Publications

2014

• Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations
  
  • Rao Zhiping
  • Siconolfi Antonio
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2014, 257 (11), pp.3978--4014. We establish a comparison principle for a Hamilton-Jacobi-Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of "-partition and minimal "-partition for intervals of definition of an integral trajectory. (10.1016/j.jde.2014.07.015)
  DOI : 10.1016/j.jde.2014.07.015
• Geometric Control Theory and sub-Riemannian Geometry
  
  • Stefani Gianna
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Sarychev Andrey
  • Sigalotti Mario
  , 2014, pp.372. This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.
• The $\Gamma$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension
  
  • Bellettini Giovanni
  • Chambolle Antonin
  • Goldman Michael
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014. In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\Gamma$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a ''cohesive'' energy, that is, whose cost depends on the actual opening of the discontinuity.
• On certain hyperelliptic signals that are natural controls for nonholonomic motion planning
  
  • Gauthier Jean-Paul
  • Monroy-Perez Felipe
  , 2014. In this paper we address the general problem of approximating, in a certain optimal way, non admissible motions of a kinematic system with nonholonomic constraints. Since this kind of problems falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics. We consider sys-tems modeled by a subriemannian Goursat structure, a particular case being the well known system of a car with trailers, along with the associated parallel parking problem. Several authors approximate the successive Lie brackets by using trigonometric functions. By contrast, we show that the more natural op-timal motions are related with closed hyperelliptic plane curves with a certain number of loops.
• A combination of algebraic, geometric and numerical methods in the contrast problem by saturation in magnetic resonance imaging
  
  • Bonnard Bernard
  • Claeys Mathieu
  • Cots Olivier
  • Jacquemard Alain
  • Martinon Pierre
  , 2014. In this article, the contrast imaging problem by saturation in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. The optimal solution can be found as an extremal solution of the Maximum Principle and analyzed with the recent advanced techniques of geometric optimal control. This leads to a numerical investigation based on shooting and continuation methods implemented in the HamPath software. The results are compared with a direct approach to the optimization problem and implemented within the Bocop toolbox. In complement lmi techniques are used to estimate a global optimum. It is completed with the analysis of the saturation problem of an ensemble of spin particles to deal with magnetic fields inhomogeneities.
• Almost sure optimal hedging strategy
  
  • Gobet Emmanuel
  • Landon Nicolas
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2014, 24 (4), pp.1652--1690. In this work, we study the optimal discretization error of stochastic integrals, in the context of the hedging error.
• Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds
  
  • Ghezzi Roberta
  • Jean Frédéric
  , 2014, 5, pp.201-218. (10.1007/978-3-319-02132-4_13)
  DOI : 10.1007/978-3-319-02132-4_13
• Avis en réponse à la saisine du 7 novembre 2013, de Madame Marie-Christine Blandin, relative à l’article de Snell et al. (Food and Chemical Toxicology, 2012)
  
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie Anne M. A.
  • Bellivier Florence
  • Berny Philippe
  • Bertheau Yves
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Coléno François
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Eychenne Nathalie
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Jestin André
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie V.
  • Lemaire Olivier O.
  • Lereclus Didier
  • Maximilien Rémi
  • Meurs Eliane
  • Moreau de Bellaing Cédric
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Parzy Daniel
  • Regnault-Roger Catherine
  • Renard Michel
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2014. Le Haut Conseil des biotechnologies (HCB) a été saisi le 7 novembre 2013 par Madame la Sénatrice Marie-Christine Blandin, en vertu de l’article L531-3 du code de l’environnement, d’une demande d’avis relative à l’article de Snell et al., intitulé «Assessment of the health impact of GM plant diets in long-term and multigenerational animal feeding trials: A literature review», publié dans la revue Food and Chemical Toxicology (Snellet al.,2012). Pour répondre aux questions de la saisine, le Comité Scientifique (CS) du HCB a constitué un groupe de travail ad hoc. A la suite du compte-rendu de ce dernier, le CS du HCB a procédé à l’examen du projet de réponse le 25 février 2014 sous la présidence de Jean-Christophe Pagès.
• Tropical Cramer Determinants Revisited
  
  • Akian Marianne
  • Gaubert Stéphane
  • Guterman Alexander
  , 2014, 616, pp.45. We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are enriched with an information of multiplicity, sign, or argument. We obtain existence or uniqueness results, which extend or refine earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss-Seidel type algorithms to solve linear systems in suitably extended tropical semirings.
• Two properties of two-velocity two-pressure models for two-phase flows
  
  • Coquel Frédéric
  • Hérard Jean-Marc
  • Saleh Khaled
  • Seguin Nicolas
  Communications in Mathematical Sciences, International Press, 2014, 12 (3). We study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form.
• Inversion of weighted Radon transforms via finite Fourier series weight approximations
  
  • Guillement Jean-Pol
  • Novikov Roman
  Inverse Problems in Science and Engineering, Taylor & Francis, 2014, 22 (5), pp.787–802. We consider weighted Radon transforms on the plane. We show that the Chang approximate inversion formula for these transforms admits a principal refinement as inversion via finite Fourier series weight approximations. We illustrate this inversion approach by numerical examples for the case of the attenuated Radon transforms in the framework of the single-photon emission computed tomography (SPECT).
• Optimization of joint p-variations of Brownian semimartingales
  
  • Gobet Emmanuel
  • Landon Nicolas
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2014, 19 (none). We study the optimization of the joint $(p^Y,p^Z)-$variations of two continuous semimartingales $(Y,Z)$ driven by the same Itô process $X$. The $p$-variations are defined on random grids made of finitely many stopping times. We establish an explicit asymptotic lower bound for our criterion, valid in rather great generality on the grids, and we exhibit minimizing sequences of hitting time form. The asymptotics is such that the spatial increments of $X$ and the number of grid points are suitably converging to 0 and $+\infty$ respectively. (10.1214/ECP.v19-2975)
  DOI : 10.1214/ECP.v19-2975
• Optimal control of leukemic cell population dynamics
  
  • Dupuis Xavier
  Mathematical Modelling of Natural Phenomena, EDP Sciences, 2014, 9 (1), pp.4-26. We are interested in optimizing the co-administration of two drugs for some acute myeloid leukemias (AML), and we are looking for in vitro protocols as a first step. This issue can be formulated as an optimal control problem. The dynamics of leukemic cell populations in culture is given by age-structured partial differential equations, which can be reduced to a system of delay differential equations, and where the controls represent the action of the drugs. The objective function relies on eigenelements of the uncontrolled model and on general relative entropy, with the idea to maximize the efficiency of the protocols. The constraints take into account the toxicity of the drugs. We present in this paper the modeling aspects, as well as theoretical and numerical results on the optimal control problem that we get. (10.1051/mmnp/20149102)
  DOI : 10.1051/mmnp/20149102
• Local properties of almost-Riemannian structures in dimension 3
  
  • Boscain Ugo
  • Charlot Grégoire
  • Gaye Moussa
  • Mason Paolo
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2014, 35 (9). A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres. We finally give some preliminary results about heat diffusion on such manifolds.
• Complexity in control-affine systems
  
  • Jean Frédéric
  • Prandi Dario
  , 2014. We will consider affine-control systems, i.e., systems in the form _ q(t) = f0(q(t)) + Xm i=1 ui (t)fi (q(t)) Here, the point q belongs to a smooth manifold M the fi 's are smooth vector fields on M u 2 L1([0;T];Rm) This type of system appears in many applications Mechanical systems Quantum control Microswimmers (Tucsnak, Alouges) Neuro-geometry of vision (Mumfor, Petitot)
• A linearized approach to worst-case design in parametric and geometric shape optimization
  
  • Allaire Grégoire
  • Dapogny Charles
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014, 24 (11), pp.2199-2257. The purpose of this article is to propose a deterministic method for optimizing a structure with respect to its worst possible behavior when a 'small' uncertainty exists over some of its features. The main idea of the method is to linearize the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, which can be rewritten as a more 'classical' function of the design, owing to standard adjoint techniques from optimal control theory. The resulting 'linearized worst-case' objective function turns out to be the sum of the initial cost function and of a norm of an adjoint state function, which is dual with respect to the considered norm over perturbations. This formal approach is very general, and can be justified in some special cases. In particular, it allows to address several problems of considerable importance in both parametric and shape optimization of elastic structures, in a unified framework. (10.1142/S0218202514500195)
  DOI : 10.1142/S0218202514500195
• A Robust Entropy-Satisfying Finite Volume Scheme for the Isentropic Baer-Nunziato Model
  
  • Coquel Frédéric
  • Hérard Jean-Marc
  • Saleh Khaled
  • Seguin Nicolas
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2014, 48 (1), pp.165-206. We construct an approximate Riemann solver for the isentropic Baer-Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions. (10.1051/m2an/2013101)
  DOI : 10.1051/m2an/2013101
• Convexities on ordered structures have their Krein--Milman theorem
  
  • Poncet Paul
  Journal of Convex Analysis, Heldermann, 2014, 21 (1), pp.89--120. We show analogues of the classical Krein-Milman theorem for several ordered algebraic structures, especially in a semilattice (non-linear) framework. In that case, subsemilattices are seen as convex subsets, and for our proofs we use arguments from continuous lattice theory and abstract convexity theory.
• 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
• VWAP execution and guaranteed VWAP
  
  • Guéant Olivier
  • Guillaume Royer
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2014, 5 (1), pp.445-471. If optimal liquidation using VWAP strategies has been considered in the literature, it has never been considered in the presence of permanent market impact and only rarely with execution costs. Moreover, only VWAP strategies have been studied and no pricing of guaranteed VWAP contract is provided. In this article, we develop a model to price guaranteed VWAP contracts in the most general framework for market impact. Numerical applications are also provided. (10.1137/130924676)
  DOI : 10.1137/130924676
• 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
• 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