rough) whereas the reachable set of a non controllable one is increased by the presence of a wall. The last part is devoted to provide a general framework to study optimal controllability of driftless swimmers. We focus on the study of optimal strokes i.e. periodic shape changes. More precisely, we are interested in the existence of optimal strokes, minimizing or maximizing various cost functionals, qualitative properties of the optimal strokes, regularity and monotony of the value functions.
• Bayesian Estimation of Probabilistic Atlas for Anatomically-Informed Functional MRI Group Analyses
  
  • Xu Hao
  • Thirion Bertrand
  • Allassonnière Stéphanie
  , 2013. Traditional analyses of Functional Magnetic Resonance Imaging (fMRI) use little anatomical information. The registration of the images to a template is based on the individual anatomy and ignores functional information; subsequently detected activations are not confined to gray matter (GM). In this paper, we propose a statistical model to estimate a probabilistic atlas from functional and T1 MRIs that summarizes both anatomical and functional information and the geometric variability of the population. Registration and Segmentation are performed jointly along the atlas estimation and the functional activity is constrained to the GM, increasing the accuracy of the atlas.
• Le groupe abélien -commutatif- défini sur les courbes elliptiques
  
  • Colonna Jean-François
  , 2013. The abelian -commutative- group defined on elliptic curves (Le groupe abélien -commutatif- défini sur les courbes elliptiques)
• HOMOGENIZATION OF COMPLEX FLOWS IN POROUS MEDIA AND APPLICATIONS
  
  • Hutridurga Ramaiah Harsha
  , 2013. Our work is a contribution to the understanding of transport of solutes in a porous medium. It has applications in groundwater contaminant transport, CO2 sequestration, underground storage of nuclear waste, oil reservoir simulations. We derive expressions for the effective Taylor dispersion taking into account convection, diffusion, heterogeneous geometry of the porous medium and reaction phenomena. Microscopic phenomena at the pore scale are upscaled to obtain effective behaviour at the observation scale. Method of two-scale convergence with drift from the theory of homogenization is employed as an upscaling technique. In the first part of our work, we consider reactions of mass exchange type, adsorption/desorption, at the fluid-solid interface of the porous medium. Starting with coupled convection-diffusion equations for bulk and surface concentrations of a single solute, coupled via adsorption isotherms, at a microscopic scale we derive effective equations at the macroscopic scale. We consider the microscopic system with highly oscillating coefficients in a strong convection regime i.e., large Péclet regime. The presence of strong convection in the microscopic model leads to the induction of a large drift in the concentration profiles. Both linear and nonlinear adsorption isotherms are considered and the results are compared. In the second part of our work we generalize our results on single component flow to multicomponent flow in a linear setting. In the latter case, the effective parameters are obtained using Factorization principle and two-scale convergence with drift. The behaviour of effective parameters with respect to Péclet number and Damköhler number are numerically studied. Freefem++ is used to perform numerical tests in two dimensions.
• Two algorithms for the discrete time approximation of Markovian backward stochastic differential equations under local conditions
  
  • Turkedjiev Plamen
  , 2013. Two discretizations of a novel class of Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes $Z$, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process $Z$ directly.Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs.These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in the time-grid.The class of BSDEs considered includes Lipschitz BSDEs with fractionally smooth terminal condition as well as quadratic BSDEs with bounded, H\"older continuous terminal condition.
• Une courbe elliptique
  
  • Colonna Jean-François
  , 2013. An elliptic curve (Une courbe elliptique)
• Slow-fast stochastic diffusion dynamics and quasi-stationary distributions for diploid populations
  
  • Coron Camille
  , 2013. We are interested in the long-time behavior of a diploid population with sexual reproduction and randomly varying population size, characterized by its genotype composition at one bi-allelic locus. The population is modeled by a $3$-dimensional birth-and-death process with competition, cooperation and Mendelian reproduction. This stochastic process is indexed by a scaling parameter $K$ that goes to infinity, following a large population assumption. When the individual birth and natural death rates are of order $K$, the sequence of stochastic processes indexed by $K$ converges toward a new slow-fast dynamics with variable population size. We indeed prove the convergence toward $0$ of a fast variable giving the deviation of the population from Hardy-Weinberg equilibrium, while the sequence of slow variables giving the respective numbers of occurrences of each allele converges toward a $2$-dimensional diffusion process that reaches $(0,0)$ almost surely in finite time. The population size and the proportion of a given allele converge toward a Wright-Fisher diffusion with stochastically varying population size and diploid selection. We insist on differences between haploid and diploid populations due to population size stochastic variability. Using a non trivial change of variables, we study the absorption of this diffusion and its long time behavior conditioned on non-extinction. In particular we prove that this diffusion starting from any non-trivial state and conditioned on not hitting $(0,0)$ admits a unique quasi-stationary distribution. We give numerical approximations of this quasi-stationary behavior in three biologically relevant cases: neutrality, overdominance, and separate niches.
• Convergent Stochastic Expectation Maximization algorithm with efficient sampling in high dimension. Application to deformable template model estimation
  
  • Allassonniere Stéphanie
  • Kuhn Estelle
  , 2013. Estimation in the deformable template model is a big challenge in image analysis. The issue is to estimate an atlas of a population. This atlas contains a template and the corresponding geometrical variability of the observed shapes. The goal is to propose an accurate algorithm with low computational cost and with theoretical guaranties of relevance. This becomes very demanding when dealing with high dimensional data which is particularly the case of medical images. We propose to use an optimized Monte Carlo Markov Chain method into a stochastic Expectation Maximization algorithm in order to estimate the model parameters by maximizing the likelihood. In this paper, we present a new Anisotropic Metropolis Adjusted Langevin Algorithm which we use as transition in the MCMC method. We first prove that this new sampler leads to a geometrically uniformly ergodic Markov chain. We prove also that under mild conditions, the estimated parameters converge almost surely and are asymptotically Gaussian distributed. The methodology developed is then tested on handwritten digits and some 2D and 3D medical images for the deformable model estimation. More widely, the proposed algorithm can be used for a large range of models in many fields of applications such as pharmacology or genetic.