Publications

2011

• 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).
• 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.
• 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
• 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
• 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
• 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.
• 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
• 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.
• Generalized impedance boundary conditions for thin dielectric coatings with variable thickness
  
  • Aslanyurek Birol
  • Haddar Houssem
  • Shahinturk Hulya
  Wave Motion, Elsevier, 2011, 48 (7), pp.681-700. (10.1016/j.wavemoti.2011.06.002)
  DOI : 10.1016/j.wavemoti.2011.06.002
• Polymorphic evolution sequence and evolutionary branching
  
  • Champagnat Nicolas
  • Méléard Sylvie
  Probability Theory and Related Fields, Springer Verlag, 2011, 151 (1-2), pp.45-94. We are interested in the study of models describing the evolution of a polymorphic population with mutation and selection in the specific scales of the biological framework of adaptive dynamics. The population size is assumed to be large and the mutation rate small. We prove that under a good combination of these two scales, the population process is approximated in the long time scale of mutations by a Markov pure jump process describing the successive trait equilibria of the population. This process, which generalizes the so-called trait substitution sequence, is called polymorphic evolution sequence. Then we introduce a scaling of the size of mutations and we study the polymorphic evolution sequence in the limit of small mutations. From this study in the neighborhood of evolutionary singularities, we obtain a full mathematical justification of a heuristic criterion for the phenomenon of evolutionary branching. To this end we finely analyze the asymptotic behavior of 3-dimensional competitive Lotka-Volterra systems. (10.1007/s00440-010-0292-9)
  DOI : 10.1007/s00440-010-0292-9
• Asymptotic behaviour of the number of the Eulerian circuits
  
  • Isaev Mikhail
  The Electronic Journal of Combinatorics, Open Journal Systems, 2011, 18 (1), pp.219. We determine the asymptotic behaviour of the number of the Eulerian circuits in undirected simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). We also prove some new properties of the Laplacian matrix.
• Exponential instability in the Gel'fand inverse problem on the energy intervals
  
  • Isaev Mikhail
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2011, 19 (3), pp.453-473. We consider the Gel'fand inverse problem and continue studies of [Mandache,2001]. We show that the Mandache-type instability remains valid even in the case of Dirichlet-to-Neumann map given on the energy intervals. These instability results show, in particular, that the logarithmic stability estimates of [Alessandrini,1988], [Novikov,Santacesaria,2010] and especially of [Novikov,2010] are optimal (up to the value of the exponent).
• A continuous semigroup of notions of independence between the classical and the free one
  
  • Benaych-Georges Florent
  • Lévy Thierry
  The Annals of Probability, Institute of Mathematical Statistics, 2011, 39 (3), pp.904-938. In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence. (10.1214/10-AOP573)
  DOI : 10.1214/10-AOP573
• A posteriori error estimates for the effective Hamiltonian of dislocation dynamics
  
  • Cacace Simone
  • Chambolle Antonin
  • Monneau Régis
  Numerische Mathematik, Springer Verlag, 2011, pp.55 p.. We study an implicit and discontinuous scheme for a non-local Hamilton-Jacobi equation modelling dislocation dynamics. For the evolution problem, we prove an a posteriori estimate of Crandall-Lions type for the error between continuous and discrete solutions. We deduce an a posteriori error estimate for the effective Hamiltonian associated to a stationary cell problem. In dimension one and under suitable assumptions, we also give improved a posteriori estimates. Numerical simulations are provided. (10.1007/s00211-011-0430-z)
  DOI : 10.1007/s00211-011-0430-z
• Stable reconstruction of generalized impedance boundary conditions
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2011, 27 (9), pp.095002. (10.1088/0266-5611/27/9/095002)
  DOI : 10.1088/0266-5611/27/9/095002
• Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces
  
  • Lefebvre-Lepot Aline
  • Maury Bertrand
  • Lefebvre Aline
  Advances in Mathematical Sciences and Applications, Gakkōtosho Co. Ltd, 2011, 21 (2), pp.535–557. We consider here a discrete system of spheres interacting through a lubrication force. This force is dissipative, and singular near contact: it behaves like the reciprocal of interparticle distance. We propose a macroscopic constitutive equation which is built as the natural continuous counterpart of this microscopic lubrication model. This model, which is of the newtonian type, relies on an elongational viscosity, which is proportional to the reciprocal of the local fluid fraction. We then establish the convergence in a weak sense of solutions to the discrete problem towards the solution to the partial differential equation which we identified as the macroscopic constitutive equation.
• Hedging and Vertical Integration in Electricity Markets
  
  • Aïd René
  • Chemla Gilles
  • Porchet Arnaud
  • Touzi Nizar
  Management Science, INFORMS, 2011, 57 (8). This paper analyzes the interactions between competitive (wholesale) spot, retail, and forward markets and vertical integration in electricity markets. We develop an equilibrium model with producers, retailers, and traders to study and quantify the impact of forward markets and vertical integration on prices, risk premia, and retail market shares. We point out that forward hedging and vertical integration are two separate mechanisms for demand and spot price risk diversification that both reduce the retail price and increase retail market shares. We show that they differ in their impact on prices and firms' utility because of the asymmetry between production and retail segments. Vertical integration restores the symmetry between producers' and retailers' exposure to demand risk, whereas linear forward contracts do not. Vertical integration is superior to forward hedging when retailers are highly risk averse. We illustrate our analysis with data from the French electricity market. (10.1287/mnsc.1110.1357)
  DOI : 10.1287/mnsc.1110.1357