Publications

2012

• Convergence of time-space adaptive algorithms for nonlinear conservation laws
  
  • Coquel F.
  • Postel M.
  • Tran Q.-H.
  IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2012, 32 (4), pp.1440 - 1483. A family of explicit adaptive algorithms is designed to solve nonlinear scalar one-dimensional conservation laws. Based on the Godunov scheme on a uniform grid, a first strategy uses the multiresolution analysis of the solution to design an adaptive grid that evolves in time according to the time-dependent local smoothness. The method is furthermore enhanced by a local time-stepping strategy. Both numerical schemes are shown to converge towards the unique entropy solution. (10.1093/imanum/drr054)
  DOI : 10.1093/imanum/drr054
• Palaeoenvironmental shifts drove the adaptive radiation of a noctuid stemborer tribe (Lepidoptera, Noctuidae, Apameini) in the Miocene
  
  • Toussaint Emmanuel F.A.
  • Condamine Fabien L.
  • Kergoat Gael
  • Capdevielle-Dulac Claire
  • Barbut Jérôme
  • Silvain Jean-François
  • Le Ru Bruno P.
  PLoS ONE, Public Library of Science, 2012, 7 (7), pp.np. Between the late Oligocene and the early Miocene, climatic changes have shattered the faunal and floral communities and drove the apparition of new ecological niches. Grassland biomes began to supplant forestlands, thus favouring a large-scale ecosystem turnover. The independent adaptive radiations of several mammal lineages through the evolution of key innovations are classic examples of these changes. However, little is known concerning the evolutionary history of other herbivorous groups in relation with this modified environment. It is especially the case in phytophagous insect communities, which have been rarely studied in this context despite their ecological importance. Here, we investigate the phylogenetic and evolutionary patterns of grass-specialist moths from the species-rich tribe Apameini (Lepidoptera, Noctuidae). The molecular dating analyses carried out over the corresponding phylogenetic framework reveal an origin around 29 million years ago for the Apameini. Ancestral state reconstructions indicate (i) a potential Palaearctic origin of the tribe Apameini associated with a major dispersal event in Afrotropics for the subtribe Sesamiina; (ii) a recent colonization from Palaearctic of the New World and Oriental regions by several independent lineages; and (iii) an ancestral association of the tribe Apameini over grasses (Poaceae). Diversification analyses indicate that diversification rates have not remained constant during the evolution of the group, as underlined by a significant shift in diversification rates during the early Miocene. Interestingly, this age estimate is congruent with the development of grasslands at this time. Rather than clade ages, variations in diversification rates among genera better explain the current differences in species diversity. Our results underpin a potential adaptive radiation of these phytophagous moths with the family Poaceae in relation with the major environmental shifts that have occurred in the Miocene. (10.1371/journal.pone.0041377)
  DOI : 10.1371/journal.pone.0041377
• 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'))
• 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.
• 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.
• 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
• 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.
• 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.
• 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
• 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
• 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
• 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
• 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.
• 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.
• 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.
• 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
• 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
• 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
• 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
• 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
• Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis
  
  • Adjé A.
  • Gaubert Stéphane
  • Goubault E.
  Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2012, 8 (1), pp.1:01, 32. We introduce a new domain for finding precise numerical invariants of pro- grams by abstract interpretation. This domain, which consists of sub-level sets of non- linear functions, generalizes the domain of linear templates introduced by Manna, Sankara- narayanan, and Sipma. In the case of quadratic templates, we use Shor's semi-definite relaxation to derive safe and computable abstractions of semantic functionals, and we show that the abstract fixpoint can be over-approximated by coupling policy iteration and semi-definite programming. We demonstrate the interest of our approach on a series of examples (filters, integration schemes) including a degenerate one (symplectic scheme). (10.2168/LMCS-8(1:1)2012)
  DOI : 10.2168/LMCS-8(1:1)2012