Publications

2012

• 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.
• 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
• 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 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
• 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
• Contacts in dimension 2, A penalization method
  
  • Pantz Olivier
  , 2012.
• 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.
• 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
• 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
• 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
• 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.
• 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.
• 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.
• 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
• 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
• 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.
• 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
• 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
• 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.
• 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