Publications

2014

• Numerical study of a macroscopic finite pulse model of the diffusion MRI signal
  
  • Li Jing-Rebecca
  • Nguyen Hang Tuan
  • Nguyen Dang Van
  • Haddar Houssem
  • Coatléven Julien
  • Le Bihan Denis
  Journal of Magnetic Resonance, Elsevier, 2014, pp.54–65. Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. The dMRI signal from a heterogeneous sample includes the contribution of the water proton magnetization from all spatial positions in a voxel. If the voxel can be spatially divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be approximated using the macroscopic Karger model, which is a system of coupled ordinary differential equations (ODEs), under the assumption that the duration of the diffusion-encoding gradient pulses is short compared to the diffusion time (the narrow pulse assumption). \soutnew{Recently, a new macroscopic ODE model of the dMRI signal, the Finite Pulse ODE (FP-ODE) model, was derived from the Bloch-Torrey partial differential equation (PDE), without the narrow pulse restriction, using periodic homogenization techniques.}{Recently, a new macroscopic model of the dMRI signal, without the narrow pulse restriction, was derived from the Bloch-Torrey partial differential equation (PDE) using periodic homogenization techniques.} \soutnew{When restricted to narrow pulses, the FP-ODE model has the same form as the Karger model.}{When restricted to narrow pulses, this new homogenized model has the same form as the Karger model.} We conduct a numerical study of the \soutnew{FP-ODE}{new homogenized} model for voxels that are made up of periodic copies of a representative volume that contains spherical and cylindrical cells of various sizes and orientations and show that the signal predicted by the \soutnew{FP-ODE}{new} model approaches the reference signal obtained by solving the full Bloch-Torrey PDE in $O(\veps^2)$, where $\veps$ is the ratio between the size of the representative volume and \soutnew{the diffusion displacement}{a measure of the diffusion length}. When the narrow gradient pulse assumption is not satisfied, the \soutnew{FP-ODE}{new homogenized} model offers a much better approximation of the full PDE signal than the Karger model. Finally, preliminary results of applying the \soutnew{FP-ODE}{new} model to a voxel that is not made up of periodic copies of a representative volume are shown and discussed. (10.1016/j.jmr.2014.09.004)
  DOI : 10.1016/j.jmr.2014.09.004
• Faster Speciation and Reduced Extinction in the Tropics Contribute to the Mammalian Latitudinal Diversity Gradient
  
  • Rolland Jonathan
  • Condamine Fabien L.
  • Jiguet Frederic
  • Morlon Hélène
  PLoS Biology, Public Library of Science, 2014, 12 (1), pp.e1001775. The increase in species richness from the poles to the tropics, referred to as the latitudinal diversity gradient, is one of the most ubiquitous biodiversity patterns in the natural world. Although understanding how rates of speciation and extinction vary with latitude is central to explaining this pattern, such analyses have been impeded by the difficulty of estimating diversification rates associated with specific geographic locations. Here, we use a powerful phylogenetic approach and a nearly complete phylogeny of mammals to estimate speciation, extinction, and dispersal rates associated with the tropical and temperate biomes. Overall, speciation rates are higher, and extinction rates lower, in the tropics than in temperate regions. The diversity of the eight most species-rich mammalian orders (covering 92% of all mammals) peaks in the tropics, except that of the Lagomorpha (hares, rabbits, and pikas) reaching a maxima in northern-temperate regions. Latitudinal patterns in diversification rates are strikingly consistent with these diversity patterns, with peaks in species richness associated with low extinction rates (Primates and Lagomorpha), high speciation rates (Diprotodontia, Artiodactyla, and Soricomorpha), or both (Chiroptera and Rodentia). Rates of range expansion were typically higher from the tropics to the temperate regions than in the other direction, supporting the ''out of the tropics'' hypothesis whereby species originate in the tropics and disperse into higher latitudes. Overall, these results suggest that differences in diversification rates have played a major role in shaping the modern latitudinal diversity gradient in mammals, and illustrate the usefulness of recently developed phylogenetic approaches for understanding this famous yet mysterious pattern. (10.1371/journal.pbio.1001775)
  DOI : 10.1371/journal.pbio.1001775
• Growth rates for persistently excited linear systems
  
  • Chitour Yacine
  • Colonius Fritz
  • Sigalotti Mario
  Mathematics of Control, Signals, and Systems, Springer Verlag, 2014, 26 (4), pp.589-616. We consider a family of linear control systems $\dot{x}=Ax+\alpha Bu$ where $\alpha$ belongs to a given class of persistently exciting signals. We seek maximal $\alpha$-uniform stabilisation and destabilisation by means of linear feedbacks $u=Kx$. We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if the pair $(A,B)$ verifies a certain Lie bracket generating condition, then the maximal rate of convergence of $(A,B)$ is equal to the maximal rate of divergence of $(-A,-B)$. We also provide more precise results in the general single-input case, where the above result is obtained under the sole assumption of controllability of the pair $(A,B)$. (10.1007/s00498-014-0131-0)
  DOI : 10.1007/s00498-014-0131-0
• Optimal feedback control of undamped wave equations by solving a HJB equation
  
  • Kröner Axel
  • Kunisch Karl
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 21 (2), pp.442 - 464. An optimal fi nite-time horizon feedback control problem for (semi linear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on nite elements lead to approximated problems governed by ODEs in high dimensional space which makes infeasible the numerical resolution by HJB approach. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The e ffect of noise is considered and numerical simulations are presented to show the relevance of the approach. (10.1051/cocv/2014033)
  DOI : 10.1051/cocv/2014033
• Image Reconstruction Via Non-Isotropic Diffusion in Dubins/Reed-Shepp- Like Control Systems
  
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Prandi Dario
  • Remizov Alexey
  , 2014.
• A macroscopic model including membrane exchange for diffusion MRI
  
  • Coatléven Julien
  • Haddar Houssem
  • Li Jing-Rebecca
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2014, 2, pp.516-546.. Diffusion Magnetic Resonance Imaging (dMRI) is a promising tool to obtain useful infor- mation on microscopic structure and has been extensively applied to biological tissues. We establish a new macroscopic model from homogenization theory to obtain the aggregate dMRI signal measured in practice in the case of intermediate water exchange across cellular membranes. Based on a particular scaling of the permeability condition modeling cellular membranes, this model accurately reproduces the memory effects observed in practice. Explicit formulae given by homogenization for the coeffcients of this model emphasize their link to the relevant physiological quantities, and the inverse problem of retrieving these coefficients from a realistic set of measurements is considered. (10.1137/130914255)
  DOI : 10.1137/130914255
• Riemann--Hilbert problem approach for two-dimensional flow inverse scattering
  
  • Agaltsov Alexey
  • Novikov Roman
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2014, 55 (10), pp.103502. We consider inverse scattering for the time-harmonic wave equation with first-order perturbation in two dimensions. This problem arises in particular in the acoustic tomography of moving fluid. We consider linearized and nonlinearized reconstruction algorithms for this problem of inverse scattering. Our nonlinearized reconstruction algorithm is based on the non-local Riemann--Hilbert problem approach. Comparisons with preceding results are given.
• 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.
• 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.
• 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
• 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.
• 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
• 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
• 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
• 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