Publications

2013

• 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
• 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
• Statistics of animal movement
  
  • Berthelot Geoffroy C.B.
  • Bansaye Vincent
  • Calenge C.
  , 2013.
• A new non linear shell modeling combining flexural and membrane effects
  
  • Pantz Olivier
  • Trabelsi Karim
  , 2013.
• A conformal mapping method in inverse obstacle scattering
  
  • Haddar Houssem
  • Kress Rainer
  Complex Variables and Elliptic Equations, Taylor & Francis, 2013. Akduman, Haddar and Kress [1, 5, 11] have employed a conformal map- ping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for elec- trostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. We propose an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to ob- tain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We present the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples.
• Second order corrector in the homogenization of a conductive-radiative heat transfer problem
  
  • Allaire Grégoire
  • Habibi Zakaria
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18 (1), pp.1-36. This paper focuses on the contribution of the so-called second order corrector in periodic homogenization applied to a conductive-radiative heat transfer problem. More precisely, heat is diffusing in a periodically perforated domain with a non-local boundary condition modelling the radiative transfer in each hole. If the source term is a periodically oscillating function (which is the case in our application to nuclear reactor physics), a strong gradient of the temperature takes place in each periodicity cell, corresponding to a large heat flux between the sources and the perforations. This effect cannot be taken into account by the homogenized model, neither by the first order corrector. We show that this local gradient effect can be reproduced if the second order corrector is added to the reconstructed solution. (10.3934/dcdsb.2013.18.1)
  DOI : 10.3934/dcdsb.2013.18.1
• Bases Mathématiques de la théorie des jeux
  
  • Laraki Rida
  • Renault Jérôme
  • Sorin Sylvain
  , 2013, pp.186. Cet ouvrage est destiné aux étudiants en master ainsi qu'aux étudiants des écoles d'ingénieurs. Les connaissances mathématiques requises sont celles d'une licence scientifique. Ce cours est consacré à une présentation des principaux concepts et outils mathématiques de la théorie des jeux stratégiques. L'accent est mis sur l'exposé et les preuves des résultats fondamentaux (minmax et opérateur valeur, équilibre de Nash et corrélé). Par ailleurs certains développements récents sont présentés : variété des équilibres, dynamiques de sélection, apprentissage et jeux répétés. L'ouvrage comporte une importante section d'exercices et corrigés.
• Exponential instability in the inverse scattering problem on the energy interval
  
  • Isaev Mikhail
  Functional Analysis and Its Applications, Springer Verlag, 2013, 47 (3), pp.8. We consider the inverse scattering problem on the energy interval in three dimensions. We are focused on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows optimality of the logarithmic stability result of [Stefanov, 1990] (up to the value of the exponent).
• Optimisation of cancer drug treatments using cell population dynamics
  
  • Billy Frédérique
  • Clairambault Jean
  • Fercoq Olivier
  , 2013, pp.265. Cancer is primarily a disease of the physiological control on cell population proliferation. Tissue proliferation relies on the cell division cycle: one cell becomes two after a sequence of molecular events that are physiologically controlled at each step of the cycle at so-called checkpoints, in particular at transitions between phases of the cycle [105]. Tissue proliferation is the main physiological process occurring in development and later in maintaining the permanence of the organism in adults, at that late stage mainly in fast renewing tissues such as bone marrow, gut and skin. (10.1007/978-1-4614-4178-6_10)
  DOI : 10.1007/978-1-4614-4178-6_10
• Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy
  
  • Novikov Roman
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013, 21 (6), pp.813–823. We prove approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy with incomplete data in dimension d ≥ 2. Our estimates are given in uniform norm for coefficient difference and related stability precision efficiently increases with increasing energy and coefficient difference regularity. In addition, our estimates are rather optimal even in the Born approximation.
• Representation, relaxation and convexity for variational problems in Wiener spaces
  
  • Chambolle Antonin
  • Goldman Michael
  • Novaga Matteo
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2013. We show convexity of solutions to a class of convex variational problems in the Gauss and in the Wiener space. An important tool in the proof is a representation formula for integral functionals in this infinite dimensional setting, that extends analogous results valid in the classical Euclidean framework.
• A Global Steering Method for Nonholonomic Systems
  
  • Chitour Yacine
  • Jean Frédéric
  • Long Ruixing
  Journal of Differential Equations, Elsevier, 2013, 254, pp.1903-1956. In this paper, we present an iterative steering algorithm for nonholonomic systems (also called driftless control-affine systems) and we prove its global convergence under the sole assumption that the Lie Algebraic Rank Condition (LARC) holds true everywhere. That algorithm is an extension of the one introduced in [21] for regular systems. The first novelty here consists in the explicit algebraic construction, starting from the original control system, of a lifted control system which is regular. The second contribution of the paper is an exact motion planning method for nilpotent systems, which makes use of sinusoidal control laws and which is a generalization of the algorithm described in [29] for chained-form systems. (10.1016/j.jde.2012.11.012)
  DOI : 10.1016/j.jde.2012.11.012
• The topological derivative in anisotropic elasticity
  
  • Bonnet Marc
  • Delgado Gabriel
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2013, 66, pp.557-586. A comprehensive treatment of the topological derivative for anisotropic elasticity is presented, with both the background material and the trial small inhomogeneity assumed to have arbitrary anisotropic elastic properties. A formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are treated; they differ mostly on details in the elastic moment tensor (EMT). Moreover, the main properties of the EMT, a critical component of the topological derivative, are studied for the fully anisotropic case. Then, the topological derivative of strain energy-based quadratic cost functionals is derived, which requires a distinct treatment. Finally, numerical experiments on the numerical evaluation of the EMT and the topological derivative of the compliance cost functional are reported. (10.1093/qjmam/hbt018)
  DOI : 10.1093/qjmam/hbt018
• State-constrained Optimal Control Problems of Impulsive Differential Equations
  
  • Forcadel Nicolas
  • Rao Zhiping
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2013, 68, pp.1--19. The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption. (10.1007/s00245-013-9193-5)
  DOI : 10.1007/s00245-013-9193-5
• Modelling microstructure noise with mutually exciting point processes
  
  • Bacry Emmanuel
  • Delattre Sylvain
  • Hoffmann Marc
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2013, 13 (1), pp.65-77. We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point pro- cesses and relies on linear self and mutually exciting stochastic inten- sities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations. (10.1080/14697688.2011.647054)
  DOI : 10.1080/14697688.2011.647054
• Linearized Cauchy Data Inversion Method for Two-Dimensional Buried Target Imaging
  
  • Ozdemir Ozgur
  • Haddar Houssem
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2013, 61 (6). We propose a novel inversion algorithm to image buried objects in inhomogeneous media from the electromagnetic data on the outer boundary. Our method is based on exploiting the Cauchy data to derive a new Born-like linearization of the inverse problem. The main advantage of this formulation is to avoid the use of the background Green function and therefore is computationally more efficient. It also provides better accuracy than classical Born approximation. In the case of stratified media, our approach can be coupled with any appropriate continuation method. We discuss here the coupling with a continuation method based on the use of approximate transmission conditions. The feasibility and robustness of our methodology is validated through numerical experiments for single and multiple targets.
• Reconstruction of a potential from the impedance boundary map
  
  • Isaev Mikhail
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2013, 1 (1), pp.5-28. We give formulas and equations for finding generalized scattering data for the Schrödinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map.
• Central limit theorems for linear statistics of heavy tailed random matrices
  
  • Benaych-Georges Florent
  • Guionnet Alice
  • Male Camille
  , 2013. We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $\alpha$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.
• Is the Distance Geometry Problem in NP?
  
  • Beeker Nathanael
  • Gaubert Stéphane
  • Glusa Christian
  • Liberti Leo
  , 2013, pp.85-93. (10.1007/978-1-4614-5128-0_5)
  DOI : 10.1007/978-1-4614-5128-0_5
• Méthodes de Monte-Carlo et processus stochastiques
  
  • Gobet Emmanuel
  , 2013, pp.258. La méthode de Monte-Carlo, qui tire son nom du fameux casino à Monaco, s’est développée de manière spectaculaire depuis 60 ans : elle figure parmi les 10 algorithmes ayant eu le plus d’influence sur le développement et la pratique de la science et de l’ingénierie au xxe siècle. En fait, il n’existe pas une méthode de Monte-Carlo mais des méthodes de Monte-Carlo. La 1re partie de l’ouvrage dresse un panorama de l’existant, puis détaille les outils de base pour la simulation de variables aléatoires, les résultats de convergence les plus courants et les techniques d’accélération des méthodes de Monte-Carlo. Puis, la 2e partie aborde la simulation des équations différentielles stochastiques (processus à évolution linéaire dérivant du mouvement brownien), dont les applications en biologie, chimie, économie, finance, géophysique, mécanique des fluides, neuroscience etc. sont importantes. L’objectif principal est le calcul d’espérance de leurs trajectoires. Cela donne, via les formules de Feynman-Kac, des solutions probabilistes aux équations aux dérivées partielles : ce lien remarquable permet de résoudre, par simulations Monte-Carlo, ces équations en toute dimension. Enfin, la 3e partie, la plus originale, traite des processus stochastiques ayant des évolutions non-linéaires (modélisant des interactions variées), comme les équations du contrôle stochastique, les diffusions branchantes, les équations stochastiques de McKean-Vlasov, avec des applications fondamentales en plein développement. Nous présentons notamment quelques idées importantes d’apprentissage statistique, dont le couplage aux méthodes de Monte-Carlo (via les régressions empiriques) conduit à des algorithmes des plus performants. Dans cet ouvrage, nous mettons en avant les grands principes de simulation efficace, avec une présentation exigeant le moins de préalables mathématiques. Le niveau prérequis à la lecture de ce cours est celui de Master 1, ou 2e année d’école d’ingénieurs. Cet ouvrage intéressera aussi des étudiants plus avancés ou des enseignants-chercheurs, souhaitant dégager l’essentiel des outils sophistiqués pour la simulation de processus stochastiques linéaires et non-linéaires.
• Minimal external representations of tropical polyhedra
  
  • Allamigeon Xavier
  • Katz R.D.
  Journal of Combinatorial Theory, Series A, Elsevier, 2013, 120 (4), pp.907-940. (10.1016/j.jcta.2013.01.011)
  DOI : 10.1016/j.jcta.2013.01.011
• Faddeev eigenfunctions for multipoint potentials
  
  • Grinevich Piotr
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2013, 1 (2), pp.76-91. We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for multipoint potentials in two and three dimensions. For single point potentials in 3D such formulas were obtained in an old unpublished work of L.D. Faddeev. For single point potentials in 2D such formulas were given recently in [P.G. Grinevich, R.G. Novikov, Physics Letters A,376,(2012),1102-1106].
• Stochastic Simulation and Monte Carlo Methods. Mathematical Foundations of Stochastic Simulation.
  
  • Talay Denis
  • Graham Carl
  , 2013, 68, pp.268.
• Shape dependent controllability of a quantum transistor
  
  • Méhats Florian
  • Privat Yannick
  • Sigalotti Mario
  , 2013, pp.1253-1258.
• A decomposition technique for pursuit evasion games with many pursuers
  
  • Festa Adriano
  • Vinter Richard
  , 2013. Here we present a decomposition technique for a class of differential games. The technique consists in a decomposition of the target set which produces, for geometrical reasons, a decomposition in the dimensionality of the problem. Using some elements of Hamilton-Jacobi equations theory, we find a relation between the regularity of the solution and the possibility to decompose the problem. We use this technique to solve a pursuit evasion game with multiple agents.