Publications

2012

• A Patchy Dynamic Programming Scheme for a Class of Hamilton-Jacobi-Bellman Equations
  
  • Cacace Simone
  • Cristiani Emiliano
  • Falcone Maurizio
  • Picarelli Athena
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (5), pp.A2625-A2649. In this paper we present a new parallel algorithm for the solution of Hamilton-Jacobi- Bellman equations related to optimal control problems. The main idea is to divide the domain of computation into subdomains following the dynamics of the control problem. This results in a rather complex geometrical subdivision, but has the advantage that every subdomain is invariant with respect to the optimal controlled vector field, so that we can compute the value function in each subdomain assigning the task to a processor and avoiding the classical transmission condition on the boundaries of the subdomains. For this specific feature the subdomains are patches in the sense introduced by Ancona and Bressan in [1]. Several examples in dimension two and three illustrate the properties of the new method. (10.1137/110841576)
  DOI : 10.1137/110841576
• 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'))
• 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.
• 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
• 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.
• 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
• 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
• 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.
• 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
• 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.
• 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
• Characterization of a local quadratic growth of the Hamiltonian for control constrained optimal control problems
  
  • Bonnans J. Frédéric
  • Osmolovskii Nikolai P.
  Dynamics of Continuous, Discrete and Impulsive Systems, University of Waterloo, Ontario, Canada, 2012, 19 (1-2), pp.1-16. We consider an optimal control problem with inequality control constraints given by smooth functions satisfying the hypothesis of linear independence of gradients of active constraints. For this problem, we formulate a generalization of strengthened Legendre condition and prove that this generalization is equivalent to the condition of a local quadratic growth of the Hamiltonian subject to control constraints.
• Transmission Eigenvalues in Inverse Scattering Theory
  
  • Cakoni Fioralba
  • Haddar Houssem
  , 2012, 60, pp.527-578. This survey aims to present the state of the art of research on the transmission eigenvalue problem focussing on three main topics, namely the discreteness of transmission eigenvalues, the existence of trans- mission eigenvalues and estimates on transmission eigenvalues, in particular, Faber-Krahn type inequalities.
• Application of the linear sampling method to identify cracks with impedance boundary conditions
  
  • Ben Hassen Fahmi
  • Boukari Yosra
  • Haddar Houssem
  Inverse Problems in Science and Engineering, Taylor & Francis, 2012, pp.1-25. We use the linear sampling method (LSM) to identify a crack with impedance boundary conditions from far-field measurements at a fixed frequency. This article extends the work of Cakoni-Colton [F. Cakoni and D. Colton, The linear sampling method for cracks, Inverse Probl. 19 (2003), pp. 279-295] where LSM has been used to reconstruct a crack with impedance boundary conditions on one side of the crack and a Dirichlet boundary condition on the other one. In addition, we present two methods to also reconstruct the impedance parameters whence the geometry is known. The first one is based on the interpretation of the indicator function produced by the LSM, while the second one is a natural approach based on the integral representation of the far-field in terms of densities on the crack geometry. The performance of the different reconstruction methods is illustrated through numerical examples in a 2D setting of the scattering problem. (10.1080/17415977.2012.686997)
  DOI : 10.1080/17415977.2012.686997
• Coherent Interferometry Algorithms for Photoacoustic Imaging
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Jugnon Vincent
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2012, 50 (5), pp.2259 - 2280. The aim of this paper is to develop new coherent interferometry (CINT) algorithms to correct the effect of an unknown cluttered sound speed (random fluctuations around a known constant) on photoacoustic images. By back-propagating the correlations between the preprocessed pressure measurements, we show that we are able to provide statistically stable photoacoustic images. The preprocessing is exactly in the same way as when we use the circular or the line Radon inversion to obtain photoacoustic images. Moreover, we provide a detailed stability and resolution analysis of the new CINT-Radon algorithms. We also present numerical results to illustrate their performance and to compare them with Kirchhoff-Radon migration functions. (10.1137/100814275)
  DOI : 10.1137/100814275
• Localization and delocalization for heavy tailed band matrices
  
  • Benaych-Georges Florent
  • Péché Sandrine
  , 2012. We consider some random band matrices with band-width $N^\mu$ whose entries are independent random variables with distribution tail in $x^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $\alpha<2(1+\mu^{-1})$, the largest eigenvalues have order $N^{(1+\mu)/\alpha}$, are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked by Soshnikov for full matrices with heavy tailed entries,i.e. when $\alpha<2$, and by Auffinger, Ben Arous and Péché when $\alpha<4$). On the other hand, when $\alpha>2(1+\mu^{-1})$, the largest eigenvalues have order $N^{\mu/2}$ and most eigenvectors of the matrix are delocalized, i.e. approximately uniformly distributed on their $N$ coordinates.
• Slow and fast scales for superprocess limits of age-structured populations
  
  • Méléard Sylvie
  • Tran Viet Chi
  Stochastic Processes and their Applications, Elsevier, 2012, 122 (1), pp.250-276. A superprocess limit for an interacting birth-death particle system modelling a population with trait and physical age-structures is established. Traits of newborn offspring are inherited from the parents except when mutations occur, while ages are set to zero. Because of interactions between individuals, standard approaches based on the Laplace transform do not hold. We use a martingale problem approach and a separation of the slow (trait) and fast (age) scales. While the trait marginals converge in a pathwise sense to a superprocess, the age distributions, on another time scale, average to equilibria that depend on traits. The convergence of the whole process depending on trait and age, only holds for finite-dimensional time-marginals. We apply our results to the study of examples illustrating different cases of trade-off between competition and senescence. (10.1016/j.spa.2011.08.007)
  DOI : 10.1016/j.spa.2011.08.007
• Large deviations of the extreme eigenvalues of random deformations of matrices
  
  • Benaych-Georges Florent
  • Guionnet Alice
  • Maïda Mylène
  Probability Theory and Related Fields, Springer Verlag, 2012. Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\ud X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.
• Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations: Convergence Result and Error Estimate
  
  • Briani Ariela
  • Camilli Fabio
  • Zidani Hasnaa
  Differential Equations and Applications, Element, 2012, 4, pp.297-317. We consider approximation schemes for monotone systems of fully nonlinear second order partial di erential equations. We rst prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of Hamilton-Jacobi-Bellman equations. Examples including nite di erence schemes and Semi-Lagrangian schemes are discussed. (10.7153/dea-04-18)
  DOI : 10.7153/dea-04-18
• The Bounded Confidence Model Of Opinion Dynamics
  
  • Gómez-Serrano Javier
  • Graham Carl
  • Boudec Jean-Yves Le
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2012, 22 (2), pp.11500072. The bounded confidence model of opinion dynamics, introduced by Deffuant et al., is a stochastic model for the evolution of [0,1]-valued opinions within a finite group of peers. We show that as time goes to infinity, the opinions evolve into a random non-interacting set of clusters, and subsequently the opinions in each cluster converge to their barycenter; the limit empirical distribution is called a partial consensus. Then, we prove a mean-field limit result: for i.i.d. initial opinions, as the number of peers increases and time is rescaled accordingly, the peers asymptotically behave as i.i.d. peers, each influenced by opinions drawn independently from the unique solution of a nonlinear integro-differential equation. As a consequence, the (random) empirical distribution process converges to this (deterministic) solution. We also show that as time goes to infinity, this solution converges to a partial consensus, and identify sufficient conditions for the limit not to depend on the initial condition, and for formation of total consensus. Finally, we show that if the equation has an initial condition with a density, then its solution has a density at all times, develop a numerical scheme to solve the corresponding functional equation of the Kac type, and show, using numerical examples, that bifurcations may occur. (10.1142/S0218202511500072)
  DOI : 10.1142/S0218202511500072
• A convergent finite element approximation for Landau–Lifschitz–Gilbert equation
  
  • Alouges François
  • Kritsikis Evaggelos
  • Toussaint Jean-Christophe
  Physica B: Condensed Matter, Elsevier, 2012, 407 (9), pp.1345-1349. (10.1016/j.physb.2011.11.031)
  DOI : 10.1016/j.physb.2011.11.031
• Faddeev eigenfunctions for point potentials in two dimensions
  
  • Grinevich Piotr
  • Novikov Roman
  Physics Letters A, Elsevier, 2012, 376, pp.1102-1106. We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for point (delta-type) potentials in two dimensions. In particular, we obtain the first explicit examples of such eigenfunctions with contour singularity in spectral parameter at a fixed real energy. (10.1016/j.physleta.2012.02.025)
  DOI : 10.1016/j.physleta.2012.02.025