Publications

2013

• 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.
• 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.
• 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.
• 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
• 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
• 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.
• 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
• Stochastic Simulation and Monte Carlo Methods. Mathematical Foundations of Stochastic Simulation.
  
  • Talay Denis
  • Graham Carl
  , 2013, 68, pp.268.
• 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.
• Shape dependent controllability of a quantum transistor
  
  • Méhats Florian
  • Privat Yannick
  • Sigalotti Mario
  , 2013, pp.1253-1258.
• 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].
• 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
• 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
• 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
• Stabilization of persistently excited linear systems
  
  • Chitour Yacine
  • Mazanti Guilherme
  • Sigalotti Mario
  , 2013, pp.85-120. This chapter presents recent developments on the stabilization of persistently excited linear systems. The first section of the chapter deals with finite-dimensional systems and gives two main results on stabilization, concerning neutrally stable systems and systems whose eigenvalues all have non-positive real parts. It also presents a result stating the existence of persistently excited systems for which the pair (A, b) is controllable but that cannot be stabilized by means of a linear state feedback. The second section presents some results for infinite-dimensional systems to the case of systems defined by a linear operator A which generates a strongly continuous contraction semigroup, with applications to Schrödinger's equation and the wave equation. The final section discusses some problems that remain open, giving some preliminary results in certain cases. (10.1002/9781118639856.ch4)
  DOI : 10.1002/9781118639856.ch4
• 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
• 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.
• 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
• 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
• On the class of graphs with strong mixing properties
  
  • Isaev Mikhail
  • Isaeva K.V
  Proceeding of MIPT, 2013, 5 (6), pp.44-54. We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph.
• 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