Publications

2013

• The Factorization method applied to cracks with impedance boundary conditions
  
  • Boukari Yosra
  • Haddar Houssem
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2013, 7 (4). We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed fre- quency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks.
• Ecologie prédictive & changement planétaire
  
  • Austerlitz Frédéric
  • Blum Michael
  • Calba Sarah
  • Chave Jérôme
  • Choisy Marc
  • Coreau Audrey
  • Devictor Vincent
  • Doyen Luc
  • Dray Stéphane
  • Duputié Anne
  • Eveillard Damien
  • Faure Denis
  • Favier Charly
  • Gaggiotti Oscar
  • Galtier Nicolas
  • Garnier Éric
  • Gimenez Olivier
  • Guis Helene
  • Herbreteau Vincent
  • Huneman Philippe
  • Jabot Franck
  • Jarne Philippe
  • Joly Dominique
  • Julliard Romain
  • Kéfi Sonia
  • Kergoat Gael
  • Lacroix Gerard
  • Lagadeuc Yvan
  • Lavorel Sandra
  • Le Gaillard Jean-François
  • Le Gall Line
  • Loreau Michel
  • Maris Virginie
  • Morand Serge
  • Morin Xavier
  • Morlon Hélène
  • Mouquet Nicolas
  • Pinay Gilles
  • Pottier Julien
  • Pradel Roger
  • Ronce Ophélie
  • Schurr Frank
  • Simonet Pascal
  • Teplitsky Céline
  • Thuiller Wilfried
  • Tran Anne-Lise
  • Venner Samuel
  , 2013, hors s\'{e}rie, pp.9-44.
• Representation, relaxation and convexity for variational problems in Wiener spaces
  
  • Chambolle Antonin
  • Goldman Michael
  • Novaga Matteo
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2013. We show convexity of solutions to a class of convex variational problems in the Gauss and in the Wiener space. An important tool in the proof is a representation formula for integral functionals in this infinite dimensional setting, that extends analogous results valid in the classical Euclidean framework.
• Modelling microstructure noise with mutually exciting point processes
  
  • Bacry Emmanuel
  • Delattre Sylvain
  • Hoffmann Marc
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2013, 13 (1), pp.65-77. We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point pro- cesses and relies on linear self and mutually exciting stochastic inten- sities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations. (10.1080/14697688.2011.647054)
  DOI : 10.1080/14697688.2011.647054
• Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy
  
  • Novikov Roman
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013, 21 (6), pp.813–823. We prove approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy with incomplete data in dimension d ≥ 2. Our estimates are given in uniform norm for coefficient difference and related stability precision efficiently increases with increasing energy and coefficient difference regularity. In addition, our estimates are rather optimal even in the Born approximation.
• State-constrained Optimal Control Problems of Impulsive Differential Equations
  
  • Forcadel Nicolas
  • Rao Zhiping
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2013, 68, pp.1--19. The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption. (10.1007/s00245-013-9193-5)
  DOI : 10.1007/s00245-013-9193-5
• Central limit theorems for linear statistics of heavy tailed random matrices
  
  • Benaych-Georges Florent
  • Guionnet Alice
  • Male Camille
  , 2013. We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $\alpha$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.
• Optimisation of cancer drug treatments using cell population dynamics
  
  • Billy Frédérique
  • Clairambault Jean
  • Fercoq Olivier
  , 2013, pp.265. Cancer is primarily a disease of the physiological control on cell population proliferation. Tissue proliferation relies on the cell division cycle: one cell becomes two after a sequence of molecular events that are physiologically controlled at each step of the cycle at so-called checkpoints, in particular at transitions between phases of the cycle [105]. Tissue proliferation is the main physiological process occurring in development and later in maintaining the permanence of the organism in adults, at that late stage mainly in fast renewing tissues such as bone marrow, gut and skin. (10.1007/978-1-4614-4178-6_10)
  DOI : 10.1007/978-1-4614-4178-6_10
• Second order corrector in the homogenization of a conductive-radiative heat transfer problem
  
  • Allaire Grégoire
  • Habibi Zakaria
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18 (1), pp.1-36. This paper focuses on the contribution of the so-called second order corrector in periodic homogenization applied to a conductive-radiative heat transfer problem. More precisely, heat is diffusing in a periodically perforated domain with a non-local boundary condition modelling the radiative transfer in each hole. If the source term is a periodically oscillating function (which is the case in our application to nuclear reactor physics), a strong gradient of the temperature takes place in each periodicity cell, corresponding to a large heat flux between the sources and the perforations. This effect cannot be taken into account by the homogenized model, neither by the first order corrector. We show that this local gradient effect can be reproduced if the second order corrector is added to the reconstructed solution. (10.3934/dcdsb.2013.18.1)
  DOI : 10.3934/dcdsb.2013.18.1
• The topological derivative in anisotropic elasticity
  
  • Bonnet Marc
  • Delgado Gabriel
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2013, 66, pp.557-586. A comprehensive treatment of the topological derivative for anisotropic elasticity is presented, with both the background material and the trial small inhomogeneity assumed to have arbitrary anisotropic elastic properties. A formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are treated; they differ mostly on details in the elastic moment tensor (EMT). Moreover, the main properties of the EMT, a critical component of the topological derivative, are studied for the fully anisotropic case. Then, the topological derivative of strain energy-based quadratic cost functionals is derived, which requires a distinct treatment. Finally, numerical experiments on the numerical evaluation of the EMT and the topological derivative of the compliance cost functional are reported. (10.1093/qjmam/hbt018)
  DOI : 10.1093/qjmam/hbt018
• Exponential instability in the inverse scattering problem on the energy interval
  
  • Isaev Mikhail
  Functional Analysis and Its Applications, Springer Verlag, 2013, 47 (3), pp.8. We consider the inverse scattering problem on the energy interval in three dimensions. We are focused on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows optimality of the logarithmic stability result of [Stefanov, 1990] (up to the value of the exponent).
• Bases Mathématiques de la théorie des jeux
  
  • Laraki Rida
  • Renault Jérôme
  • Sorin Sylvain
  , 2013, pp.186. Cet ouvrage est destiné aux étudiants en master ainsi qu'aux étudiants des écoles d'ingénieurs. Les connaissances mathématiques requises sont celles d'une licence scientifique. Ce cours est consacré à une présentation des principaux concepts et outils mathématiques de la théorie des jeux stratégiques. L'accent est mis sur l'exposé et les preuves des résultats fondamentaux (minmax et opérateur valeur, équilibre de Nash et corrélé). Par ailleurs certains développements récents sont présentés : variété des équilibres, dynamiques de sélection, apprentissage et jeux répétés. L'ouvrage comporte une importante section d'exercices et corrigés.
• Reconstruction of a potential from the impedance boundary map
  
  • Isaev Mikhail
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2013, 1 (1), pp.5-28. We give formulas and equations for finding generalized scattering data for the Schrödinger equation in open bounded domain at fixed energy from the impedance boundary map (or Robin-to-Robin map). Combining these results with results of the inverse scattering theory we obtain efficient methods for reconstructing potential from the impedance boundary map.
• A Global Steering Method for Nonholonomic Systems
  
  • Chitour Yacine
  • Jean Frédéric
  • Long Ruixing
  Journal of Differential Equations, Elsevier, 2013, 254, pp.1903-1956. In this paper, we present an iterative steering algorithm for nonholonomic systems (also called driftless control-affine systems) and we prove its global convergence under the sole assumption that the Lie Algebraic Rank Condition (LARC) holds true everywhere. That algorithm is an extension of the one introduced in [21] for regular systems. The first novelty here consists in the explicit algebraic construction, starting from the original control system, of a lifted control system which is regular. The second contribution of the paper is an exact motion planning method for nilpotent systems, which makes use of sinusoidal control laws and which is a generalization of the algorithm described in [29] for chained-form systems. (10.1016/j.jde.2012.11.012)
  DOI : 10.1016/j.jde.2012.11.012
• Linearized Cauchy Data Inversion Method for Two-Dimensional Buried Target Imaging
  
  • Ozdemir Ozgur
  • Haddar Houssem
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2013, 61 (6). We propose a novel inversion algorithm to image buried objects in inhomogeneous media from the electromagnetic data on the outer boundary. Our method is based on exploiting the Cauchy data to derive a new Born-like linearization of the inverse problem. The main advantage of this formulation is to avoid the use of the background Green function and therefore is computationally more efficient. It also provides better accuracy than classical Born approximation. In the case of stratified media, our approach can be coupled with any appropriate continuation method. We discuss here the coupling with a continuation method based on the use of approximate transmission conditions. The feasibility and robustness of our methodology is validated through numerical experiments for single and multiple targets.
• Spatiotemporal Dynamic Simulation of Acute Perfusion/Diffusion Ischemic Stroke Lesions Evolution: A Pilot Study Derived from Longitudinal MR Patient Data
  
  • Rekik Islem
  • Allassonnière Stéphanie
  • Durrleman Stanley
  • Carpenter Trevor
  • Wardlaw Joanna M
  Computational and Mathematical Methods in Medicine, Hindawi Publishing Corporation, 2013. The spatiotemporal evolution of stroke lesions, from acute injury to final tissue damage, is complex. Diffusion-weighted (DWI) and perfusion-weighted (PWI) imaging is commonly used to detect early ischemic changes and attempts to distinguish between permanently damaged and salvageable tissues. To date, 2D and 3D measures of diffusion/perfusion regions at individual timepoints have been widely used but may underestimate the true lesion spatio-temporal dynamics. Currently there is no spatio-temporal 4D dynamic model that simulates the continuous evolution of ischemic stroke from MR images. We determined whether a 4D current-based diffeomorphic model, developed in the field of statistical modeling for measuring the variability of anatomical surfaces, could estimate patient-specific spatio-temporal continuous evolution for MR PWI (measured as mean transit time, (MTT)) and DWI lesions. In our representative pilot sample, the model fitted the data well. Our dynamic analysis of lesion evolution showed different patterns; for example, some DWI/PWI dynamic changes corresponded with DWI lesion expansion into PWI lesions, but other patterns were much more complex and diverse. There was wide variation in the time when the final tissue damage was reached after stroke for DWI and MTT (10.1155/2013/283593)
  DOI : 10.1155/2013/283593
• Normal forms and invariants for 2-dimensional almost-Riemannian structures
  
  • Boscain Ugo
  • Charlot Grégoire
  • Ghezzi Roberta
  Differential Geometry and its Applications, Elsevier, 2013, 31 (1), pp.41-62. 2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are "complete" in the sense that they permit to recognize locally isometric structures. The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel. Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization (10.1016/j.difgeo.2012.10.001)
  DOI : 10.1016/j.difgeo.2012.10.001
• Asymptotic enumeration of Eulerian circuits for graphs with strong mixing properties
  
  • Isaev Mikhail
  Izvestiya RAN. Serya Matematicheskaya, 2013, 77 (6), pp.45-70. We prove an asymptotic formula for the number of Eulerian circuits for graphs with strong mixing properties and with vertices having even degrees. The exact value is determined up to the multiplicative error $O(n^{-1/2+\varepsilon})$, where $n$ is the number of vertices
• New global stability estimates for monochromatic inverse acoustic scattering
  
  • Isaev Mikhail
  • Novikov Roman
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2013, 45 (3), pp.1495-1504. We give new global stability estimates for monochromatic inverse acoustic scattering. These estimates essentially improve estimates of [P. Hahner, T. Hohage, SIAM J. Math. Anal., 33(3), 2001, 670-685] and can be considered as a solution of an open problem formulated in the aforementioned work.
• Analysis of the factorization method for a general class of boundary conditions
  
  • Chamaillard Mathieu
  • Chaulet Nicolas
  • Haddar Houssem
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2013. We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the farfield operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the testing function. (10.1515/jip-2013-0013)
  DOI : 10.1515/jip-2013-0013
• Path Planning and Ground Control Station Simulator for UAV
  
  • Ajami Alain
  • Balmat Jean-François
  • Gauthier Jean-Paul
  • Maillot Thibault
  , 2013, pp.1-13. no abstract
• Partition-Based Conditional Density Estimation
  
  • Cohen Serge X.
  • Le Pennec Erwan
  ESAIM: Probability and Statistics, EDP Sciences, 2013, 17, pp.672--697. (10.1051/ps/2012017)
  DOI : 10.1051/ps/2012017
• Optimally swimming stokesian robots
  
  • Alouges François
  • Desimone Antonio
  • Heltai Luca
  • Lefebvre-Lepot Aline
  • Merlet Benoît
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2013, 18 (5), pp.1189-1215. We study self-propelled stokesian robots composed of assemblies of balls, in dimensions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow's theorem in an analytic framework, similar to what has been done in [3] for an axisymmetric system swimming along the axis of symmetry. We generalize the analyticity result given in [3] to the situation where the swimmers can move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail. (10.3934/dcdsb.2013.18.1189)
  DOI : 10.3934/dcdsb.2013.18.1189
• Direct competition results from strong competiton for limited resource
  
  • Mirrahimi Sepideh
  • Perthame Benoît
  • Wakano Joe Yuichiro
  Journal of Mathematical Biology, Springer, 2013, pp.0303-6812. We study a model of competition for resource through a chemostat-type model where species consume the common resource that is constantly supplied. We assume that the species and resources are characterized by a continuous trait. As already proved, this model, although more complicated than the usual Lotka-Volterra direct competition model, describes competitive interactions leading to concentrated distributions of species in continuous trait space. Here we assume a very fast dynamics for the supply of the resource and a fast dynamics for death and uptake rates. In this regime we show that factors that are independent of the resource competition become as important as the competition efficiency and that the direct competition model is a good approximation of the chemostat. Assuming these two timescales allows us to establish a mathematically rigorous proof showing that our resource-competition model with continuous traits converges to a direct competition model. We also show that the two timescales assumption is required to mathematically justify the corresponding classic result on a model consisting of only finite number of species and resources (MacArthur, R. Theor. Popul. Biol. 1970:1, 1-11). This is performed through asymptotic analysis, introducing different scales for the resource renewal rate and the uptake rate. The mathematical difficulty relies in a possible initial layer for the resource dynamics. The chemostat model comes with a global convex Lyapunov functional. We show that the particular form of the competition kernel derived from the uptake kernel, satisfies a positivity property which is known to be necessary for the direct competition model to enjoy the related Lyapunov functional. (10.1007/s00285-013-0659-5)
  DOI : 10.1007/s00285-013-0659-5
• Preliminary control variates to improve empirical regression methods
  
  • Benzineb Tarik
  • Gobet Emmanuel
  Monte Carlo Methods and Applications, De Gruyter, 2013, 19 (4), pp.331--354. We design a variance reduction method to reduce the estimation error in regression problems. It is based on an appropriate use of other known regression functions. Theoretical estimates are supporting this improvement and numerical experiments are illustrating the efficiency of the method.