Publications

2012

• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. A pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Non-parametric kernel estimation for symmetric Hawkes processes. Application to high frequency financial data
  
  • Bacry Emmanuel
  • Muzy Khalil Dayri Jean-François
  The European Physical Journal B: Condensed Matter and Complex Systems, Springer-Verlag, 2012, 85 (5), pp.1--12. We define a numerical method that provides a non-parametric estimation of the kernelshape in symmetric multivariate Hawkes processes. This method relies on second orderstatistical properties of Hawkes processes that relate the covariance matrix of theprocess to the kernel matrix. The square root of the correlation function is computedusing a minimal phase recovering method. We illustrate our method on some examples andprovide an empirical study of the estimation errors. Within this framework, we analyzehigh frequency financial price data modeled as 1D or 2D Hawkes processes. We find slowlydecaying (power-law) kernel shapes suggesting a long memory nature of self-excitationphenomena at the microstructure level of price dynamics. (10.1140/epjb/e2012-21005-8)
  DOI : 10.1140/epjb/e2012-21005-8
• Modified Lees-Edwards Boundary conditions and viscous contact for numerical simulations of particles in a shear flow
  
  • Verdon Nicolas
  • Lefebvre-Lepot Aline
  • Laure Patrice
  • Lobry Laurent
  Revue Européenne de Mécanique Numérique/European Journal of Computational Mechanics, Hermès / Paris : Lavoisier, 2012, 121 (3-6 / Special Issue: French Conference on Computational Mechanics 2011: selected contributions), pp.397-406. We present a way to handle contacts between rigid particles in shear flow. The influence of such a modeling is shown by studying an example with 13 particles in 3D. Studying a concentrated suspension in 2D, we demonstrate that contact modelling as well as choice of boundary conditions influences the macroscopic properties of the suspension. (10.1080/17797179.2012.714851)
  DOI : 10.1080/17797179.2012.714851
• Volume-constrained minimizers for the prescribed curvature problem in periodic media
  
  • Goldman Michael
  • Novaga Matteo
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2012. We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a suitable Wulff Shape, when the volume tends to infinity.
• Min-max spaces and complexity reduction in min-max expansions
  
  • Gaubert Stéphane
  • Mceneaney W.M.
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2012, 65 (3), pp.315--348. Idempotent methods have been found to be extremely helpful in the numerical solution of certain classes of nonlinear control problems. In those methods, one uses the fact that the value function lies in the space of semiconvex functions (in the case of maximizing controllers), and approximates this value using a truncated max-plus basis expansion. In some classes, the value function is actually convex, and then one specifically approximates with suprema (i.e., max-plus sums) of affine functions. Note that the space of convex functions is a max-plus linear space, or moduloid. In extending those concepts to game problems, one finds a different function space, and different algebra, to be appropriate. Here we consider functions which may be represented using infima (i.e., min-max sums) of max-plus affine functions. It is natural to refer to the class of functions so represented as the min-max linear space (or moduloid) of max-plus hypo-convex functions. We examine this space, the associated notion of duality and min-max basis expansions. In using these methods for solution of control problems, and now games, a critical step is complexity-reduction. In particular, one needs to find reduced-complexity expansions which approximate the function as well as possible. We obtain a solution to this complexity-reduction problem in the case of min-max expansions. (10.1007/s00245-011-9158-5)
  DOI : 10.1007/s00245-011-9158-5
• Tropical bounds for the eigenvalues of structured matrices
  
  • Akian Marianne
  • Gaubert Stéphane
  • Sharify M.
  , 2012. We establish several inequalities of log-majorization type, relating the moduli of the eigenvalues of a complex matrix or matrix polynomial with the tropical eigenvalues of auxiliary matrix polynomials. This provides bounds which can be computed by combinatorial means. We consider in particular structured matrices and obtain bounds depending on the norms of block submatrices and on the pattern (graph structure) of the matrix.
• Solving multi-stage stochastic mixed integer linear programs by the dual dynamic programming approach
  
  • Cen Zhihao
  , 2012. We consider a model of medium-term commodity contracts management. Randomness takes place only in the prices on which the commodities are exchanged, whilst state variable is multi-dimensional, and decision variable is integer. In our previous article, we proposed an algorithm based on the quantization of random process and a dual dynamic programming type approach to solve the continuous relaxation problem. In this paper, we study the multi-stage stochastic mixed integer linear program (SMILP) and show the difficulty when using dual programming type algorithm. We propose an approach based on the cutting plane method combined with the algorithm in our previous article, which gives an upper and a lower bound of the optimal value and a sub-optimal integer solution. Finally, a numerical test on a real problem in energy market is provided.
• Perspective Shape from Shading: Ambiguity Analysis and Numerical Approximations
  
  • Breuß Michael
  • Cristiani Emiliano
  • Durou Jean-Denis
  • Falcone Maurizio
  • Vogel Oliver
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2012, 5 (1), pp.311--342. In this paper we study a perspective model for shape from shading and its numerical approximation. We show that an ambiguity still persists, although the model with light attenuation factor has previously been shown to be well-posed under appropriate assumptions. Analytical results revealing the ambiguity are complemented by various numerical tests. Moreover, we present convergence results for two iterative approximation schemes. The first is based on a finite difference discretization, whereas the second is based on a semi-Lagrangian discretization. The convergence results are obtained in the general framework of viscosity solutions of the underlying partial differential equation. In addition to these theoretical and numerical results, we propose an algorithm for reconstructing discontinuous surfaces, making it possible to obtain results of reasonable quality even for complex scenes. To this end, we solve the constituting equation on a previously segmented input image, using state constraint boundary conditions at the segment borders. (10.1137/100815104)
  DOI : 10.1137/100815104
• ITD Interpolation and Personalization for Binaural Synthesis Using Spherical Harmonics
  
  • Aussal Matthieu
  • Alouges Francois
  • Katz Brian F. G.
  , 2012, pp.04:01-10.
• Approximate Models for Wave Propagation Across Thin Periodic Interfaces
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2012, 98 (1), pp.28-71. This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy. (10.1016/j.matpur.2012.01.003)
  DOI : 10.1016/j.matpur.2012.01.003
• Identification of small inclusions from multistatic data using the reciprocity gap concept
  
  • Haddar Houssem
  • Mdimagh Ridha
  Inverse Problems, IOP Publishing, 2012, 28 (4), pp.045011, 19. We consider the problem of identifying small inclusions (or point sources) from multistatic Cauchy data at given surface measurements associated with harmonic waves at a fixed frequency. We employ the reciprocity gap sampling method to recover the location of the inclusions and identify their equivalent dielectric properties. As opposed to the case of extended obstacles, no approximation argument is needed in the theoretical justification of the method. These aspects are numerically validated through multiple numerical experiments associated with small inclusions. (10.1088/0266-5611/28/4/045011)
  DOI : 10.1088/0266-5611/28/4/045011
• Homogenization of reactive flows in porous media and competition between bulk and surface diffusion
  
  • Allaire Grégoire
  • Hutridurga Harsha
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2012, 77, pp.788-815. In this work, we study the convection and diffusion of a solute in a porous medium in the presence of a linear chemical reaction of adsorption/desorption on the pore surfaces. The mathematical model is a system of two coupled convection-diffusion equations, one in the bulk of the saturated fluid flowing in the porous medium, the other on the pore surface, at the interface with the solid part of the porous medium. The coupling takes place through a linear reaction term expressing the exchange of mass between the bulk concentration and the surface concentration. By a method of two-scale asymptotic expansion with drift we obtain the homogenized problem in a moving frame. We rigorously justify our upscaling approach by using the notion of two-scale convergence with drift. Some 2-d numerical tests are performed in order to study the effect of variations of the adsorption rate constant and surface molecular diffusion on the effective dispersion tensor.
• Representation formula for stochastic Schrödinger evolution equations and applications
  
  • de Bouard Anne
  • Fukuizumi Reika
  Nonlinearity, IOP Publishing, 2012, 25 (11), pp.2993-3022. We prove a representation formula for solutions of Schrödinger equations with potentials multiplied by a temporal real-valued white noise in the Stratonovich sense. Using this formula, we obtain a dispersive estimate which allows us to study the Cauchy problem in L2 or in the energy space of model equations arising in Bose-Einstein condensation or in fiber optics. Our results also give a justification of diffusion-approximation for stochastic nonlinear Schrödinger equations. (10.1088/0951-7715/25/11/2993)
  DOI : 10.1088/0951-7715/25/11/2993
• Error estimates for the logarithmic barrier method in stochastic linear quadratic optimal control problems
  
  • Bonnans Joseph Frédéric
  • Silva Francisco J.
  Systems and Control Letters, Elsevier, 2012, 61 (1), pp.143-147. We consider a linear quadratic stochastic optimal control problem whith non-negativity control constraints. The latter are penalized with the classical logarithmic barrier. Using a duality argument and the stochastic minimum principle, we provide an error estimate for the solution of the penalized problem which is the natural extension of the well known estimate in the deterministic framework.
• Un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'MandelBulb')
  
  • Colonna Jean-François
  , 2012. A pseudo-octonionic Mandelbrot set (a 'MandelBulb') (Un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'MandelBulb'))
• Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed or localized vectors
  
  • Benaych-Georges Florent
  • Cabanal-Duvillard Thierry
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2012, 9 (2), pp.685-715. The celebrated Marchenko-Pastur theorem gives the asymptotic spectral distribution of sums of random, independent, rank-one projections. Its main hypothesis is that these projections are more or less uniformly distributed on the first grassmannian, which implies for example that the corresponding vectors are delocalized, i.e. are essentially supported by the whole canonical basis. In this paper, we propose a way to drop this delocalization assumption and we generalize this theorem to a quite general framework, including random projections whose corresponding vectors are localized, i.e. with some components much larger than the other ones. The first of our two main examples is given by heavy tailed random vectors (as in a model introduced by Ben Arous and Guionnet or as in a model introduced by Zakharevich where the moments grow very fast as the dimension grows). Our second main example is given by vectors which are distributed as the Brownian motion on the unit sphere, with localized initial law. Our framework is in fact general enough to get new correspondences between classical infinitely divisible laws and some limit spectral distributions of random matrices, generalizing the so-called Bercovici-Pata bijection.
• Transmission eigenvalues for inhomogeneous media containing obstacles
  
  • Cakoni Fioralba
  • Cossonnière Anne
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2012, 6 (3), pp.373-398. (10.3934/ipi.2012.6.373)
  DOI : 10.3934/ipi.2012.6.373
• Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition
  
  • Geiss Christel
  • Geiss Stefan
  • Gobet Emmanuel
  Stochastic Processes and their Applications, Elsevier, 2012, 122 (5), pp.2078--2116. We relate the $L_p$-variation, $2\le p < \infty$, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.