Publications

2010

• 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 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.
• Unique solvability of equations of motion for ferrofluids
  
  • Amirat Youcef
  • Hamdache Kamel
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2010, 73, pp.471-494.
• Explicit polyhedral approximation of the Euclidean ball
  
  • Bonnans J. Frederic
  • Lebelle M.
  RAIRO - Operations Research, EDP Sciences, 2010, 44 (1), pp.45-60. We discuss the problem of computing points of IRn whose convex hull contains the Euclidean ball, and is contained in a small multiple of it. Given a polytope containing the Euclidean ball, we introduce its successor obtained by intersection with all tangent spaces to the Euclidean ball, whose normals point towards the vertices of the polytope. Starting from the L-infinity ball, we discuss the computation of the two first successors, and give a complete analysis in the case when n = 6. (10.1051/ro/2010003)
  DOI : 10.1051/ro/2010003
• 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
• 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
• 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.
• 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
• 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
• 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
• Conformal mapping and impedance tomography
  
  • Haddar Houssem
  • Kress Rainer
  Inverse Problems, IOP Publishing, 2010, 26 (7), pp.074002, 18. (10.1088/0266-5611/26/7/074002)
  DOI : 10.1088/0266-5611/26/7/074002
• Inverse impedance boundary problem via the conformal mapping method: the case of small impedances
  
  • Ben Hassen Fehmi
  • Boukari Yosra
  • Haddar Houssem
  Revue Africaine de Recherche en Informatique et Mathématiques Appliquées, African Society in Digital Science, 2010, 13, pp.47-62.
• 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