Publications

2014

• Plane-like minimizers and differentiability of the stable norm
  
  • Chambolle Antonin
  • Goldman Michael
  • Novaga Matteo
  The Journal of Geometric Analysis, Springer, 2014. In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.
• Two-Level Domain Decomposition Methods for Highly Heterogeneous Darcy Equations. Connections with Multiscale Methods
  
  • Dolean Victorita
  • Jolivet Pierre
  • Nataf Frédéric
  • Spillane Nicole
  • Xiang Hua
  Oil & Gas Science and Technology - Revue d'IFP Energies nouvelles, Institut Français du Pétrole (IFP), 2014, 69 (4), pp.731-752. Multiphase, compositional porous media flow models lead to the solution of highly heterogeneous systems of Partial Differential Equations (PDE). We focus on overlapping Schwarz type methods on parallel computers and on multiscale methods. We present a coarse space [Nataf F., Xiang H., Dolean V., Spillane N. (2011) SIAM J. Sci. Comput. 33, 4, 1623-1642] that is robust even when there are such heterogeneities. The two-level domain decomposition approach is compared to multiscale methods. (10.2516/ogst/2013206)
  DOI : 10.2516/ogst/2013206
• Estimator selection in the Gaussian setting
  
  • Baraud Yannick
  • Giraud Christophe
  • Huet Sylvie
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2014, pp.to appear. We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $\sigma^{2}$. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $\FF$ of estimators of $f$ based on $Y$ and, with the same data $Y$, aim at selecting an estimator among $\FF$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown. We establish a non-asymptotic risk bound for the selected estimator. As particular cases, our approach allows to handle the problems of aggregation and model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. We also derive oracle-type inequalities when $\FF$ consists of linear estimators. For illustration, we carry out two simulation studies. One aims at comparing our procedure to cross-validation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.
• Tropical Fourier–Motzkin elimination, with an application to real-time verification
  
  • Allamigeon Xavier
  • Legay Axel
  • Fahrenberg Uli
  • Katz Ricardo
  • Gaubert Stéphane
  International Journal of Algebra and Computation, World Scientific Publishing, 2014, 24 (5), pp.569 - 607. We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata). (10.1142/S0218196714500258)
  DOI : 10.1142/S0218196714500258
• A minimum effort optimal control problem for the wave equation.
  
  • Kröner Axel
  • Kunisch Karl
  Computational Optimization and Applications, Springer Verlag, 2014, 57 (1), pp.241-270. A minimum effort optimal control problem for the undamped waveequation is considered which involves L∞–control costs. Since the problem isnon-differentiable a regularized problem is introduced. Uniqueness of the solu-tion of the regularized problem is proven and the convergence of the regularizedsolutions is analyzed. Further, a semi-smooth Newton method is formulatedto solve the regularized problems and its superlinear convergence is shown.Thereby special attention has to be paid to the well-posedness of the Newtoniteration. Numerical examples confirm the theoretical results.
• Exploring diffusion across permeable barriers at high gradients. I. Narrow pulse approximation
  
  • Grebenkov Denis S
  • Nguyen Dang Van
  • Li Jing-Rebecca
  Journal of Magnetic Resonance, Elsevier, 2014, pp.153–163. (10.1016/j.jmr.2014.07.013)
  DOI : 10.1016/j.jmr.2014.07.013
• Multi-phase structural optimization via a level set method
  
  • Allaire Grégoire
  • Dapogny Charles
  • Delgado Gabriel
  • Michailidis Georgios
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20, pp.576-611. We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing "grey" regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases. (10.1051/cocv/2013076)
  DOI : 10.1051/cocv/2013076
• On conjugate times of LQ optimal control problems
  
  • Agrachev Andrei
  • Rizzi Luca
  • Silveira Pavel
  Journal of Dynamical and Control Systems, Springer Verlag, 2014, 21 (4), pp.625-641. Motivated by the study of linear quadratic optimal control problems, we consider a dynamical system with a constant, quadratic Hamiltonian, and we characterize the number of conjugate times in terms of the spectrum of the Hamiltonian vector field $\vec{H}$. We prove the following dichotomy: the number of conjugate times is identically zero or grows to infinity. The latter case occurs if and only if $\vec{H}$ has at least one Jordan block of odd dimension corresponding to a purely imaginary eigenvalue. As a byproduct, we obtain bounds from below on the number of conjugate times contained in an interval in terms of the spectrum of $\vec{H}$. (10.1007/s10883-014-9251-6)
  DOI : 10.1007/s10883-014-9251-6
• The horofunction boundary and isometry group of the Hilbert geometry
  
  • Walsh Cormac
  , 2014, 22. The horofunction boundary is a means of compactifying metric spaces that was introduced by Gromov in the 1970s. We describe explicitly the horofunction boundary of the Hilbert geometry, and sketch how it may be used to study the isometry group of this space.
• Characterization of Glioma Microcirculation and Tissue Features Using Intravoxel Incoherent Motion Magnetic Resonance Imaging in a Rat Brain Model
  
  • Iima M.
  • Reynaud O.
  • Tsurugizawa T.
  • Ciobanu Luisa
  • Li Jing-Rebecca
  • Geffroy F.
  • Djemai B.
  • Umehana M.
  • Le Bihan Denis
  Investigative Radiology, Lippincott, Williams & Wilkins, 2014, pp.485–490. (10.1097/RLI.0000000000000040)
  DOI : 10.1097/RLI.0000000000000040
• Weak approximation of averaged diffusion processes
  
  • Gobet Emmanuel
  • Miri Mohammed
  Stochastic Processes and their Applications, Elsevier, 2014, 124, pp.475--504. We derive expansion results in order to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approximation is based on the use of proxys with normal distribution or log-normal distribution, so that the expansion terms are explicit. We provide non asymptotic error bounds, which justifies the expansion accuracy as the time or the diffusion coefficients are small in a suitable sense.
• Policy iteration for stochastic zero-sum games
  
  • Akian Marianne
  , 2014. 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.
• Variational Curvature Flows
  
  • Chambolle Antonin
  • Morini Massimiliano
  • Ponsiglione Marcello
  , 2014. Variational Curvature Flows
• Découverte et analyse de la biodiversité : les moyens actuels
  
  • Elias Marianne
  • Condamine Fabien
  Mémoires de la SEF, 2014 (9), pp.23-39. – Depuis l’antiquité, la diversité des formes biologiques fascine, et fait l’objet de descriptions, d’analyses et classifications. Alors que pendant des siècles la caractérisation de la biodiversité a reposé sur des critères morphologiques, l’avènement et le développement rapide des techniques moléculaires et des méthodes analytiques associées ont permis de développer des approches complexes et intégratives pour étudier la biodiversité, tout en incorporant les données issues de la morphologie. Ces approches se classent en trois catégories : 1) Les approches microévolutives de type “génétique des populations”, basées sur de nombreux marqueurs moléculaires répartis dans tout le génome, et qui renseignent sur la structure fine de la biodiversité en étudiant les processus de spéciation (formation des espèces). 2) Les approches de type barcoding, qui permettent de caractériser génétiquement un individu et de lui assigner un nom d’espèce sur la base de la variation de la séquence d’ADN d’un ou de plusieurs gènes. Éventuellement, cette approche permet de définir les limites d’espèces. Combiné à la morphologie, le barcoding moléculaire a conduit à la taxonomie intégrative. 3) Les approches macroévolutives de type “phylogénies” (établies sur des données moléculaires et/ou morphologiques), qui reconstruisent les relations de parenté entre espèces et sont donc non seulement informatives sur la systématique du groupe étudié, mais également sur les processus évolutifs qui en ont structuré la biodiversité (biogéographie historique, rôle de caractères particuliers dans la diversification du groupe, variation des taux de diversification dans le temps ou entre lignées...). Ces trois approches, qui peuvent être combinées entre elles ainsi qu’aux approches morphologiques, ont contribué à mieux caractériser et comprendre les facteurs à l’origine de la biodiversité, par exemple en mettant en évidence des espèces cryptiques (des espèces qui se ressemblent tellement qu’elles étaient considérées comme une seule et même espèce), notamment chez les Insectes; en bouleversant les classifications traditionnelles; ou en remettant en cause le paradigme de l’isolement géographique dans la formation des espèces. Bien que la contribution de ces approches modernes soit notable, elles restent cependant très complexes et perfectibles. Plus important encore, il est essentiel aujourd’hui d’intégrer des experts de divers domaines pour appréhender sous divers angles et mieux comprendre cette biodiversité.