Publications

2013

• A Formula for Popp’s Volume in Sub-Riemannian Geometry
  
  • Barilari Davide
  • Rizzi Luca
  Analysis and Geometry in Metric Spaces, Versita, 2013, 1. For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp's volume is essentially the unique volume with such a property. (10.2478/agms-2012-0004)
  DOI : 10.2478/agms-2012-0004
• Optimally swimming stokesian robots
  
  • Alouges François
  • Desimone Antonio
  • Heltai Luca
  • Lefebvre-Lepot Aline
  • Merlet Benoît
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18 (5), pp.1189-1215. We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow's theorem in an analytic framework, similar to what has been done in [3] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [3] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail. (10.3934/dcdsb.2013.18.1189)
  DOI : 10.3934/dcdsb.2013.18.1189
• Path Planning and Ground Control Station Simulator for UAV
  
  • Ajami Alain
  • Balmat Jean-François
  • Gauthier Jean-Paul
  • Maillot Thibault
  , 2013, pp.1-13. no abstract
• On the control of spin-boson systems
  
  • Boscain Ugo
  • Mason Paolo
  • Panati Gianluca
  • Sigalotti Mario
  , 2013, pp.2110-2115. (10.23919/ecc.2013.6669621)
  DOI : 10.23919/ecc.2013.6669621
• Numerical approximation of Nash equilibria for a class of non-cooperative differential games
  
  • Cacace Simone
  • Cristiani Emiliano
  • Falcone Maurizio
  , 2013, 16, pp.45-58. In this paper we propose a numerical method to obtain an approximation of Nash equilibria for multi-player non-cooperative games with a special structure. We consider the infinite horizon problem in a case which leads to a system of Hamilton-Jacobi equations. The numerical method is based on the Dynamic Programming Principle for every equation and on a global fixed point iteration. We present the numerical solutions of some two-player games in one and two dimensions. The paper has an experimental nature, but some features and properties of the approximation scheme are discussed.
• A general Hamilton-Jacobi framework for nonlinear state-constrained control problems
  
  • Altarovici Albert
  • Bokanowski Olivier
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (2), pp.337--357. The paper deals with deterministic optimal control problem with state constraints and non-linear dynamics. It is known for such a problem that the value function is in general discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumption, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypass the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach. (10.1051/cocv/2012011)
  DOI : 10.1051/cocv/2012011
• Stabilization of two-dimensional persistently excited linear control systems with arbitrary rate of convergence
  
  • Chitour Yacine
  • Mazanti Guilherme
  • Sigalotti Mario
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2013, 51 (2), pp.801-823. We study the control system $\dot x = A x + \alpha(t) b u$ where the pair $(A, b)$ is controllable, $x \in \mathbb R^2$, $u \in \mathbb R$ is a scalar control and the unknown signal $\alpha: \mathbb R_+ \to [0, 1]$ is $(T,\mu)$-persistently exciting (PE), i.e., there exists $T \geq \mu > 0$ such that, for all $t \in \mathbb R_+$, $\int_t^{t + T} \alpha(s) ds \geq \mu$. We are interested in the stabilization problem of this system by a linear state feedback $u = - K x$. In this paper, we positively answer a question asked in \cite{YacineMario} and prove the following: Assume that the class of $(T,\mu)$-PE signals is restricted to those which are $M$\nobreakdash-Lipschitzian, where $M>0$ is a positive constant. Then, given any $C>0$, there exists a linear state feedback $u = - K x$ where $K$ only depends on $(A,b)$ and $T,\mu,M$ so that, for every $M$-Lipschitzian $(T,\mu)$-PE signal, the rate of exponential decay of the time-varying system $\dot x = (A -\alpha(t) bK)x $ is greater than $C$. (10.1137/110848153)
  DOI : 10.1137/110848153
• Stability estimates for determination of potential from the impedance boundary map
  
  • Isaev Mikhail
  • Novikov Roman
  Algebra and Analysis, 2013, 25 (1), pp.37-63. We study the impedance boundary map (or Robin-to-Robin map) for the Schrodinger equation in open bounded demain at fixed energy in multidimensions. We give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map.
• A mesh evolution algorithm based on the level set method for geometry and topology optimization
  
  • Allaire Grégoire
  • Dapogny Charles
  • Frey Pascal
  Structural and Multidisciplinary Optimization, Springer Verlag, 2013, 48 (4), pp.711-715. An approach for structural optimization is proposed, which combines the versatility of the level set method for handling large deformations and topology changes with the accurate description of the geometry provided by an exact mesh of the shape. The key ingredients of this method are efficient algorithms for (i) moving a level set function on an unstructured mesh, (ii) remeshing the surface corresponding to the zero level set and (iii) simultaneously adaptating the volumic mesh which fits to this surfacic mesh. (10.1007/s00158-013-0929-2)
  DOI : 10.1007/s00158-013-0929-2
• New approximations in local volatility models
  
  • Gobet Emmanuel
  • Suleiman Ali
  , 2013, pp.305--330. For general time-dependent local volatility models, we propose new approximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well.
• On the Well-Posedness , Stability And Accuracy Of An Asymptotic Model For Thin Periodic Interfaces In Electromagnetic Scattering Problems
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We analyze the well-posedness and stability properties of a parameter dependent problem that models the reflection and transmission of electromagnetic waves at a thin and rapidly oscillating interface. The latter is modeled using approximate interface conditions that can be derived using asymptotic expansion of the exact solution with respect to the small parameter (proportional to the periodicity length of oscillations and the width of the interface). The obtained uniform stability results are then used to analyze the accuracy (with respect to the small parameter) of the proposed model.
• Coupling techniques for nonlinear hyperbolic equations. III. Well-balanced approximation of thick interfaces
  
  • Boutin Benjamin
  • Coquel Frédéric
  • LeFloch Philippe G.
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2013, 51 (2), pp.1108-1133. We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems, based on the so-called thin interface model, which uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation but introduce a new framework for coupling problems, now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe first that the Cauchy problem is well-posed. Our main achievements in the present paper are, on one hand, the design of a new well-balanced finite volume scheme which is adapted to the thick interface model and, on the other hand, a proof of the convergence of this scheme toward the unique entropy solution to the Cauchy problem for a large class of nonlinear hyperbolic equations. Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used instead. Our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the entropy dissipation of the proposed discrete scheme. (10.1137/120865768)
  DOI : 10.1137/120865768
• How pilots fly: An inverse optimal control problem approach
  
  • Maillot Thibault
  • Serres Ulysse
  • Gauthier Jean-Paul
  • Ajami Alain
  , 2013, pp.1792-1797. (10.1109/CDC.2013.6760142)
  DOI : 10.1109/CDC.2013.6760142
• Quantifying the Mutational Meltdown in Diploid Populations
  
  • Coron Camille
  • Méléard Sylvie
  • Porcher Emmanuelle
  • Robert Alexandre
  The American Naturalist, University of Chicago Press, 2013, 181 (5), pp.623-636. Mutational meltdown, in which demographic and genetic processes mutually reinforce one another to accelerate the extinction of small populations, has been poorly quantified despite its potential importance in conservation biology. Here we present a model-based framework to study and quantify the mutational meltdown in a finite diploid population that is evolving continuously in time and subject to resource competition. We model slightly deleterious mutations affecting the population demographic parameters and study how the rate of mutation fixation increases as the genetic load increases, a process that we investigate at two timescales: an ecological scale and a mutational scale. Unlike most previous studies, we treat population size as a random process in continuous time. We show that as deleterious mutations accumulate, the decrease in mean population size accelerates with time relative to a null model with a constant mean fixation time. We quantify this mutational meltdown via the change in the mean fixation time after each new mutation fixation, and we show that the meltdown appears less severe than predicted by earlier theoretical work. We also emphasize that mean population size alone can be a misleading index of the risk of population extinction, which could be better evaluated with additional information on demographic parameters. (10.1086/670022)
  DOI : 10.1086/670022
• Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations
  
  • Amirat Youcef
  • Hamdache Kamel
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 33 (8), pp.3289-3320. In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field. (10.3934/dcds.2013.33.3289)
  DOI : 10.3934/dcds.2013.33.3289
• Some limit theorems for Hawkes processes and application to financial statistics
  
  • Bacry Emmanuel
  • Delattre Sylvain
  • Hoffmann Marc
  • Muzy Jean-François
  Stochastic Processes and their Applications, Elsevier, 2013, 123 (7), pp.2475 - 2499. Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms. (10.1016/j.spa.2013.04.007)
  DOI : 10.1016/j.spa.2013.04.007
• Log-normal continuous cascade model of asset returns: aggregation properties and estimation
  
  • Bacry Emmanuel
  • Kozhemyak Alexey
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2013, 13 (5), pp.795-818. no abstract (10.1080/14697688.2011.647411)
  DOI : 10.1080/14697688.2011.647411
• The 2d-Directed Spanning Forest is almost surely a tree
  
  • Coupier David
  • Tran Viet Chi
  Random Structures and Algorithms, Wiley, 2013, 42 (1), pp.59-72. We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF. (10.1002/rsa.20400)
  DOI : 10.1002/rsa.20400
• A geometric approach for convexity in some variational problem in the Gauss space
  
  • Goldman Michael
  Rendiconti del Seminario Matematico della Università di Padova, University of Padua / European Mathematical Society, 2013. In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
• A numerical method for kinetic equations with discontinuous equations : application to mathematical modeling of cell dynamics
  
  • Aymard Benjamin
  • Clément Frédérique
  • Coquel Frédéric
  • Postel Marie
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2013, 35 (6), pp.27 pages. Abstract: In this work, we propose a numerical method to handle discontinuous fluxes arising in transport-like equations. More precisely, we study hyperbolic PDEs with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous. A dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces. The validation of the method is done on 1D and 2D toy problems for which exact solutions are available, allowing us to do a thorough convergence study. We then apply the method to a biological model focusing on complex cell dynamics, that initially motivated this study, and illustrates the full potentialities of the scheme. (10.1137/120904238)
  DOI : 10.1137/120904238
• A new non linear shell modeling combining flexural and membrane effects
  
  • Pantz Olivier
  • Trabelsi Karim
  , 2013.
• An improved time domain linear sampling method for Robin and Neumann obstacles
  
  • Haddar Houssem
  • Lechleiter Armin
  • Marmorat Simon
  Applicable Analysis, Taylor & Francis, 2013, pp.1-22. We consider inverse obstacle scattering problems for the wave equation with Robin or Neu- mann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular so- lutions to the wave equation. We analyze this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation. (10.1080/00036811.2013.772583)
  DOI : 10.1080/00036811.2013.772583
• Statistical models for deformable templates in image and shape analysis (Modèles statistiques d'atlas déformables pour l'analyse d'images et de formes)
  
  • Allassonnière Stéphanie
  • Bigot Jérémie
  • Glaunès Joan Alexis
  • Maire Florian
  • Richard Frederic J.P.
  Annales Mathématiques Blaise Pascal, Université Blaise-Pascal - Clermont-Ferrand, 2013, 20 (1), pp.1-35. Les données de grande dimensions sont de plus en plus fréquemment collectées dans de nombreux domaines d'application. Il devient alors particulièrement important d'être capable d'extraire des caractéristiques significatives de ces bases de données. Le modèle d'atlas déformable (Deformable template model) est un outil maintenant répandu pour atteindre ce but. Cet article présente un panorama des aspects statistiques de ce modèle ainsi que ses généralisations. Nous décrivons les différents cadres mathématiques permettant de prendre en compte des types variés de données et de déformations. Nous rappelons les propriétés théoriques de convergence des estimateurs et des algorithmes permettant l'estimation de ces caractéristiques. Nous terminons cet article par la présentation de quelques résultats publiés utilisant des données réelles. (10.5802/ambp.320)
  DOI : 10.5802/ambp.320
• Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Jing Wenjia
  • Kang Hyeonbae
  • Wahab Abdul
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2013, 6 (4), pp.2174 - 2212. (10.1137/120899303)
  DOI : 10.1137/120899303
• Statistics of animal movement
  
  • Berthelot Geoffroy C.B.
  • Bansaye Vincent
  • Calenge C.
  , 2013.