Publications

2013

• 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.
• 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).
• 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
• On the robust superhedging of measurable claims
  
  • Possamaï Dylan
  • Royer Guillaume
  • Touzi Nizar
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (95), pp.1-13. The problem of robust hedging requires to solve the problem of superhedging under a nondominated family of singular measures. Recent progress was achieved by van Handel, Neufeld, and Nutz. We show that the dual formulation of this problem is valid in a context suitable for martingale optimal transportation or, more generally, for optimal transportation under controlled stochastic dynamics (10.1214/ECP.v18-2739)
  DOI : 10.1214/ECP.v18-2739
• Daphnias: from the individual based model to the large population equation
  
  • Metz J.A.J.
  • Tran Viet Chi
  Journal of Mathematical Biology, Springer, 2013, 66 (4-5), pp.915--933. The class of deterministic 'Daphnia' models treated by Diekmann et~al. (J Math Biol 61: 277--318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23: 114--135, 1983) and Diekmann et~al. (Nieuw Archief voor Wiskunde 4: 82--109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population ('Daphnia') and an unstructured resource ('algae'). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and throught their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et~al., l.c.). (10.1007/s00285-012-0619-5)
  DOI : 10.1007/s00285-012-0619-5
• Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem in a Heterogeneous Domain
  
  • Allaire Grégoire
  • Habibi Zakaria
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2013, 45 (3), pp.1136-1178. We are interested in the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order than the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection and radiative transfer in the fluid part (the cylinders). A non-local boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a 3D to 2D asymptotic analysis since the radiative transfer in the limit cell problem is purely two-dimensional. Eventually, we provide some 3D numerical results in order to show the convergence and the computational advantages of our homogenization method.
• Asymptotic enumeration of Eulerian orientations for graphs with strong mixing properties
  
  • Isaev Mikhail
  • Kseniia Isaeva
  Journal of Applied and Industrial Mathematics / Sibirskii Zhurnal Industrial'noi Matematiki and Diskretnyi Analiz i Issledovanie Operatsii, MAIK Nauka/Interperiodica, 2013, 20 (6), pp.40-58. We prove an asymptotic formula for the number of Eulerian orientations 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+epsilon}), where n is the number of vertices.
• On the extinction of Continuous State Branching Processes with catastrophes
  
  • Bansaye Vincent
  • Pardo Millan Juan Carlos
  • Smadi Charline
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18, pp.1-31. We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection. (10.1214/EJP.v18-2774)
  DOI : 10.1214/EJP.v18-2774
• Time-reversal in visco-elastic media.
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Wahab Abdul
  European Journal of Applied Mathematics, Cambridge University Press (CUP), 2013 (24), pp.565-600. In this paper, we consider the problem of reconstructing sources in a homogeneous viscoelastic medium from wavefield measurements. We first present a modified time-reversal imaging algorithm based on a weighted Helmholtz decomposition and justify mathematically that it provides a better approximation than by simply time reversing the displacement field, where artifacts due to the coupling betwe en the pressure and shear waves appear. Then, we investigate the source inverse problem in an elastic attenuating medium. We provide a regularized time-reversal imagin g which corrects the attenuation effect at the first order. The results of this paper yie ld the fundamental tools for solving imaging problems in elastic media using cross correl ation techniques
• A Hamilton-Jacobi approach to junction problems and application to traffic flows
  
  • Imbert Cyril
  • Monneau Régis
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (01), pp.pp 129-166. This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ''junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems. (10.1051/cocv/2012002)
  DOI : 10.1051/cocv/2012002
• Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions
  
  • Isaev Mikhail
  Journal of Mathematics, Hindawi Publishing Corp., 2013, pp.DOI:10.1155/2013/318154. We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimension $d\geq 3$. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension $d=2$ is also given. (10.1155/2013/318154)
  DOI : 10.1155/2013/318154
• Semi-infinite paths of the two dimensional radial spanning tree
  
  • Baccelli François
  • Coupier David
  • Tran Viet Chi
  Advances in Applied Probability, Applied Probability Trust, 2013, 45 (4), pp.895-916. We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius $r$ grows sublinearly with $r$. Then, we prove that in each (deterministic) direction, there exists with probability one a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there are more than one semi-infinite paths is dense in $[0,2\pi)$. It corresponds to possible asymptotic directions of competition interfaces. We show that the RST can be decomposed in at most five infinite subtrees directly connected to the root. The interfaces separating these subtrees are studied and simulations are provided. (10.1239/aap/1386857849)
  DOI : 10.1239/aap/1386857849
• Aircraft classification with a low resolution infrared sensor
  
  • Lefebvre Sidonie
  • Allassonniere Sidonie
  • Jakubowicz Jérémie
  • Lasne Thomas
  • Moulines Éric
  Machine Vision and Applications, Springer Verlag, 2013, 24 (1), pp.175-186. Existing computer simulations of aircraft infrared signature (IRS) do not account for dispersion induced by uncertainty on input parameters, such as aircraft aspect angles and meteorological conditions. As a result, they are of little use to quantify the detection performance of IR optronic systems: in this case, the scenario encompasses a lot of possible situations that must indeed be considered, but cannot be individually simulated. In this paper, we focus on low resolution infrared sensors and we propose a methodological approach for predicting simulated IRS dispersion of an aircraft, and performing a classification of different aircraft on the resulting set of low resolution infrared images. It is based on a quasi-Monte Carlo survey of the code output dispersion, and on a maximum likelihood classification taking advantage of Bayesian dense deformable template models estimation. This method is illustrated in a typical scenario, i.e., a daylight air-to-ground full-frontal attack by a generic combat aircraft flying at low altitude, over a database of 30,000 simulated aircraft images. Assuming a spatially white noise background model, classification performance is very promising, and appears to be more accurate than more classical state of the art techniques (such as kernel-based support vector classifiers). (10.1007/s00138-012-0437-1)
  DOI : 10.1007/s00138-012-0437-1
• A homogenization approach for the motion of motor proteins
  
  • Mirrahimi Sepideh
  • Souganidis Panagiotis E.
  Nonlinear Differential Equations and Applications, Springer Verlag, 2013, 20, pp.129-147. We consider the asymptotic behavior of an evolving weakly coupled Fokker-Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are respectively proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile. (10.1007/s00030-012-0156-3)
  DOI : 10.1007/s00030-012-0156-3
• On the asymptotics of a Robin eigenvalue problem
  
  • Cakoni Fioralba
  • Chaulet Nicolas
  • Haddar Houssem
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2013, 351, pp.517-521. The considered Robin problem can formally be seen as a small perturbation of a Dirichlet problem. However, due to the sign of the impedance value, its associated eigenvalues converge point-wise to −∞ as the perturbation goes to zero. We prove in this case that Dirichlet eigenpairs are the only accumulation points of the Robin eigenpairs with normalized eigenvectors. We then provide a criterion to select accumulating sequences of eigenvalues and eigenvectors and exhibit their full asymptotic with respect to the small parameter. (10.1016/j.crma.2013.07.022)
  DOI : 10.1016/j.crma.2013.07.022
• A Formula for Popp’s Volume in Sub-Riemannian Geometry
  
  • Barilari Davide
  • Rizzi Luca
  Analysis and Geometry in Metric Spaces, Versita, 2013, 1. For an equiregular sub-Riemannian manifold M, Popp's volume is a smooth volume which is canonically associated with the sub-Riemannian structure, and it is a natural generalization of the Riemannian one. In this paper we prove a general formula for Popp's volume, written in terms of a frame adapted to the sub-Riemannian distribution. As a first application of this result, we prove an explicit formula for the canonical sub-Laplacian, namely the one associated with Popp's volume. Finally, we discuss sub-Riemannian isometries, and we prove that they preserve Popp's volume. We also show that, under some hypotheses on the action of the isometry group of M, Popp's volume is essentially the unique volume with such a property. (10.2478/agms-2012-0004)
  DOI : 10.2478/agms-2012-0004
• 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
• Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant
  
  • Lafitte-Godillon Pauline
  • Raschel Kilian
  • Tran Viet Chi
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2013, 73 (2), pp.700-722. In this paper, we study a flower population in which self-reproduction is not permitted. Individuals are diploid, {that is, each cell contains two sets of chromosomes}, and {distylous, that is, two alleles, A and a, can be found at the considered locus S}. Pollen and ovules of flowers with the same genotype at locus S cannot mate. This prevents the pollen of a given flower to fecundate its {own} stigmata. Only genotypes AA and Aa can be maintained in the population, so that the latter can be described by a random walk in the positive quadrant whose components are the number of individuals of each genotype. This random walk is not homogeneous and its transitions depend on the location of the process. We are interested in the computation of the extinction probabilities, {as} extinction happens when one of the axis is reached by the process. These extinction probabilities, which depend on the initial condition, satisfy a doubly-indexed recurrence equation that cannot be solved directly. {Our contribution is twofold : on the one hand, we obtain an explicit, though intricate, solution through the study of the PDE solved by the associated generating function. On the other hand, we provide numerical results comparing stochastic and deterministic approximations of the extinction probabilities. (10.1137/120864258)
  DOI : 10.1137/120864258
• 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
• 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.
• 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.
• 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
• 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
• 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