Publications

2013

• Surface integral formulation of the interior transmission problem
  
  • Cossonnière Anne
  • Haddar Houssem
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2013, 25 (3), pp.341-376.. We consider a surface integral formulation of the so-called interior transmission problem that appears in the study of inverse scattering problems from dielectric inclusions. In the case where the magnetic permeability contrast is zero, the main originality of our approach consists in still using classical potentials for the Helmholtz equation but in weaker trace space solutions. One major outcome of this study is to establish Fredholm properties of the problem for relaxed assumptions on the material coefficients. For instance we allow the contrast to change sign inside the medium. We also show how one can retrieve discreteness results for transmission eigenvalues in some particular situations. (10.1216/JIE-2013-25-3-341)
  DOI : 10.1216/JIE-2013-25-3-341
• Time reversal in viscoelastic media
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Wahab Abdul
  European Journal of Applied Mathematics, Cambridge University Press (CUP), 2013, 24, pp.565-600. In this paper, we consider the problem of reconstructing sources in a homogeneous viscoelastic medium from wavefield measurements using time-reversal algorithms. Our motivation is the recent advances on hybrid methods in biomedical imaging. We first present a modified time-reversal imaging algorithm based on a weighted Helmholtz decomposition and justify mathematically that it provides a better approximation than by simply time reversing the displacement field. Then, we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imaging which corrects the attenuation effect at the first order.
• The Factorization method applied to cracks with impedance boundary conditions
  
  • Boukari Yosra
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2013, 7 (4). We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed fre- quency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks.
• Ecologie prédictive & changement planétaire
  
  • Austerlitz Frédéric
  • Blum Michael
  • Calba Sarah
  • Chave Jérôme
  • Choisy Marc
  • Coreau Audrey
  • Devictor Vincent
  • Doyen Luc
  • Dray Stéphane
  • Duputié Anne
  • Eveillard Damien
  • Faure Denis
  • Favier Charly
  • Gaggiotti Oscar
  • Galtier Nicolas
  • Garnier Éric
  • Gimenez Olivier
  • Guis Helene
  • Herbreteau Vincent
  • Huneman Philippe
  • Jabot Franck
  • Jarne Philippe
  • Joly Dominique
  • Julliard Romain
  • Kéfi Sonia
  • Kergoat Gael
  • Lacroix Gerard
  • Lagadeuc Yvan
  • Lavorel Sandra
  • Le Gaillard Jean-François
  • Le Gall Line
  • Loreau Michel
  • Maris Virginie
  • Morand Serge
  • Morin Xavier
  • Morlon Hélène
  • Mouquet Nicolas
  • Pinay Gilles
  • Pottier Julien
  • Pradel Roger
  • Ronce Ophélie
  • Schurr Frank
  • Simonet Pascal
  • Teplitsky Céline
  • Thuiller Wilfried
  • Tran Anne-Lise
  • Venner Samuel
  , 2013, hors s\'{e}rie, pp.9-44.
• An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations
  
  • Bokanowski Olivier
  • Garcke Jochen
  • Griebel Michael
  • Klompmaker Irene
  Journal of Scientific Computing, Springer Verlag, 2013, 55, pp.pp. 575-605. We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d = 8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem. (10.1007/s10915-012-9648-x)
  DOI : 10.1007/s10915-012-9648-x
• Tumor Growth Parameters Estimation and Source Localization From a Unique Time Point: Application to Low-grade Gliomas
  
  • Rekik Islem
  • Allassonnière Stéphanie
  • Clatz Olivier
  • Geremia Ezequiel
  • Stretton Erin
  • Delingette Hervé
  • Ayache Nicholas
  Computer Vision and Image Understanding, Elsevier, 2013, 117 (3), pp.238--249. Coupling time series of MR Images with reaction-di usion-based models has provided interesting ways to better understand the proliferative-invasive as- pect of glial cells in tumors. In this paper, we address a di erent formulation of the inverse problem: from a single time point image of a non-swollen brain tumor, estimate the tumor source location and the di usivity ratio between white and grey matter, while exploring the possibility to predict the further extent of the observed tumor at later time points in low-grade gliomas. The synthetic and clinical results show the stability of the located source and its varying distance from the tumor barycenter and how the estimated ratio controls the spikiness of the tumor. (10.1016/j.cviu.2012.11.001)
  DOI : 10.1016/j.cviu.2012.11.001
• 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
• 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
• 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.
• Lipschitz classification of almost-Riemannian distances on compact oriented surfaces
  
  • Boscain Ugo
  • Charlot Grégoire
  • Ghezzi Roberta
  • Sigalotti Mario
  The Journal of Geometric Analysis, Springer, 2013, 23, pp.438-455. Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot--Caratheodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyse the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labelled graph associated with it. (10.1007/s12220-011-9262-4)
  DOI : 10.1007/s12220-011-9262-4
• Shape dependent controllability of a quantum transistor
  
  • Méhats Florian
  • Privat Yannick
  • Sigalotti Mario
  , 2013, pp.1253-1258.
• 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.
• 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.
• 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
• 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.
• 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
• 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.
• 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
• 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
• 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
• 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).
• 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.
• 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.
• 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.