Publications

2010

• 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
• Optimal control of a parabolic equation with time-dependent state constraints
  
  • Bonnans J. Frederic
  • Jaisson Pascal
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2010, 48 (7), pp.4550-4571. In this paper we study the optimal control problem of the heat equation by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied at each time. Using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations, we show that both the control and multiplier are continuous in time. Under some natural geometric hypotheses, we can prove that extended polyhedricity holds, allowing to obtain no-gap second-order optimality conditions, that characterize quadratic growth. An expansion of the value function and of approximate solutions can be computed for a directional perturbation of the r.h.s. of the state equation.
• A sampling method for inverse scattering in the time domain
  
  • Chen Qiang
  • Haddar Houssem
  • Lechleiter Armin
  • Monk Peter
  Inverse Problems, IOP Publishing, 2010, 26 (8), pp.085001, 17. (10.1088/0266-5611/26/8/085001)
  DOI : 10.1088/0266-5611/26/8/085001
• Asymptotic models for scattering from unbounded media with high conductivity
  
  • Haddar Houssem
  • Lechleiter Armin
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2010, 44 (6), pp.1295-1317. (10.1051/m2an/2010029)
  DOI : 10.1051/m2an/2010029
• On the determination of Dirichlet or transmission eigenvalues from far field data
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2010, 348 (7-8), pp.379-383. (10.1016/j.crma.2010.02.003)
  DOI : 10.1016/j.crma.2010.02.003
• Measurability of optimal transportation and strong coupling of martingale measures
  
  • Fontbona Joaquin
  • Guérin Hélène
  • Méléard Sylvie
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2010, 15, pp.124-133. We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with specified covariance measure. This is done by pushing forward one martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures. (10.1214/ECP.v15-1534)
  DOI : 10.1214/ECP.v15-1534
• Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media
  
  • Allaire Grégoire
  • Brizzi Robert
  • Mikelic Andro
  • Piatnitski Andrey
  Chemical Engineering Science, Elsevier, 2010, 65, pp.2292-2300. In this work we study reactive flows through porous media. We suppose dominant Peclet's number, dominant Damköhler's number and general linear reactions at the pore boundaries. Our goal is to obtain the dispersion tensor and the upscaled model. We introduce the multiple scale expansions with drift for the problem and use this technique to upscale the reactive flow equations. Our result is illustrated with numerical simulations for the dispersion tensor.
• Spectral Volumetric Integral Equation Methods for Acoustic Medium Scattering in a Planar Homogeneous 3D Waveguide
  
  • Lechleiter Armin
  • Nguyen Dinh Liem
  , 2010. Scattering of acoustic waves from an inhomogeneous medium can be described by the Lippmann-Schwinger integral equation. For scattering problems in free space, Vainikko proposed a fast spectral solution method that exploits the convolution structure of this equation's integral operator by using the fast Fourier transform. In a planar 3--dimensional waveguide, the integral operator of the Lippmann-Schwinger integral equation fails to be a convolution. In this paper, we show that the separable structure of the kernel nevertheless allows to construct fast spectral collocation methods similar to Vainikko's technique. The numerical analysis of this method requires smooth material parameters; if the material parameters are, say, discontinuous, no theoretical statement on convergence is available. We show how to construct a Galerkin variant of Vainikko's method for which a rigorous convergence analysis is available even for discontinuous materials. For several distant scattering objects inside the 3--dimensional waveguide this discretization technique leads to a computational domain consisting of one large box containing all scatterers, and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects.
• Towards a general convergence theory for inexact Newton regularizations
  
  • Lechleiter Armin
  • Rieder Andreas
  Numerische Mathematik, Springer Verlag, 2010, 114 (3), pp.521--548. (10.1007/s00211-009-0256-0)
  DOI : 10.1007/s00211-009-0256-0
• Approximation of Liquid-Vapor Phase Transition for Compressible Fluids with Tabulated EOS
  
  • Faccanoni Gloria
  • Kokh Samuel
  • Allaire Grégoire
  Comptes Rendus de l'Academie des Sciences. Série IV, Physique, Astronomie, Elsevier, 2010, 348 (7-8), pp.473-478. This Note investigates the approximation of phase change in compressible fluids with complex equation of state (EOS). Assuming a local and instantaneous equilibrium with respect to phasic pressures, temperatures and chemical potentials when both phases are present leads to the classical de nition of an equilibrium EOS for the two-phase medium. Unfortunately, there is no explicit expression of the equilibrium EOS in most cases. We propose simple means to approximate the equilibrium EOS when both phases are governed by very general EOS, including tabulated ones. We present a relaxation type numerical algorithm based on this approximation for simulating two-phase flows with phase change. (10.1016/j.crma.2010.01.012)
  DOI : 10.1016/j.crma.2010.01.012
• Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint
  
  • Hermant Audrey
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2010, 61 (1), pp.85-127. This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points. (10.1007/s00245-009-9076-y)
  DOI : 10.1007/s00245-009-9076-y
• The interior transmission problem for regions with cavities
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2010, 42 (1), pp.145-162. (10.1137/090754637)
  DOI : 10.1137/090754637
• 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 boundaryconditions. 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.
• 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
• 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.
• 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
• 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
• 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
• 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
• 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.
• 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.
• 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 global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions
  
  • Novikov Roman
  • Santacesaria Matteo
  J. Inv. Ill-Posed Problems, 2010, 18, pp.765-785. We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain. (10.1515/JIIP.2011.003)
  DOI : 10.1515/JIIP.2011.003
• 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
• What happens after a default : The conditional density approach
  
  • El Karoui Nicole
  • Jeanblanc M.
  • Jiao Y.
  Stochastic Processes and their Applications, Elsevier, 2010, 120 (7), pp.1011-1032. We present a general model for default times, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow us to compute the price of defaultable claims, except in the case where the immersion property is satisfied. We propose in this paper a density approach for default times. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and on the immersion property. (10.1016/j.spa.2010.02.003)
  DOI : 10.1016/j.spa.2010.02.003