Publications

2011

• Zoom sur une représentation bidimensionnelle de l'ensemble de Mandelbrot, avec visualisation des arguments
  
  • Colonna Jean-François
  , 2011. Bidimensional zoom in on the Mandelbrot set with display of the arguments (Zoom sur une représentation bidimensionnelle de l'ensemble de Mandelbrot, avec visualisation des arguments)
• 5-étoile récursive
  
  • Colonna Jean-François
  , 2011. Recursive 5-star (5-étoile récursive)
• Pentagone récursif
  
  • Colonna Jean-François
  , 2011. Recursive pentagon (Pentagone récursif)
• Application of convex lexicographical optimization to the balance of GRTgaz gas grid
  
  • Bonnans Joseph Frederic
  • Paraisy Ruben
  • Veyrat Sébastien
  • Adam Soizic
  Journal of Global Optimization, Springer Verlag, 2011, 49 (3), pp.415--423.
• Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains
  
  • Abboud Toufic
  • Joly Patrick
  • Rodríguez Jerónimo
  • Terrasse Isabelle
  Journal of Computational Physics, Elsevier, 2011, 230 (15), pp.5877-5907. This work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities. To do so, we propose to combine a discretization based on a discontinuous Galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain. The coupling formula enforces a discrete energy identity ensuring the stability under the usual CFL condition in the interior. Moreover, the scheme allows to use a smaller time step in the interior domain yielding to quasi-optimal discretization parameters for both methods. The aliasing phenomena introduced by the local time stepping are reduced by a post-processing by averaging in time obtaining a stable and second order consistent (in time) coupling algorithm. We compute the numerical rate of convergence of the method for an academic problem. The numerical results show the feasibility of the whole discretization procedure. © 2011 Elsevier Inc. (10.1016/j.jcp.2011.03.062)
  DOI : 10.1016/j.jcp.2011.03.062
• Large time asymptotics for the Grinevich-Zakharov potentials
  
  • Kazeykina Anna
  • Novikov Roman
  Bulletin des Sciences Mathématiques, Elsevier, 2011, 135 (4), pp.374-382. In this article we show that the large time asymptotics for the Grinevich-Zakharov rational solutions of the Novikov-Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum of localized travel waves (solitons). (10.1016/j.bulsci.2011.02.003)
  DOI : 10.1016/j.bulsci.2011.02.003
• Absence of exponentially localized solitons for the Novikov--Veselov equation at negative energy
  
  • Kazeykina Anna
  • Novikov Roman
  Nonlinearity, IOP Publishing, 2011, 24, pp.1821-1830. We show that Novikov--Veselov equation (an analog of KdV in dimension 2 + 1) does not have exponentially localized solitons at negative energy. (10.1088/0951-7715/24/6/007)
  DOI : 10.1088/0951-7715/24/6/007
• New global stability estimates for the Gel'fand-Calderon inverse problem
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2011, 27 (1), pp.015001 (21pp). We prove new global stability estimates for the Gel'fand-Calderon inverse problem in 3D. For sufficiently regular potentials this result of the present work is a principal improvement of the result of [G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal. 27 (1988), 153-172]. (10.1088/0266-5611/27/1/015001)
  DOI : 10.1088/0266-5611/27/1/015001
• Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide
  
  • Arens Tilo
  • Gintides Drossos
  • Lechleiter Armin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2011, 71 (3), pp.753--772. (10.1137/100806333)
  DOI : 10.1137/100806333
• The singular values and vectors of low rank perturbations of large rectangular random matrices
  
  • Benaych-Georges Florent
  • Rao Nadakuditi Raj
  , 2011. In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate projections of the corresponding singular vectors of the perturbed matrix. As in the prequel, where we considered the eigenvalue aspect of the problem, the non-random limiting value is shown to depend explicitly on the limiting singular value distribution of the unperturbed matrix via an integral transforms that linearizes rectangular additive convolution in free probability theory. The large matrix limit of the extreme singular values of the perturbed matrix differs from that of the original matrix if and only if the singular values of the perturbing matrix are above a certain critical threshold which depends on this same aforementioned integral transform. We examine the consequence of this singular value phase transition on the associated left and right singular eigenvectors and discuss the finite $n$ fluctuations above these non-random limits.
• Les outils stochastiques des marchés financiers
  
  • El Karoui Nicole
  • Gobet Emmanuel
  , 2011, pp.238. Depuis 40 ans, les outils mathématiques probabilistes ont montré leur rôle central dans le développement d’outils d’aide à la décision pour les marchés financiers. Ils offrent un cadre méthodologique robuste de modélisation et calcul des risques associés aux produits dérivés, ces fameux instruments financiers qui dépendent de manière plus ou moins complexe d’autres produits financiers plus simples (actions, indices, taux de change, taux d’intérêt, matières premières ...). Cet ouvrage se veut être une introduction aux outils stochastiques de la finance de marché, et à leurs utilisations dans la gestion dynamique des produits dérivés. Pour le développement des outils probabilistes du calcul stochastique, nous suivons une approche élémentaire à la Föllmer, qui permettra à un lecteur ayant juste des bases de probabilité de rentrer plus facilement dans le sujet. Pour autant, cette grande simplification permet de traiter de manière complète des applications aux options (simples ou exotiques) sur actions, à la modélisation des taux d’intérêt ou du risque de crédit. À travers l’expérience de la crise financière actuelle, nous expliquons l’importance des hypothèses sous-tendant l’utilisation de ces outils en salle de marché.
• An adaptive high-gain observer for wastewater treatment systems
  
  • Lafont Frédéric
  • Busvelle Eric
  • Gauthier Jean-Paul
  Journal of Process Control, Elsevier, 2011, 21, pp.893-900.
• Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh
  
  • Allaire Grégoire
  • Dapogny Charles
  • Frey Pascal
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2011, 349 (17-18), pp.999--1003. We propose a method for structural optimization that relies on two alternative descriptions of shapes: on the one hand, they are exactly meshed so that mechanical evaluations by finite elements are accurate; on the other hand, we resort to a level-set characterization to describe their deformation along the shape gradient. The key ingredient is a meshing algorithm for building a mesh, suitable for numerical computations, out of a piecewise linear level-set function on an unstructured mesh. Therefore, our approach is at the same time a geometric optimization method (since shapes are exactly meshed) and a topology optimization method (since the topology of successive shapes can change thanks to the power of the level-set method).
• On adaptive stratification
  
  • Etoré Pierre
  • Fort Gersende
  • Jourdain Benjamin
  • Moulines Éric
  Annals of Operations Research, Springer Verlag, 2011, 189 (1), pp.127-154. This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (\ie\ when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators. (10.1007/s10479-009-0638-9)
  DOI : 10.1007/s10479-009-0638-9
• Deterministic state constrained optimal control problems without controllability assumptions
  
  • Bokanowski Olivier
  • Forcadel Nicolas
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2011, 17 (4), pp.pp. 995-1015. In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position $x$. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems. (10.1051/cocv/2010030)
  DOI : 10.1051/cocv/2010030
• Electromagnetic Wave Scattering from Rough Penetrable Layers
  
  • Haddar Houssem
  • Lechleiter Armin
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2011, 43, pp.2418-2443. We consider scattering of time-harmonic electromagnetic waves from an unbounded penetrable dielectric layer mounted on a perfectly conducting infinite plate. This model describes for instance propagation of monochromatic light through dielectric photonic assemblies mounted on a metal plate. We give a variational formulation for the electromagnetic scattering problem in a suitable Sobolev space of functions defined in an unbounded domain containing the dielectric structure. Further, we derive a Rellich identity for a solution to the variational formulation. For simple material configurations and under suitable non-trapping and smoothness conditions, this integral identity allows to prove an a-priori estimate for such a solution. A-priori estimates for solutions to more complicated material configurations are then shown using a perturbation approach. While the estimates derived from the Rellich identity show that the electromagnetic rough surface scattering problem has at most one solution, a limiting absorption argument finally implies existence of a solution to the problem. (10.1137/100783613)
  DOI : 10.1137/100783613
• Branching Feller diffusion for cell division with parasite infection
  
  • Bansaye Vincent
  • Tran Viet Chi
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2011, 8, pp.95-127. We describe the evolution of the quantity of parasites in a population of cells which divide in continuous-time. The quantity of parasites in a cell follows a Feller diffusion, which is splitted randomly between the two daughter cells when a division occurs. The cell division rate may depend on the quantity of parasites inside the cell and we are interested in the cases of constant or monotone division rate. We first determine the asymptotic behavior of the quantity of parasites in a cell line, which follows a Feller diffusion with multiplicative jumps. We then consider the evolution of the infection of the cell population and give criteria to determine whether the proportion of infected cells goes to zero (recovery) or if a positive proportion of cells becomes largely infected (proliferation of parasites inside the cells).
• Absence of exponentially localized solitons for the Novikov-Veselov equation at positive energy
  
  • Novikov Roman
  Physics Letters A, Elsevier, 2011, 375, pp.1233-1235. In this note we show that the Novikov-Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) has no exponentially localized solitons ( in the two-dimensional sense).
• Weighted Radon transforms for which the Chang approximate inversion formula is precise
  
  • Novikov Roman
  Uspekhi Mat. Nauk, 2011, 66 (2), pp.237-238. We describe all weighted Radon transforms on the plane for which the Chang approximate inversion formula is precise. Some subsequent results, including the Cormack type inversion for these transforms, are also given.
• Differential games and Zubov's method
  
  • Grüne Lars
  • Serea Oana
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2011, 49 (9), pp.2349-2377. In this paper we provide generalizations of Zubov's equation to differential games without Isaacs' condition. We show that both generalizations of Zubov's equation (which we call min-max and max-min Zubov equation, respectively) possess unique viscosity solutions which characterize the respective controllability domains. As a consequence, we show that under the usual Isaacs condition the respective controllability domains as well as the local controllability assumptions coincide. (10.1137/100787829)
  DOI : 10.1137/100787829
• Adaptive High-Gain observers with an application to wastewater treatment plants
  
  • Methnani Salowa
  • Damak Tarak
  • Toumi Ahmed
  • Lafont Frédéric
  • Gauthier Jean-Paul
  , 2011. no abstract
• Sampling the Fermi statistics and other conditional product measures
  
  • Gaudilliere Alexandre
  • Reygner Julien
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2011, 47, pp.790-812. Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the temperature as well as the energy values and degeneracies of the energy levels we give an explicit upper bound with leading term km(ln k) for the mixing time of the dynamics. We obtain such construction and upper bound as a special case of a general result on (non-homogeneous) products of ultra log-concave measures (like binomial or Poisson laws) with a global constraint. As a consequence of this general result we also obtain a disorder-independent upper bound on the mixing time of a simple exclusion process on the complete graph with site disorder. This general result is based on an elementary coupling argument and extended to (non-homogeneous) products of log-concave measures.
• Control of the bilinear Schrödinger equation for fully coupling potentials
  
  • Caponigro Marco
  • Boscain Ugo
  • Chambrion Thomas
  • Sigalotti Mario
  , 2011, pp.to appear. We present a general result of approximate controllability for the bilinear Schrödinger equation (with wave function varying in the unit sphere of an infinite dimensional Hilbert space), under the hypothesis that the Schrödinger operator has discrete spectrum and that the control potential couples all eigenstates. The control method is based on a tracking procedure for the Galerkin approximations, lifted in SU(n). The method allows to estimate the L 1 norm of the control laws achieving controllability.
• Explicit characterization of the support of non-linear inclusions
  
  • Lechleiter Armin
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2011, 5 (3), pp.675--694. (10.3934/ipi.2011.5.675)
  DOI : 10.3934/ipi.2011.5.675
• Optimal Design of Low-contrast Two-phase Structures For the Wave Equation
  
  • Allaire Grégoire
  • Kelly Alex
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2011, 21 (7), pp.1499--1538. This paper is concerned with the following optimal design problem:¯nd the distribution of two phases in a given domain that minimizes an objective function evaluated through the solution of a wave equation. This type of optimization problem is known to be ill-posed in the sense that it generically does not admit a minimizer among classical admissible designs. Its relaxation could be found, in principle, through homogenization theory but, unfortunately, it is not always explicit, in particular for objective functions depending on the solution gradient. To circumvent this di±culty, we make the simplifying assumption that the two phases have a low constrast. Then, a second-order asymptotic expansion with respect to the small amplitude of the phase coe±cients yields a simpli¯ed optimal design problem which is amenable to relaxation by means of H-measures. We prove a general existence theorem in a larger class of composite materials and propose a numerical algorithm to compute minimizers in this context. As in the case of an elliptic state equation, the optimal composites are shown to be rank-one laminates. However, the proof that relaxation and small-amplitude limit commute is more delicate than in the elliptic case. (10.1142/S0218202511005477)
  DOI : 10.1142/S0218202511005477