Publications

2012

• 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.
• 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
• 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
• 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
• 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
• Existence and qualitative properties of isoperimetric sets in periodic media
  
  • Chambolle Antonin
  • Goldman Michael
  • Novaga Matteo
  , 2013. We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties.
• The Interior Transmission Eigenvalue Problem for Absorbing Media
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2012, 28 (4), pp.045005. In recent years, the transmission eigenvalue problem has been extensively studied for non-absorbing media. In this paper, we initiate the study of this problem for absorbing media. In particular, we show that, in the case of absorbing media, transmission eigenvalues form a discrete set, exist for sufficiently small absorption and for spherically stratified media exist without this assumption. For constant index of refraction, we also obtain regions in the complex plane where the transmission eigenvalues cannot exist and obtain a priori estimates for real transmission eigenvalues. (10.1088/0266-5611/28/4/045005)
  DOI : 10.1088/0266-5611/28/4/045005
• 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
• Instability in the Gel'fand inverse problem at high energies
  
  • Isaev Mikhail
  Applicable Analysis, Taylor & Francis, 2012, pp.DOI:10.1080/00036811.2012.731501. We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceeding stability results on inverse problems of such a type. (10.1080/00036811.2012.73150)
  DOI : 10.1080/00036811.2012.73150
• 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
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Singular Forward-Backward Stochastic Differential Equations and Emissions Derivatives
  
  • Carmona René
  • Delarue François
  • Espinosa Gilles-Edouard
  • Touzi Nizar
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 23, pp.1086--1128. We introduce two simple models of forward-backward stochastic differential equations with a singular terminal condition and we explain how and why they appear naturally as models for the valuation of CO2 emission allowances. Single phase cap-and-trade schemes lead readily to terminal conditions given by indicator functions of the forward component, and using fine partial differential equations estimates, we show that the existence theory of these equations, as well as the properties of the candidates for solution, depend strongly upon the characteristics of the forward dynamics. Finally, we give a first order Taylor expansion and show how to numerically calibrate some of these models for the purpose of CO2 option pricing.
• 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
• Credit Risk with asymmetric information on the default threshold
  
  • Hillairet Caroline
  • Jiao Ying
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2012, 84 (2-3), pp.183-198. We study the impact of asymmetric information in a general credit model where the default is triggered when a fundamental diff usion process of the firm passes below a random threshold. Inspired by some recent technical default events during the fi nancial crisis, we consider the role of the firm's managers who choose the level of the default threshold and have complete information. However, other investors on the market only have partial observations either on the process or on the threshold. We specify the accessible information for di fferent types of investors. Besides the framework of progressive enlargement of fi ltrations usually adopted in the credit risk modelling, we also combine the results on initial enlargement of filtrations to deal with the uncertainty on the default threshold. We consider several types of investors who have di fferent information levels and we compute the default probabilities in each case. Numerical illustrations show that the insiders who have extra information on the default threshold obtain better estimations of the default probability compared to the standard market investors. (10.1080/17442508.2011.575944)
  DOI : 10.1080/17442508.2011.575944
• Tropical polyhedra are equivalent to mean payoff games
  
  • Akian Marianne
  • Gaubert Stéphane
  • Guterman A.
  International Journal of Algebra and Computation, World Scientific Publishing, 2012, 22 (1), pp.1250001, 43. We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory. (10.1142/S0218196711006674)
  DOI : 10.1142/S0218196711006674
• Large liquidity expansion of super-hedging costs
  
  • Possamaï Dylan
  • Soner Mete H.
  • Touzi Nizar
  Asymptotic Analysis, IOS Press, 2012, 79 (1-2), pp.45-64. We consider a financial market with liquidity cost as in Çetin, Jarrow and Protter [2004], where the supply function S{\epsilon}(s,{\nu}) depends on a parameter {\epsilon}\geq0 with S0(s,{\nu})=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [2010], of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of {\epsilon}. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option. (10.3233/ASY-2011-1089)
  DOI : 10.3233/ASY-2011-1089
• Consistency result for a non monotone scheme for anisotropic mean curvature flow
  
  • Bonnetier Eric
  • Bretin Elie
  • Chambolle Antonin
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2012, 14 (1), pp.1-35. In this paper, we propose a new scheme for anisotropic motion by mean curvature in $R^d$. The scheme consists of a phase-field approximation of the motion, where the nonlinear diffusive terms in the corresponding anisotropic Allen-Cahn equation are linearized in the Fourier space. In real space, this corresponds to the convolution with a specific kernel of the form $$K_{\phi,t}(x)=F^{−1}[e^{−4\pi^2t\phi^0(ξ)}](x).$$ We analyse the resulting scheme, following the work of Ishii-Pires-Souganidis on the convergence of the Bence-Merriman-Osher algorithm for isotropic motion by mean curvature. The main difficulty here, is that the kernel $K_{\phi,t}$ is not positive and that its moments of order 2 are not in $L^1(R^d)$. Still, we can show that in one sense the scheme is consistent with the anisotropic mean curvature flow. (10.4171/IFB/272)
  DOI : 10.4171/IFB/272
• Disentangling dispersal, vicariance and adaptive radiation patterns: A case study using armyworms in the pest genus Spodoptera (Lepidoptera: Noctuidae)
  
  • Kergoat Gael G.
  • Prowell Dorothy P.
  • Le Ru Bruno P.
  • Mitchell Andrew
  • Dumas Pascaline
  • Clamens Anne Laure
  • Condamine Fabien L.
  • Silvain Jean-Francois
  Molecular Phylogenetics and Evolution, Elsevier, 2012, 65 (3), pp.855-870. Thanks to the recent development of integrative approaches that combine dated phylogenies with models of biogeographic evolution, it is becoming more feasible to assess the roles of dispersal and vicariance in creating complex patterns of geographical distribution. However, the historical biogeography of taxa with good dispersal abilities, like birds or flying insects, still remains largely unknown because of the lack of complete phylogenies accompanied by robust estimates of divergence times. In this study, we investigate the evolution and historical biogeography of the globally distributed pest genus Spodoptera (Lepidoptera: Noctuidae) using complete taxon sampling and an extensive set of analyses. Through the analysis of a combined morphological and molecular dataset, we provide the first robust phylogenetic framework for this widespread and economically important group of moths. Historical biogeography approaches indicate that dispersal events have been the driving force in the biogeographic history of the group. One of the most interesting findings of this study is the probable occurrence of two symmetric long-distance dispersal events between the Afrotropical and the Neotropical region, which appear to have occurred in the late Miocene. Even more remarkably, our dated phylogenies reveal that the diversification of the clade that includes specialist grass feeders has followed closely the expansion of grasslands in the Miocene, similar to the adaptive radiation of specialist grazing mammals during the same period. (10.1016/j.ympev.2012.08.006)
  DOI : 10.1016/j.ympev.2012.08.006
• Quasi-stationary distributions and population processes
  
  • Méléard Sylvie
  • Villemonais Denis
  Probability Surveys, Institute of Mathematical Statistics (IMS), 2012, 9, pp.340-410. This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures. (10.1214/11-PS191)
  DOI : 10.1214/11-PS191
• Domain theory and mirror properties in inverse semigroups
  
  • Poncet Paul
  Semigroup Forum, Springer Verlag, 2012, 84 (3), pp.434-446. Inverse semigroups are a class of semigroups whose structure induces a compatible partial order. This partial order is examined so as to establish mirror properties between an inverse semigroup and the semilattice of its idempotent elements, such as continuity in the sense of domain theory. (10.1007/s00233-012-9392-4)
  DOI : 10.1007/s00233-012-9392-4
• Tropical linear-fractional programming and parametric mean payoff games
  
  • Gaubert Stéphane
  • Katz R.D.
  • Sergeev S.N.
  Journal of Symbolic Computation, Elsevier, 2012, 47 (12), pp.1447-1478. Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to study the tropical analogue of the classical linear-fractional programming problem. We construct an associated parametric mean payoff game problem, and show that the optimality of a given point, or the unboundedness of the problem, can be certified by exhibiting a strategy for one of the players having certain infinitesimal properties (involving the value of the game and its derivative) that we characterize combinatorially. We use this idea to design a Newton-like algorithm to solve tropical linear-fractional programming problems, by reduction to a sequence of auxiliary mean payoff game problems (10.1016/j.jsc.2011.12.049)
  DOI : 10.1016/j.jsc.2011.12.049