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
• A la recherche de l'inégalité perdue
  
  • Allaire Grégoire
  Matapli, Société de Mathématiques Appliquées et Industrielles (SMAI), 2012 (98), pp.52-64. A l'occasion du centenaire de la disparition de Henri Poincaré on s'intéresse à l'origine de la célèbre inégalité de Poincaré, si utile dans l'analyse de certaines équations aux dérivées partielles. Comme souvent en mathématiques les bons résultats ne sont pas attribués aux bonnes personnes... En fait, cette inégalité est probablement due à Neumann, Schwarz ou Scheeffer et on doit plutôt à Henri Poincaré une variante, appelée communément inégalité de Poincaré-Wirtinger. Sa démonstration originale est très intéressante car elle est constructive, au contraire d'une démonstration ultérieure par contradiction, utilisant le théorème de compacité de Rellich, que l'on trouve désormais dans la plupart des ouvrages sur les équations aux dérivées partielles. La motivation de cette inégalité est l'étude des valeurs propres et fonctions propres du Laplacien.
• 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.
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• 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
• On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (3), pp.A1824-A1848. We consider the inverse obstacle scattering problem of determining both the shape and the "equiva- lent impedance" from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface differential operator (refer to as generalized impedance boundary condition) as opposed to a scalar function. The generalized impedance boundary condition can be seen as a more accurate model for effective impedances and is widely used in the scattering problem for thin coatings. Our approach is based on a least square optimization technique. A major part of our analysis is to characterize the derivative of the cost function with respect to the boundary and this complex surface impedance configuration. In particular, we provide an extension of the notion of shape derivative to the case where the involved impedance parameters do not need to be surface traces of given functions, which leads (in general) to a non-vanishing tangential boundary perturbation. The efficiency of considering this type of derivative is illustrated by several 2D numerical experiments based on a (classical) steepest descent method. The feasibility of retrieving both the shape and the impedance parameters is also discussed in our numerical experiments. (10.1137/110850347)
  DOI : 10.1137/110850347
• Asymptotic and non asymptotic approximations for option valuation
  
  • Bompis Romain
  • Gobet Emmanuel
  , 2012, pp.80. We give a broad overview of approximation methods to derive analytical formulas for accurate and quick evaluation of option prices. We compare different approaches, from the theoretical point of view regarding the tools they require, and also from the numerical point of view regarding their performances. In the case of local volatility models with general time-dependency, we derive new formulas using the local volatility function at the mid-point between strike and spot: in general, our approximations outperform previous ones by Hagan and Henry-Labordère. We also provide approximations of the option delta.
• A general stochastic model for sporophytic self-incompatibility
  
  • Billiard Sylvain
  • Tran Viet Chi
  Journal of Mathematical Biology, Springer, 2012, 64 (1-2), pp.163-210. Disentangling the processes leading populations to extinction is a major topic in ecology and conservation biology. The difficulty to find a mate in many species is one of these processes. Here, we investigate the impact of self-incompatibility in flowering plants, where several inter-compatible classes of individuals exist but individuals of the same class cannot mate. We model pollen limitation through different relationships between mate availability and fertilization success. After deriving a general stochastic model, we focus on the simple case of distylous plant species where only two classes of individuals exist. We first study the dynamics of such a species in a large population limit and then, we look for an approximation of the extinction probability in small populations. This leads us to consider inhomogeneous random walks on the positive quadrant. We compare the dynamics of distylous species to self-fertile species with and without inbreeding depression, to obtain the conditions under which self-incompatible species could be less sensitive to extinction while they can suffer more pollen limitation. (10.1007/s00285-011-0410-z)
  DOI : 10.1007/s00285-011-0410-z
• Analysis of exposure–response of CI-945 in patients with epilepsy: application of novel mixed hidden Markov modeling methodology
  
  • Delattre Maud
  • Savic Radojka M.
  • Miller Raymond
  • Karlsson Mats O.
  • Lavielle Marc
  Journal of Pharmacokinetics and Pharmacodynamics, Springer Verlag, 2012, 39 (3), pp.263 - 271. We propose to describe exposure–response relationship of an antiepileptic agent, using mixed hidden Markov modeling methodology, to reveal additional insights in the mode of the drug action which the novel approach offers. Daily seizure frequency data from six clinical studies including patients who received gabapentin were available for the analysis. In the model, seizure frequencies are governed by underlying unobserved disease activity states. Individual neighbouring states are dependent, like in reality and they exhibit their own dynamics with patients transitioning between low and high disease states, according to a set of transition probabilities. Our methodology enables estimation of unobserved disease dynamics and daily seizure frequencies in all disease states. Additional modes of drug action are achievable: gabapentin may influence both daily seizure frequencies and disease state dynamics. Gabapentin significantly reduced seizure frequencies in both disease activity states; however it did not significatively affect disease dynamics. Mixed hidden Markov modeling is able to mimic dynamics of seizure frequencies very well. It offers novel insights into understanding disease dynamics in epilepsy and gabapentin mode of action. (10.1007/s10928-012-9248-2)
  DOI : 10.1007/s10928-012-9248-2
• Computing estimates of material properties from transmission eigenvalues
  
  • Giorgi Giovanni
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2012, 28 (5), pp.055009, 23. This work is motivated by inverse scattering problems, those problems where one is interested in reconstructing the shape and the material properties of an inclusion from electromagnetic farfield measurements. More precisely, we are interested in complementing the so-called sampling methods by providing an estimate of the material properties of the sought inclusion. We use for this purpose a measure of the first transmission eigenvalue. Our method is then based on computing the desired estimate by reformulating the so-called interior transmission eigenvalue problem as an eigenvalue problem for the material coefficients. We will restrict ourselves to the two-dimensional setting of the problem and treat the cases of both transverse electric and transverse magnetic polarizations. We present a number of numerical experiments that validate our methodology for homogeneous and inhomogeneous inclusions and backgrounds. We also treat the case of a background with absorption and the case of scatterers with multiple connected components of different refractive indices. (10.1088/0266-5611/28/5/055009)
  DOI : 10.1088/0266-5611/28/5/055009
• Sensitivity analysis of energy contracts management problem by stochastic programming techniques
  
  • Cen Zhihao
  • Bonnans J. Frederic
  • Christel Thibault
  , 2012, 12 (2012), pp.447-471.. 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. In our previous article, we proposed an algorithm to deal with such problem, based on quantization of random process and a dual dynamic programming type approach. We obtained accurate estimates of the optimal value and a suboptimal strategy from this algorithm. In this paper, we analyse the sensitivity with respect to parameters driving the price model. We discuss the estimate of marginal price based on the Danskin's theorem. Finally, some numerical results applied to realistic energy market problems have been performed. Comparisons between results obtained by our algorithm and other classical methods are provided and evidence the accuracy of the estimate of marginal prices.
• Contacts in dimension 2, A penalization method
  
  • Pantz Olivier
  , 2012.
• On 2-step, corank 2 nilpotent sub-Riemannian metrics
  
  • Barilari Davide
  • Boscain Ugo
  • Gauthier Jean-Paul
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2012, 50 (1), pp.559-582. In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.
• Adiabatic control of the Schrödinger equation via conical intersections of the eigenvalues
  
  • Boscain Ugo
  • Chittaro Francesca
  • Mason Paolo
  • Sigalotti Mario
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2012, 57 (8), pp.1970-1983. In this paper, we present a constructive method to control the bilinear Schrödinger equation via two controls. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, and if the latter are conical (as it happens generically). In this framework, we are able to spread on several levels connected by conical intersections a state initially concentrated in a single energy level. We provide sharp estimates on the dependence of the error with respect to the controllability time. Moreover, we identify some special curves in the space of controls that improve the precision of the adiabatic approximation, when passing through conical intersections, with respect to classical adiabatic theory. (10.1109/TAC.2012.2195862)
  DOI : 10.1109/TAC.2012.2195862
• Noise Source Localization in an Attenuating Medium
  
  • Bretin Elie
  • Ammari Habib
  • Garnier Josselin
  • Wahab Abdul
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2012, 72 (1), pp.317 - 336. (10.1137/11083191X)
  DOI : 10.1137/11083191X
• 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