Publications

2010

• Imaging of periodic dielectrics
  
  • Lechleiter Armin
  BIT Numerical Mathematics, Springer Verlag, 2010, 50 (1), pp.59--83. (10.1007/s10543-010-0255-7)
  DOI : 10.1007/s10543-010-0255-7
• The step-harmonic potential
  
  • Rizzi Luca
  • Piattella Oliver
  • Cacciatori Sergio
  • Gorini Vittorio
  American Journal of Physics, American Association of Physics Teachers, 2010, pp.19. We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We then consider the propagation of a wave packet reflected by the harmonic barrier and obtain an expression for the interaction time as a function of the peak energy. For high energies we recover the classical half-period limit. (10.1119/1.3379290)
  DOI : 10.1119/1.3379290
• Homogenization of nonlinear reaction-diffusion equation with a large reaction term
  
  • Allaire Grégoire
  • Piatnitski Andrey
  Annali dell'Universita di Ferrara, Springer Verlag, 2010, 56, pp.141-161.
• Construction of minimization sequences for shape optimization
  
  • Trabelsi Karim
  • Pantz Olivier
  , 2010, pp.278 -283. (10.1109/MMAR.2010.5587222)
  DOI : 10.1109/MMAR.2010.5587222
• Les agriculteurs entre clôtures et passerelles
  
  • Dubuisson-Quellier Sophie
  • Giraud Christophe
  , 2010, pp.111-129. Les mondes agricoles ont été longtemps caractérisés dans les représentations savantes ou communes par une certaine clôture sociale. Le groupe socioprofessionnel des agriculteurs était considéré comme l'un de ceux dont la reproduction s'appuie le plus sur l'héritage (encore aujourd'hui 85% des agriculteurs ont un père agriculteur) et sur l'homogamie (87% des conjointes d'agriculteurs en 1959 avaient une origine agricole). Aujourd'hui, ces mondes agricoles évoluent, sous l'effet d'une porosité plus grande avec d'autres mondes du travail mais aussi d'une plus grande sensibilité aux débats contemporains.
• Modeling and Simulation of Nucleate Boiling
  
  • Faccanoni Gloria
  • Kokh Samuel
  • Allaire Grégoire
  , 2012, pp.49-53. This work investigates the modelization and simulation of liquid-vapor phase change in compressible flows. Each phase is modeled as a compressible fluid equipped with its own Equation of State (EOS). We suppose that inter-phase equilibrium processes in the medium operate at a short timescale compared to the other physical phenomena such as convection or thermal diffusion. This assumption provides an implicit definition of an equilibrium EOS for the two-phase medium. Within this framework, mass transfer is the result of local and instantaneous equilibria between both phases. The overall model is strictly hyperbolic. We examine properties of the equilibrium EOS and we propose a discretization strategy based on a Finite-Volume relaxation method. This method allows to cope with the implicit definition of the equilibrium EOS, even when the model involves complex EOSs for the pure phases, including tabulated ones. We present two-dimensional numerical simulations that shows that the model is able to reproduce mechanism such as nucleation.
• COMPETITIVE OR WEAK COOPERATIVE STOCHASTIC LOTKA-VOLTERRA SYSTEMS CONDITIONED TO NON-EXTINCTION
  
  • Cattiaux Patrick
  • Méléard Sylvie
  Journal of Mathematical Biology, Springer, 2010, 60 (6), pp.797-829. We are interested in the long time behavior of a two-type density-dependent biological population conditioned to non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka-Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned to non extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a $d$-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species. (10.1007/s00285-009-0285-4)
  DOI : 10.1007/s00285-009-0285-4
• High-order accurate thin layer approximations for time-domain electromagnetics, Part II: transmission layers
  
  • Chun Sun
  • Haddar Houssem
  • Hesthaven Jan S.
  Journal of Computational and Applied Mathematics, Elsevier, 2010, 234 (8), pp.2587--2608. (10.1016/j.cam.2010.03.022)
  DOI : 10.1016/j.cam.2010.03.022
• Homogenization approach to the dispersion theory for reactive transport through porous media
  
  • Allaire Grégoire
  • Mikelic Andro
  • Piatnitski Andrey
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2010, 42 (1), pp.125-144.
• An efficient data structure to solve front propagation problems
  
  • Bokanowski Olivier
  • Cristiani Emiliano
  • Zidani Hasnaa
  Journal of Scientific Computing, Springer Verlag, 2010, 42 (2), pp.251--273. In this paper we develop a general efficient sparse storage technique suitable to coding front evolutions in d>= 2 space dimensions. This technique is mainly applied here to deal with deterministic target problems with constraints, and solve the associated minimal time problems. To this end we consider an Hamilton-Jacobi-Bellman equation and use an adapted anti-diffusive Ultra-Bee scheme. We obtain a general method which is faster than a full storage technique. We show that we can compute problems that are out of reach by full storage techniques (because of memory). Numerical experiments are provided in dimension d=2,3,4. (10.1007/s10915-009-9329-6)
  DOI : 10.1007/s10915-009-9329-6
• Uniform estimates for metastable transition times in a coupled bistable system
  
  • Barret Florent
  • Bovier Anton
  • Méléard Sylvie
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2010, 15 (12). We consider a coupled bistable N-particle system driven by a Brownian noise, with a strong coupling corresponding to the synchronised regime. Our aim is to obtain sharp estimates on the metastable transition times between the two stable states, both for fixed N and in the limit when N tends to infinity, with error estimates uniform in N. These estimates are a main step towards a rigorous understanding of the metastable behavior of infinite dimensional systems, such as the stochastically perturbed Ginzburg-Landau equation. Our results are based on the potential theoretic approach to metastability.
• The nonlinear Schrödinger equation with white noise dispersion
  
  • de Bouard Anne
  • Debussche Arnaud
  Journal of Functional Analysis, Elsevier, 2010, 259 (5), pp.1300-1321. Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of this work is to prove that this latter equation is globally well posed in $L^2$ or $H^1$. The main ingredient is the generalization of the classical Strichartz estimates. Additionally, we justify rigorously the formal limit described above. (10.1016/j.jfa.2010.04.002)
  DOI : 10.1016/j.jfa.2010.04.002
• A variational method for wave scattering from penetrable rough layers
  
  • Lechleiter Armin
  • Ritterbusch Sebastian
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2010, 75 (3), pp.366--391. (10.1093/imamat/hxp040)
  DOI : 10.1093/imamat/hxp040
• Progress on the strong Eshelby's conjecture and extremal structures for the elastic moment tensor
  
  • Ammari Habib
  • Capdeboscq Yves
  • Kang Hyeonbae
  • Lee Hyundae
  • Milton Graeme W.
  • Zribi Habib
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2010, 94 (1), pp.93--106. (10.1016/j.matpur.2010.01.003)
  DOI : 10.1016/j.matpur.2010.01.003
• Identification of generalized impedance boundary conditions in inverse scattering problems
  
  • Bourgeois Laurent
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2010, 4 (1), pp.19-38. In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact subset of a finite dimensional space. (10.3934/ipi.2010.4.19)
  DOI : 10.3934/ipi.2010.4.19
• Numerical Algorithms for Perspective Shape from Shading
  
  • Breuss Michael
  • Cristiani Emiliano
  • Durou Jean-Denis
  • Falcone Maurizio
  • Vogel Oliver
  Kybernetika, Institute of Information Theory and Automation, 2010, 46, pp.207--225. The Shape-From-Shading (SFS) problem is a fundamental and classic problem in computer vision. It amounts to compute the 3-D depth of objects in a single given 2-D image.This is done by exploiting information about the illumination and the image brightness.We deal with a recent model for Perspective SFS (PSFS) for Lambertian surfaces. It is defined by a Hamilton–Jacobi equation and complemented by state constraints boundary conditions. In this paper we investigate and compare three state-of-the-art numerical approaches. We begin with a presentation of the methods. Then we discuss the use of some acceleration techniques, including cascading multigrid, for all the tested algorithms. The main goal of our paper is to analyze and compare recent solvers for the PSFS problem proposed in the literature.
• Two-Dimensional Almost-Riemannian Structures with Tangency Points
  
  • Agrachev Andrei
  • Boscain Ugo
  • Charlot Grégoire
  • Ghezzi Roberta
  • Sigalotti Mario
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire, EMS, 2010, 27 (3), pp.793-307. 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 study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including 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 classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points. (10.1016/j.anihpc.2009.11.011)
  DOI : 10.1016/j.anihpc.2009.11.011
• A fast time stepping method for evaluating fractional integrals
  
  • Li Jing-Rebecca
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2010, 31 (6), pp.4696--4714. (10.1137/080736533)
  DOI : 10.1137/080736533
• L^1-error estimate for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1.
  
  • Bokanowski Olivier
  • Forcadel Nicolas
  • Zidani Hasnaa
  Mathematics of Computation, American Mathematical Society, 2010, 79 (271), pp.1395--1426. The goal of this paper is to study some numerical approximations of particular Hamilton-Jacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. We investigate two anti-diffusive numerical schemes, the first one is based on the Ultra-Bee scheme and the second one is based on the Fast Marching Method. We prove the convergence and derive $L^1$-error estimates for both schemes. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions.
• Reachability and minimal times for state constrained nonlinear problems without any controllability assumption
  
  • Bokanowski Olivier
  • Forcadel Nicolas
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2010, 48 (7), pp.pp. 4292-4316. We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent state constraints, as well as a target problem for a two-player game with state constraints. Our method gives a good framework for numerical approximations, and some numerical illustrations are included in the paper.
• Carathéodory, Helly and the others in the max-plus world
  
  • Gaubert S.
  • Meunier Frédéric
  Discrete and Computational Geometry, Springer Verlag, 2010, 43 (3), pp.648-662. Carathéodory's, Helly's and Radon's theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem-Sierksma's conjecture-although still open for the usual convexity, is shown to be true in the max-plus setting. © 2009 Springer Science+Business Media, LLC. (10.1007/s00454-009-9207-x)
  DOI : 10.1007/s00454-009-9207-x
• Merton Problem with Taxes: Characterization, computation and Approximation
  
  • Ben Tahar Imen
  • Touzi Nizar
  • Soner Mete H.
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2010, 1, pp.366-395. We formulate a computationally tractable extension of the classical Merton optimal consumptioninvestment problem to include the capital gains taxes. This is the continuous-time version of the model introduced by Dammon, Spatt, and Zhang [Rev. Financ. Stud., 14 (2001), pp. 583-616]. In this model the tax basis is computed as the average cost of the stocks in the investor's portfolio. This average rule introduces only one additional state variable, namely the tax basis. Since the other tax rules such as the first in first out rule require the knowledge of all past transactions, the average model is computationally much easier. We emphasize the linear taxation rule, which allows for tax credits when capital gains losses are experienced. In this context wash sales are optimal, and we prove it rigorously. Our main contributions are a first order explicit approximation of the value function of the problem and a unique characterization by means of the corresponding dynamic programming equation. The latter characterization builds on technical results isolated in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Control Optim., 46 (2007), pp. 1779-1801]. We also suggest a numerical computation technique based on a combination of finite differences and the Howard iteration algorithm. Finally, we provide some numerical results on the welfare consequences of taxes and the quality of the first order approximation. (10.1137/080742178)
  DOI : 10.1137/080742178
• Lipschitz solutions of optimal control problems with state constraints of arbitrary order
  
  • Bonnans J. Frederic
  Mathematics and its Applications: Annals of the Academy of Romanian Scientists, Academy of Romanian Scientists Publishing House, 2010, 2 (1), pp.78-98. In this paper we generalize to an arbitrary order, under minimal hypotheses, some sufficient conditions for Lipschitz continuity of the solution of a state constrained optimal control problems. The proof combines the approach by Hager in 1979 for dealing with first-order state constraints, and the high-order alternative formulation of the optimality conditions.
• Initialization of the shooting method via the Hamilton-Jacobi-Bellman approach
  
  • Cristiani Emiliano
  • Martinon Pierre
  Journal of Optimization Theory and Applications, Springer Verlag, 2010, 146 (2), pp.321-346. The aim of this paper is to investigate from the numerical point of view the possibility of coupling the Hamilton-Jacobi-Bellman (HJB) approach and the Pontryagin's Minimum Principle (PMP) to solve some control problems. We show that an approximation of the value function computed by the HJB method on rough grids can be used to obtain a good initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered. The CPU time for the proposed method is less than four minutes up to dimension four, without code parallelization. (10.1007/s10957-010-9649-6)
  DOI : 10.1007/s10957-010-9649-6
• A matrix interpolation between classical and free max operations: I. The univariate case
  
  • Benaych-Georges Florent
  • Cabanal-Duvillard Thierry
  Journal of Theoretical Probability, Springer, 2010, 23 (2), pp.447-465. Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings. (10.1007/s10959-009-0210-1)
  DOI : 10.1007/s10959-009-0210-1