Publications

2008

• Stability of quasi-static crack evolution through dimensional reduction
  
  • Babadjian Jean-François
  , 2010, 21, pp.1-13. This paper deals with quasi-static crack growth in thin films. We show that, when the thickness of the film tends to zero, any three-dimensional quasi-static crack evolution converges to a two-dimensional one, in a sense related to the Gamma-convergence of the associated total energy. We extend the prior analysis of [2] by adding conservative body and surface forces which allow us to remove the boundedness assumption on the deformation field (10.1007/978-90-481-9195-6_1)
  DOI : 10.1007/978-90-481-9195-6_1
• L'ensemble de Julia dans le corps des quaternions calcule pour A=(0,1,0,0) -ou 'la danseuse d'Yr'
  
  • Colonna Jean-François
  , 2008. The quaternionic Julia set computed with A=(0,1,0,0) (L'ensemble de Julia dans le corps des quaternions calcule pour A=(0,1,0,0) -ou 'la danseuse d'Yr'-)
• Criterion for Bose-Einstein condensation in traps and self-bound systems
  
  • Yamada T.
  • Funaki Y.
  • Horiuchi H.
  • Röpke G.
  • Schuck P.
  • Tohsaki A.
  Physical Review A : Atomic, molecular, and optical physics [1990-2015], American Physical Society, 2008, 78 (3), pp.035603. The internal one-particle density matrix is discussed for Bose-Einstein condensates with finite number of particles in a harmonic trap. The outcome of the digonalization of the density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii-type internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi-type internal density matrix leads to an ideal condensate. We give a criterion for the choice of the internal coordinates: In the macroscopic limit the internal density matrix should have eigenvalues and eigenfunctions of an ideal Bose-Einstein condensate, this being a very physical condition for cases where the system is also an ideal Bose condensation in the laboratory frame. One choice fulfilling this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass do not satisfy the condition. Based on our criterion, a general definition of the internal one-particle density matrix is presented in a self-bound system, consisting of interacting bosons. (10.1103/PhysRevA.78.035603)
  DOI : 10.1103/PhysRevA.78.035603
• On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions
  
  • Benaych-Georges Florent
  , 2008. n this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko-Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.
• Noeuds '7-trefle' toriques
  
  • Colonna Jean-François
  , 2008. 7-foil torus knots (Noeuds '7-trefle' toriques)
• Texture semi-fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2008. texture semi-fractale tridimensionnelle
• Continuité en topologie symplectique.
  
  • Humiliere Vincent
  , 2008. Dans cette thèse, nous étudions divers problèmes issus de la topologie symplectique où la topologie C° intervient. Nous étudions diverses complétions de l'espace des applications hamiltoniennes, puis appliquons cette étude aux équations d'Hamilton-Jacobi. Nous abordons ensuite le problème de l'extension du morphisme de Calabi à des groupes d'homéomorphismes. Enfin, nous nous intéressons à la rigidité C° du crochet de Poisson et à l'extension au cadre C° de la notion de représentation hamiltonienne.
• Optimal lifetime consumption and investment under a drawdown constraint
  
  • Elie Romuald
  • Touzi Nizar
  Finance and Stochastics, Springer Verlag (Germany), 2008, 12 (3), pp.299 - 330. We consider the infinite-horizon optimal consumption-investment problem under a drawdown constraint, i.e., when the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the with constant coefficients. For a general class of utility functions, we provide the value function in explicit form and derive closed-form expressions for the optimal consumption and investment strategy.
• On the Infrared Problem for the Dressed Non-Relativistic Electron in a Magnetic Field
  
  • Amour Laurent
  • Faupin Jérémy
  • Grebert Benoit
  • Guillot Jean-Claude
  , 2009, pp.1-24. We consider a non-relativistic electron interacting with a classical magnetic field pointing along the $x_3$-axis and with a quantized electromagnetic field. The system is translation invariant in the $x_3$-direction and we consider the reduced Hamiltonian $H(P_3)$ associated with the total momentum $P_3$ along the $x_3$-axis. For a fixed momentum $P_3$ sufficiently small, we prove that $H(P_3)$ has a ground state in the Fock representation if and only if $E'(P_3)=0$, where $P_3 \mapsto E'(P_3)$ is the derivative of the map $P_3 \mapsto E(P_3) = \inf \sigma (H(P_3))$. If $E'(P_3) \neq 0$, we obtain the existence of a ground state in a non-Fock representation. This result holds for sufficiently small values of the coupling constant.
• Règles de décision pour la gestion du risque : Application à la gestion hebdomadaire de la production électrique.
  
  • Apparigliato Romain
  , 2008. Nous étudions dans cette thèse le problème de gestion du risque physique en production électrique à l'horizon hebdomadaire. Dans un premier temps, nous nous intéressons à l'intégration de l'aléa d'apport hydraulique dans la gestion locale d'une vallée hydraulique. Cette approche est menée à l'aide de l'optimisation robuste et de règles de décision linéaires. Les résultats de multiples modes de simulation montrent que ces approches permettent de réduire notoirement les déversés en comparaison des modèles déterministes appliqués en exploitation, moyennant une faible augmentation du coût. La deuxième problématique traitée est la gestion active de la marge de production, définie comme l'écart entre l'offre totale et la demande totale, compte tenu des aléas affectant le système électrique. Il s'agit de déterminer quelles décisions optimales prendre, selon un certain critère économique, pour se couvrir contre un risque trop élevé de non-satisfaction de la demande dans au moins 99% des situations. Pour cela, une formulation inédite en boucle ouverte, basée sur le processus stochastique de marge de production et des contraintes en probabilité est proposée. Pour les besoins de cette formulation, nous générons des scénarios à l'aide de techniques plus réalistes qu'en exploitation. Enfin, une résolution moins anticipative est étudiée en appliquant l'heuristique «Programmation Stochastique avec Règles de Décision Constantes par Morceaux» introduite par Thénié et Vial. Les premiers résultats sont très encourageants en comparaison avec les modèles en boucle ouverte.
• Mathematical modeling in elastography.
  
  • Kozhemyak Anastasia
  , 2008. The advent of advanced imaging has significantly improved the quality of medical supervision of patients. The terms non-invasive imaging allows physicians to make more accurate diagnoses and earlier and prescribing methods of treatment more efficient and fairer. Multiple imaging modalities are currently employed or are under study. In this thesis, we study three techniques emerging biomedical imaging: imaging • Magneto-acoustic; • Thermographic imaging; • endotomographie by electrical impedance. For each of these three techniques, we propose mathematical models and we present new methods of reconstruction in medical imaging.
• Uncovering latent singularities from multifractal scaling laws in mixed asymptotic regime. Application to turbulence
  
  • Muzy J.-F.
  • Bacry Emmanuel
  • Baile Rachel
  • Poggi Philippe
  EPL - Europhysics Letters, European Physical Society / EDP Sciences / Società Italiana di Fisica / IOP Publishing, 2008, pp.82, 60007. In this paper we revisit an idea originally proposed by Mandelbrot about the possibility to observe "negative dimensions" in random multifractals. For that purpose, we define a new way to study scaling where the observation scale l and the total sample length L are, respectively, going to zero and to infinity. This "mixed" asymptotic regime is parametrized by an exponent χ that corresponds to Mandelbrot "supersampling exponent". In order to study the scaling exponents in the mixed regime, we use a formalism introduced in the context of the physics of disordered systems relying upon traveling wave solutions of some non-linear iteration equation. Within our approach, we show that for random multiplicative cascade models, the parameter χ can be interpreted as a negative dimension and, as anticipated by Mandelbrot, allows one to uncover the "hidden" negative part of the singularity spectrum, corresponding to "latent" singularities. We illustrate our purpose on synthetic cascade models. When applied to turbulence data, this formalism allows us to distinguish two popular phenomenological models of dissipation intermittency: We show that the mixed scaling exponents agree with a log-normal model and not with log-Poisson statistics. (10.1209/0295-5075/82/60007)
  DOI : 10.1209/0295-5075/82/60007
• On the imaging of two-dimensional thin inclusions by a MUSIC-type algorithm from boundary measurements
  
  • Park Won-Kwang
  • Ammari Habib
  • Lesselier Dominique
  , 2008, pp.Session 7 Inverse Problems and Imaging. A technique for reconstructing penetrable single thin, curve-like smooth inclusion in a homogeneous medium is proposed. We present an asymptotic formula for the scattering amplitude by plane waves to build a MUSIC-type algorithm. This algorithm operated at a fixed nonzero frequency yields the shape of the inclusion from scattered far-fields in addition to estimates of the length of the supporting curve. Numerical experiments in both cases demonstrated how the proposed technique behaves.
• Numerical Simulation with Finite Volume of Dynamic Liquid-Vapor Phase Transition
  
  • Faccanoni Gloria
  • Kokh Samuel
  • Allaire Grégoire
  , 2008, pp.391-398. The present work investigates the simulation of phase transition in compressible fluids. We postulate a local and instantaneous equilibrium with respect to phasic pressures, temperatures and chemical potentials when both phases are present. This hypothesis leads to the definition of an equilibrium equation of state (EOS) for the two-phase medium. While this thermodynamical assumption is classical, there is no explicit expression of the equilibrium EOS in the general case. We propose simple means to approximate this EOS when both phases are governed by a "Stiffened Gas" EOS. Then we provide an implementation of this method within a Finite-Volume numerical scheme thanks to a two-step relaxation strategy.
• Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework
  
  • Benabid Anas
  • Bensusan Harry
  • El Karoui Nicole
  , 2008. In this paper, a study of a stochastic volatility model for asset pricing is described. Originally presented by J. Da Fonseca, M. Grasselli and C. Tebaldi, the Wishart volatility model identifies the volatility of the asset as the trace of a Wishart process. Contrary to a classic multifactor Heston model, this model allows to add degrees of freedom with regard to the stochastic correlation. Thanks to its flexibility, this model enables a better fit of market data than the Heston model. Besides, the Wishart volatility model keeps a clear interpretation of its parameters and conserves an efficient tractability. Firstly, we recall the Wishart volatility model and we present a Monte Carlo simulation method in sight of the evaluation of complex options. Regarding stochastic volatility models, implied volatility surfaces of vanilla options have to be obtained for a short time. The aim of this article is to provide an accurate approximation method to deal with asymptotic smiles and to apply this procedure to the Wishart volatility model in order to well understand it and to make its calibration easier. Inspired by the singular perturbations method introduced by J.P Fouque, G. Papanicolaou, R. Sircar and K. Solna, we suggest an efficient procedure of perturbation for affine models that provides an approximation of the asymptotic smile (for short maturities and for a two-scale volatility). Thanks to the affine properties of the Wishart volatility model, the perturbation of the Riccati equations furnishes the expected approximations. The convergence and the robustness of the procedure are analyzed in practice but not in theory. The resulting approximations allow a study of the parameters influence and can also be used as a calibration tool for a range of parameters.
• SCALES: an environmental model to measure effects of spatial and inter-annual agricultural landscape evolution on the soil erosion sensitivity. A case study in bocage area (Basse-Normandie, France).
  
  • Le Gouée Patrick
  • Marie Maxime
  • Delahaye Daniel
  • Bermond Michaël
  • Douvinet Johnny
  , 2008, pp.67-68. En Basse-Normandie, l'érosion hydrique fragilise depuis plus de 30 ans la fertilité des sols cultivés et la qualité de la ressource en eau. La lutte contre l'érosion des sols, engagée par les acteurs et gestionnaires locaux, nécessite de disposer préalablement d'un diagnostic d'échelle fine de l'aléa érosion des sols. Expérimenté à l'échelle du Calvados (5500 km²), le modèle SCALES (Le Gouée and Delahaye, 2007) a pour vocation de devenir un outil de grande précision dans l'identification et la localisation des parcelles agricoles potentiellement sensibles à l'érosion. La détermination de l'aléa érosion repose sur des données d'entrée caractérisant les 4 facteurs agissant sur l'érosion des sols (Wischmeier and Smith, 1978). L'estimation de l'aléa repose tout d'abord sur la conversion des données d'entrée en niveaux de pression (0 : pas de pression, 5 très forte pression). Les niveaux des données d'entrée sont ensuite combinés selon une démarche additive. Cette démarche est également appliquée entre niveaux combinés et niveaux de données d'entrée afin de mesurer la Sensibilité Potentielle des Milieux (SPM) à l'érosion mais aussi entre niveaux combinés pour générer les valeurs de l'aléa érosion. Le modèle montre que deux situations conduisent à l'absence d'aléa 0 : lorsque les pentes sont inférieures à 1% (niveau 0) et lorsque les terres sont occupées par des prairies permanentes (niveau 0 des pratiques agricoles). La structuration hiérarchique du modèle SCALES souligne un classement des données d'entrée qui repose à la fois sur des positions scientifiques connues, sur des dires d'expert et sur des données empiriques recueillies pendant 2 ans sur le terrain concernant la genèse du ruissellement érosif. L'une des spécificités du modèle tient au choix de l'unité spatiale d'intégration des données : l'îlot PAC ou la parcelle agricole. La prise en compte par le modèle de l'organisation parcellaire de l'espace agricole et de son évolution inter-annuelle (rotations culturales, gestion des intercultures) permet de mesurer l'impact des changements d'état rapides des paysages agricoles sur la sensibilité des milieux à l'érosion des sols (Mignolet et al., 2003). En outre, l'intégration dans SCALES des dynamiques agricoles et paysagères actuelles conduit à développer des simulations soulignant les conséquences environnementales positives ou négatives des changements attendus de l'organisation des paysages agricoles. Pour illustrer ces deux aspects du modèle SCALES, nous proposons une étude de cas dans un espace bocager en pleine recomposition agricole et paysagère (bocage virois). En effet, depuis le milieu des années 80, les trajectoires d'évolution des exploitations agricoles de cette zone sont caractérisées par un mouvement d'agrandissement foncier et une forte intensification fourragère. Ce mouvement se traduit par un recul important des prairies permanentes en grande partie compensée par une augmentation des surfaces en maïs-fourrage et en cultures de vente (céréales et colza). Un travail d'enquête de terrain exhaustive auprès des agriculteurs a permis de comprendre la structuration spatiale des territoires d'exploitation et l'organisation des pratiques agricoles à différentes échelles spatiales et temporelles. Ainsi, en accord avec Renard (1972) et Soulard et al.,(2004), nous observons que les déterminants de la mobilité spatiale des fonctionnalités productives dans l'espace parcellaire local s'expliquent par la structuration de la mosaïque des territoires d'exploitation liée à la taille, au morcellement parcellaire, à la distance des parcelles au siège d'exploitation. L'objectif de la communication est de démontrer la plus-value apportée par notre modèle à propos de l'évaluation actuelle et prospective des impacts environnementaux de l'organisation spatiale et de l'évolution inter-annuelle de paysages agricoles contrôlés par ces déterminants à partir de l'étude de cas d'un bassin versant du bocage virois.
• Approximate conditions replacing thin layers
  
  • Poignard Clair
  • Dular Patrick
  • Perrussel Ronan
  • Krähenbühl Laurent
  • Nicolas Laurent
  • Schatzman Michelle
  IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 2008, 44 (6), pp.1154 -- 1157. We provide a rigorous asymptotic method to build approximate conditions equivalent to a thin layer. With these conditions the electromagnetic field may be computed in domains such as a biological cell, without meshing the membrane. The influence of the membrane is replaced by an appropriate condition on the inner domain, while in the thin layer, the approximate field is explicitly known. We give error estimates, which validate our asymptotic method, and we present a few numerical results performed with the finite element method. (10.1109/TMAG.2007.916154)
  DOI : 10.1109/TMAG.2007.916154
• Texture semi-fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2008. Texture semi-fractale tridimensionnelle
• Direct elastic imaging of a small inclusion
  
  • Ammari Habib
  • Calmon Pierre
  • Iakovleva Ekaterina
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2008, 1 (2), pp.169-187. In this paper we consider the problem of locating a small three-dimensional elastic inclusion, using arrays of elastic source transmitters and receivers. This procedure yields the multi-static response matrix that is characteristic of the elastic inclusion. We show how the eigenvalue structure of this matrix can be employed within the framework of a non-iterative method of MUSIC (standing for MUltiple Signal Classification)-type in order to retrieve the elastic inclusion. We illustrate our reconstruction procedure with a variety of computational examples from synthetic, noiseless, and severely noisy data. (10.1137/070696076)
  DOI : 10.1137/070696076
• On the existence of some ARCH($\infty$) processes
  
  • Douc Randal
  • Roueff François
  • Soulier Philippe
  Stochastic Processes and their Applications, Elsevier, 2008, 118 (5), pp.755-761. A new sufficient condition for the existence of a stationary causal solution of an ARCH($\infty$) equation is provided. This condition allows to consider polynomially decaying coefficients, so that it can be applied to the so-called FIGARCH processes, whose existence is thus proved. (10.1016/j.spa.2007.06.002)
  DOI : 10.1016/j.spa.2007.06.002
• Crack initiation in brittle materials.
  
  • Chambolle Antonin
  • Giacomini Alessandro
  • Ponsiglione Marcello
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2008, 188 (2), pp.309-349. In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith's type energy. We prove that, during a load process through a time dependent boundary datum of the type t.g(x) and in absence of strong singularities (this is the case of homogeneous isotropic materials) the crack initiation is brutal, i.e., a big crack appears after a positive time. On the contrary, in presence of a point of strong singularity, a crack will depart from that point at the initial time of loading and with zero velocity. We prove these facts (largely expected by the experts of material science) for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for a variant of the Mumford Shah functional. We prove that if the uncracked configuration relative to a boundary displacement has at most uniformly weak singularities, then configurations with cracks short enough have a higher energy. (10.1007/s00205-007-0080-6)
  DOI : 10.1007/s00205-007-0080-6
• LARGE DEVIATIONS FOR RANDOM SPECTRAL MEASURES AND SUM RULES
  
  • Gamboa Fabrice
  • Rouault Alain
  , 2008. We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\beta$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit.
• The allelic partition for coalescent point processes
  
  • Lambert Amaury
  , 2008. Assume that individuals alive at time $t$ in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of $n$ DNA sequences belonging to distinct individuals, we consider the number $S_n$ of polymorphic sites (sites at which at least two sequences differ), and the number $A_n$ of distinct haplotypes (sequences differing at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the mutation pattern associated to coalescent point processes under this assumption. Here, $S_n$ and $A_n$ grow linearly as $n$ grows, with explicit rate. However, when the branch lengths have infinite expectation, $S_n$ grows more rapidly, e.g. as $n \ln(n)$ for critical birth--death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by $k$ individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth--death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher logarithmic series.
• La cite et les astres (un hommage a Arthur C. Clarke)
  
  • Colonna Jean-François
  , 2008. The City and the Stars (a Tribute to Arthur C. Clarke) (La cite et les astres (un hommage a Arthur C. Clarke))
• La cité et les astres (un hommage à Arthur C. Clarke)
  
  • Colonna Jean-François
  , 2008. The City and the Stars (a Tribute to Arthur C. Clarke) (La cite et les astres (un hommage à Arthur C. Clarke))