Publications

2013

• On the asymptotics of a Robin eigenvalue problem
  
  • Cakoni Fioralba
  • Chaulet Nicolas
  • Haddar Houssem
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2013, 351, pp.517-521. The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to −∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter. (10.1016/j.crma.2013.07.022)
  DOI : 10.1016/j.crma.2013.07.022
• Daphnias: from the individual based model to the large population equation
  
  • Metz J.A.J.
  • Tran Viet Chi
  Journal of Mathematical Biology, Springer, 2013, 66 (4-5), pp.915--933. The class of deterministic 'Daphnia' models treated by Diekmann et~al. (J Math Biol 61: 277--318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114--135, 1983) and Diekmann et~al. (Nieuw Archief voor Wiskunde 4: 82--109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et~al., l.c.). (10.1007/s00285-012-0619-5)
  DOI : 10.1007/s00285-012-0619-5
• Time-reversal in visco-elastic 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. 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, where artifacts due to the coupling betwe en the pressure and shear waves appear. Then, we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imagin g which corrects the attenuation effect at the first order. The results of this paper yie ld the fundamental tools for solving imaging problems in elastic media using cross correl ation techniques
• On the extinction of Continuous State Branching Processes with catastrophes
  
  • Bansaye Vincent
  • Pardo Millan Juan Carlos
  • Smadi Charline
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.1-31. We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection. (10.1214/EJP.v18-2774)
  DOI : 10.1214/EJP.v18-2774
• Asymptotic enumeration of Eulerian orientations for graphs with strong mixing properties
  
  • Isaev Mikhail
  • Kseniia Isaeva
  Journal of Applied and Industrial Mathematics / Sibirskii Zhurnal Industrial'noi Matematiki and Diskretnyi Analiz i Issledovanie Operatsii, MAIK Nauka/Interperiodica, 2013, 20 (6), pp.40-58. We prove an asymptotic formula for the number of Eulerian orientations for graphs with strong mixing properties and with vertices having even degrees. The exact value is determined up to the multiplicative error O(n^{-1+epsilon}), where n is the number of vertices.
• 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
• 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.
• 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.
• 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
• 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).
• 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.
• 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
• 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.
• 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.
• 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
• 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
• 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.
• 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
• 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.
• Actuator and sensor fault detection, isolation and identification in nonlinear dynamical systems, with an application to a waste water treatment plant
  
  • Methnani Salowa
  • Lafont Frédéric
  • Gauthier Jean-Paul
  • Damak Tarak
  • Toumi Ahmed
  , 2013. no abstract
• Mathématiques: l'explosion continue
  
  • Anantharaman Nalini
  • de Bouard Anne
  • Lagoutière Frédéric
  • Gegout-Petit Anne
  • Ollivier Yann
  • Santambrogio Filippo
  • Bardet Jean-Marc
  , 2013, pp.1-180.
• Transmission eigenvalues
  
  • Cakoni Fioralba
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2013, 29 (10), pp.100201. In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission eigenvalue problem. The need to answer these questions became important after a series of papers by Cakoni et al [5], and Cakoni et al [6] suggesting that these transmission eigenvalues could be used to obtain qualitative information about the material properties of the scattering object from far-field data. The first answer to the existence of transmission eigenvalues in the general case was given in 2008 when Päivärinta and Sylvester showed the existence of transmission eigenvalues for the index of refraction sufficiently large [7] followed in 2010 by the paper of Cakoni et al who removed the size restriction on the index of refraction [8]. More importantly, in the latter it was shown that transmission eigenvalues yielded qualitative information on the material properties of the scattering object and Cakoni et al established in [9] that transmission eigenvalues could be determined from the Tikhonov regularized solution of the far-field equation. Since the appearance of these papers there has been an explosion of interest in the transmission eigenvalue problem (we refer the reader to our recent survey paper [10] for a detailed account of the developments in this field up to 2012) and the papers in this special issue are representative of the myriad directions that this research has taken. Indeed, we are happy to see that many open theoretical and numerical questions raised in [10] have been answered (totally or partially) in the contributions of this special issue: the existence of transmission eigenvalues with minimal assumptions on the contrast, the numerical evaluation of transmission eigenvalues, the inverse spectral problem, applications to non-destructive testing, etc. In addition to these topics, many other new investigations and research directions have been proposed as we shall see in the brief content summary below. A number of papers in this special issue are concerned with the question of existence of transmission eigenvalues and the structure of the associated transmission eigenfunctions. The three papers by respectively Robbiano [11], Blasten and Päivärinta [12], and Lakshtanov and Vainberg [13] provide new complementary results on the existence of transmission eigenvalues for the scalar problem under weak assumptions on the (possibly complex valued) refractive index that mainly stipulates that the contrast does not change sign on the boundary. It is interesting here to see three different new methods to obtain these results. On the other hand, the paper by Bonnet-Ben Dhia and Chesnel [14] addresses the Fredholm properties of the interior transmission problem when the contrast changes sign on the boundary, exhibiting cases where this property fails. Using more standard approaches, the existence and structure of transmission eigenvalues are analyzed in the paper by Delbary [15] for the case of frequency dependent materials in the context of Maxwell's equations, whereas the paper by Vesalainen [16] initiates the study of the transmission eigenvalue problem in unbounded domains by considering the transmission eigenvalues for Schrödinger equation with non-compactly supported potential. The paper by Monk and Selgas [17] addresses the case where the dielectric is mounted on a perfect conductor and provides some numerical examples of the localization of associated eigenvalues using the linear sampling method. A series of papers then addresses the question of localization of transmission eigenvalues and the associated inverse spectral problem for spherically stratified media. More specifically, the paper by Colton and Leung [18] provides new results on complex transmission eigenvalues and a new proof for uniqueness of a solution to the inverse spectral problem, whereas the paper by Sylvester [19] provides sharp results on how to locate all the transmission eigenvalues associated with angular independent eigenfunctions when the index of refraction is constant. The paper by Gintides and Pallikarakis [20] investigates an iterative least square method to identify the spherically stratified index of refraction from transmission eigenvalues. On the characterization of transmission eigenvalues in terms of far-field measurements, a promising new result is obtained by Kirsch and Lechleiter [21] showing how one can identify the transmission eigenvalues using the eigenvalues of the scattering operator which are available in terms of measured scattering data. In the paper by Kleefeld [22], an accurate method for computing transmission eigenvalues based on a surface integral formulation of the interior transmission problem and numerical methods for nonlinear eigenvalue problems is proposed and numerically validated for the scalar problem in three dimensions. On the other hand, the paper by Sun and Xu [23] investigates the computation of transmission eigenvalues for Maxwell's equations using a standard iterative method associated with a variational formulation of the interior transmission problem with an emphasis on the effect of anisotropy on transmission eigenvalues. From the perspective of using transmission eigenvalues in non-destructive testing, the paper by Cakoni and Moskow [24] investigates the asymptotic behavior of transmission eigenvalues with respect to small inhomogeneities. The paper by Nakamura and Wang [25] investigates the linear sampling method for the time dependent heat equation and analyses the interior transmission problem associated with this equation. Finally, in the paper by Finch and Hickmann [26], the spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. We hope that this collection of papers will stimulate further research in the rapidly growing area of transmission eigenvalues and inverse scattering theory. (10.1088/0266-5611/29/10/100201)
  DOI : 10.1088/0266-5611/29/10/100201
• First and second order optimality conditions for optimal control problems of state constrained integral equations
  
  • Bonnans J. Frédéric
  • de La Vega Constanza
  • Dupuis Xavier
  Journal of Optimization Theory and Applications, Springer Verlag, 2013, 159 (1), pp.1-40. This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient conditions. (10.1007/s10957-013-0299-3)
  DOI : 10.1007/s10957-013-0299-3
• Perron--Frobenius theorem for nonnegative multilinear forms and extensions
  
  • Friedland S.
  • Gaubert Stéphane
  • Han L.
  Linear Algebra and its Applications, Elsevier, 2013, 438 (2), pp.738-749. We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector. (10.1016/j.laa.2011.02.042)
  DOI : 10.1016/j.laa.2011.02.042
• 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.