Publications

2010

• Localization of high frequency waves propagating in a locally periodic medium
  
  • Allaire Grégoire
  • Friz Luis
  Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, Royal Society of Edinburgh, 2010, 140A, pp.897-926. We study the homogenization and localization of high frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale √ ε. We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e., of the corresponding Bloch eigenvalue. Upon rescaling at size √ ε we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.
• Quadratic growth conditions in optimal control problems
  
  • Bonnans Joseph Frederic
  • Osmolovskii Nikolai P.
  Contemporary mathematics, American Mathematical Society, 2010, 514, pp.85--98.
• Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices
  
  • Benaych-Georges Florent
  • Guionnet Alice
  • Maïda Mylène
  , 2010. Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X_n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.
• Second-order analysis of optimal control problems with control and initial-final state constraints
  
  • Bonnans J. Frederic
  • Osmolovskii Nikolai P.
  Journal of Convex Analysis, Heldermann, 2010, 17 (3), pp.885-913. This paper provides an analysis of Pontryagine minima satisfying a quadratic growth condition, for optimal control problems of ordinary differential equations with constraints on initial-final state, as well as control constraints satisfying the uniform positive linear independence condition.
• Application of a coupled FV/FE multiscale method to cement media
  
  • Abballe Th.
  • Allaire Grégoire
  • Laucoin E.
  • Montarnal Ph.
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2010, 5 (3), pp.603-615. We present here some results provided by a multiscale resolution method using both Finite Volumes and Finite Elements. This method is aimed at solving very large diffusion problems with highly oscillating coefficients. As an illustrative example, we simulate models of cement media, where very strong variations of diffusivity occur. As a by-product of our simulations, we compute the effective diffusivities of these media. After a short introduction, we present a theorical description of our method. Numerical experiments on a two dimensional cement paste are presented subsequently. The third section describes the implementation of our method in the calculus code MPCube and its application to a sample of mortar. Finally, we discuss strengths and weaknesses of our method, and present our future works on this topic. (10.3934/nhm.2010.5.603)
  DOI : 10.3934/nhm.2010.5.603
• The existence of an infinite discrete set of transmission eigenvalues
  
  • Cakoni Fioralba
  • Gintides Drossos
  • Haddar Houssem
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2010, 42 (1), pp.237-255. (10.1137/090769338)
  DOI : 10.1137/090769338
• Preprocessing the Reciprocity Gap Sampling Method in Buried-Object Imaging Experiments
  
  • Ozdemir Ozgur
  • Haddar Houssem
  IEEE Geoscience and Remote Sensing Letters, IEEE - Institute of Electrical and Electronics Engineers, 2010, 7 (4), pp.756 -760. (10.1109/LGRS.2010.2047003)
  DOI : 10.1109/LGRS.2010.2047003
• Convergence of a non-monotone scheme for Hamilton-Jacobi-Bellman equations with discontinuous data
  
  • Bokanowski Olivier
  • Megdich Nadia
  • Zidani Hasnaa
  Numerische Mathematik, Springer Verlag, 2010, 115 (1), pp.1--44. On étudie un schéma non monotone pour l'équation Hamilton Jacobi Bellman du premier ordre, en dimension 1. Le schéma considèré est lié au schéma anti-diffusif, appellé UltraBee, proposé par Bokanowski-Zidani (publié en 2007 dans J. Sci. Compt.). Ici, on prouve la convergence, en norme $L^1$, à l'ordre 1, pour une condition initiale discontinue. Le caractère anti-diffusif du schéma est illustré par quelques exemples numériques. (10.1007/s00211-009-0271-1)
  DOI : 10.1007/s00211-009-0271-1