Publications

2015

• Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost
  
  • Bokanowski Olivier
  • Picarelli Athena
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2015, 71 (1), pp.125--163. This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach. (10.1007/s00245-014-9255-3)
  DOI : 10.1007/s00245-014-9255-3
• Identification of magnetic deposits in 2-D axisymmetric eddy current models via shape optimization
  
  • Jiang Zixian
  • Haddar Houssem
  • Lechleiter Armin
  • El-Guedri Mabrouka
  Inverse Problems in Science and Engineering, Taylor & Francis, 2015. The non-destructive control of steam generators is an essential task for the safe and failure-free operation of nuclear power plants. Due to magnetite particles in the cooling water of the plants, a frequent source for failures are magnetic deposits in the cooling loop of steam generators. From eddy current signals measured inside a U-tube in the steam generator, we propose and analyze a regularized shape optimization algorithm to identify magnetic deposits outside the U-tube with either known or unknown physical properties. Motivated by the cylindrical geometry of the U-tubes we assume an axisymmetric problem setting, reducing Maxwell's equations to a 2-D elliptic eddy current problem. The feasibility of the proposed algorithms is illustrated via numerical examples demonstrating in particular the stability of the method with respect to noise.
• A Convergent Data Completion Algorithm Using Surface Integral Equations
  
  • Boukari Yosra
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2015, pp.21. We propose and analyze a data completion algorithm based on the representation of the solution in terms of surface integral operators to solve the Cauchy problem for the Helmholtz or the Laplace equations. The proposed method is non iterative and intrinsically handle the case of noisy and incompatible data. In order to cope with the ill-posedness of the problem, our formulation is compatible with standard regularization methods associated with linear ill posed inverse problems and leads to convergent scheme. We numerically validate our method with different synthetic examples using a Tikhonov regularization.
• A deterministic approximation method in shape optimization under random uncertainties
  
  • Allaire Grégoire
  • Dapogny Charles
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2015, 1, pp.83-143. This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number N of random variables, and using first-or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions. (10.5802/smai-jcm.5)
  DOI : 10.5802/smai-jcm.5
• On the control of spin-boson systems
  
  • Boscain Ugo
  • Mason Paolo
  • Panati Gianluca
  • Sigalotti Mario
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2015, 56 (9). In this paper, we study the so-called spin-boson system, namely, a two-level system in interaction with a distinguished mode of a quantized bosonic field. We give a brief description of the controlled Rabi and Jaynes-Cummings models and we discuss their appearance in the mathematics and physics literature. We then study the controllability of the Rabi model when the control is an external field acting on the bosonic part. Applying geometric control techniques to the Galerkin approximation and using perturbation theory to guarantee non-resonance of the spectrum of the drift operator, we prove approximate controllability of the system, for almost every value of the interaction parameter. (C) 2015 AIP Publishing LLC. (10.1063/1.4929543)
  DOI : 10.1063/1.4929543
• Definable Zero-Sum Stochastic Games
  
  • Bolte Jérôme
  • Gaubert Stéphane
  • Vigeral Guillaume
  Mathematics of Operations Research, INFORMS, 2015, 40 (1), pp.171-191. Definable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally subanalytic Shapley operator. %Showing the definability of the Shapley operator in full generality appears thus as a complex and challenging issue. } Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given. (10.1287/moor.2014.0666)
  DOI : 10.1287/moor.2014.0666
• Mathematical Methods in Elasticity Imaging
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Kang Hyeonbae
  • Lee Hyundae
  • Wahab Abdul
  , 2015, pp.240 p..
• A Priori Error Estimate of a Multiscale Finite Element Method for Transport Modeling
  
  • Ouaki Franck
  • Allaire Grégoire
  • Desroziers Sylvain
  • Enchéry Guillaume
  SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, Springer, 2015, 67 (1), pp.1-37. This work proposes an \textit{a priori} error estimate of a multiscale finite element method to solve convection-diffusion problems where both velocity and diffusion coefficient exhibit strong variations at a scale which is much smaller than the domain of resolution. In that case, classical discretization methods, used at the scale of the heterogeneities, turn out to be too costly. Our method, introduced in~\cite{ECCOMAS}, aims at solving this kind of problems on coarser grids with respect to the size of the heterogeneities by means of particular basis functions. These basis functions are defined using cell problems and are designed to reproduce the variations of the solution on an underlying fine grid. Since all cell problems are independent from each other, these problems can be solved in parallel, which makes the method very efficient when used on parallel architectures. This article focuses on the proof of an \textit{a priori} error estimate of this method.
• Configuration Tracking for Mechanical Systems by Kinematic Reduction and Fast Oscillating Controls
  
  • Barbero-Liñán M.
  • Sigalotti Mario
  , 2015.
• Controllability of spin-boson systems
  
  • Boscain Ugo
  • Mason Paolo
  • Panati Gianluca
  • Sigalotti Mario
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2015, 56. In this paper we study the so-called spin-boson system, namely {a two-level system} in interaction with a distinguished mode of a quantized bosonic field. We give a brief description of the controlled Rabi and Jaynes--Cummings models and we discuss their appearance in the mathematics and physics literature. We then study the controllability of the Rabi model when the control is an external field acting on the bosonic part. Applying geometric control techniques to the Galerkin approximation and using perturbation theory to guarantee non-resonance of the spectrum of the drift operator, we prove approximate controllability of the system, for almost every value of the interaction parameter.
• Second order mean field games with degenerate diffusion and local coupling
  
  • Cardaliaguet Pierre
  • Graber J.
  • Porretta Alessio
  • Tonon Daniela
  Nonlinear Differential Equations and Applications, Springer Verlag, 2015, 22 (5), pp.1287-1317. We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.
• Hawkes Processes in Finance
  
  • Bacry Emmanuel
  • Mastromatteo Iacopo
  • Muzy Jean-François
  Market microstructure and liquidity, World scientific publishing company, 2015, 01 (01), pp.1550005. no abstract (10.1142/S2382626615500057)
  DOI : 10.1142/S2382626615500057
• Dispersal is a major driver of the latitudinal diversity gradient of Carnivora
  
  • Rolland Jonathan
  • Condamine Fabien L.
  • Beeravolu Reddy Champak
  • Jiguet Frédéric
  • Morlon Hélène
  Global Ecology and Biogeography, Wiley, 2015, 24 (9), pp.1059 - 1071. <strong>Aim</strong> Understanding the relative contribution of diversification rates (speciation and extinction) and dispersal in the formation of the latitudinal diversity gradient - the decrease in species richness with increasing latitude - is a main goal of biogeography. The mammalian order Carnivora, which comprises 286 species, displays the traditional latitudinal diversity gradient seen in almost all mammalian orders. Yet the processes driving high species richness in the tropics may be fundamentally different in this group from that in other mammalian groups. Indeed, a recent study suggested that in Carnivora, unlike in all other major mammalian orders, net diversification rates are not higher in the tropics than in temperate regions. Our goal was thus to understand the reasons why there are more species of Carnivora in the tropics. <strong>Location</strong> World-wide. <strong>Methods</strong> We reconstructed the biogeographical history of Carnivora using a time-calibrated phylogeny of the clade comprising all terrestrial species and dispersal-extinction-cladogenesis models. We also analysed a fossil dataset of carnivoran genera to examine how the latitudinal distribution of Carnivora varied through time. <strong>Results</strong> Our biogeographical analyses suggest that Carnivora originated in the East Palaearctic (i.e. Central Asia, China) in the early Palaeogene. Multiple independent lineages dispersed to low latitudes following three main paths: toward Africa, toward India/Southeast Asia and toward South America via the Bering Strait. These dispersal events were probably associated with local extinctions at high latitudes. Fossil data corroborate a high-latitude origin of the group, followed by late dispersal events toward lower latitudes in the Neogene. <strong>Main conclusions</strong> Unlike most other mammalian orders, which originated and diversified at low latitudes and dispersed out of the tropics', Carnivora originated at high latitudes, and subsequently dispersed southward. Our study provides an example of combining phylogenetic and fossil data to understand the generation and maintenance of global-scale geographical variations in species richness. (10.1111/geb.12354)
  DOI : 10.1111/geb.12354
• Axisymmetric eddy current inspection of highly conducting thin layers via asymptotic models
  
  • Haddar Houssem
  • Jiang Zixian
  Inverse Problems, IOP Publishing, 2015. Thin copper deposits covering the steam generator tubes can blind eddy current probes in non-destructive testings of problematic faults and are therefore important to be identified. Existing methods based on shape reconstruction using eddy current signals encounter difficulties of high numerical costs due to the layer's small thickness and high conductivity. In this article, we approximate the axisymmetric eddy current problem with some appropriate asymptotic models using effective transmission conditions representing the thin deposits. In these models, the geometrical information related to the deposit is transformed into parameter coefficients on a fictitious interface. Standard iterative inversion algorithm is then applied to the asymptotic models in order to reconstruct the thickness of the thin copper layers. Numerical tests both validating the asymptotic model and benefit of the inversion procedure are provided.
• Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Meng Shixu
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2015, pp.26. In this paper we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of boundary integral equa- tion, and assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point.
• Unveiling the Diversification Dynamics of Australasian Predaceous Diving Beetles in the Cenozoic
  
  • Toussaint Emmanuel F.A.
  • Condamine Fabien L.
  • Hawlitschek Oliver
  • Watts Chris H.
  • Porch Nick
  • Hendrich Lars
  • Balke Michael
  Systematic Biology, Oxford University Press (OUP), 2015, 64 (1), pp.3-24. During the Cenozoic, Australia experienced major climatic shifts that have had dramatic ecological consequences for the modern biota. Mesic tropical ecosystems were progressively restricted to the coasts and replaced by arid-adapted floral and faunal communities. Whilst the role of aridification has been investigated in a wide range of terrestrial lineages, the response of freshwater clades remains poorly investigated. To gain insights into the diversification processes underlying a freshwater radiation, we studied the evolutionary history of the Australasian predaceous diving beetles of the tribe Hydroporini (147 described species). We used an integrative approach including the latest methods in phylogenetics, divergence time estimation, ancestral character state reconstruction, and likelihood-based methods of diversification rate estimation. Phylogenies and dating analyses were reconstructed with molecular data from seven genes (mitochondrial and nuclear) for 117 species (plus 12 outgroups). Robust and well-resolved phylogenies indicate a late Oligocene origin of Australasian Hydroporini. Biogeographic analyses suggest an origin in the East Coast region of Australia, and a dynamic biogeographic scenario implying dispersal events. The group successfully colonized the tropical coastal regions carved by a rampant desertification, and also colonized groundwater ecosystems in Central Australia. Diversification rate analyses suggest that the ongoing aridification of Australia initiated in the Miocene contributed to a major wave of extinctions since the late Pliocene probably attributable to an increasing aridity, range contractions and seasonally disruptions resulting from Quaternary climatic changes. When comparing subterranean and epigean genera, our results show that contrasting mechanisms drove their diversification and therefore current diversity pattern. The Australasian Hydroporini radiation reflects a combination of processes that promoted both diversification, resulting from new ecological opportunities driven by initial aridification, and a subsequent loss of mesic adapted diversity due to increasing aridity. (10.1093/sysbio/syu067)
  DOI : 10.1093/sysbio/syu067
• Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion
  
  • Belaouar Radoin
  • de Bouard Anne
  • Debussche Arnaud
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2015, 3 (1), pp.103-132. This article is devoted to the numerical study of a nonlinear Schrödinger equation in which the coefficient in front of the group velocity dispersion is multiplied by a real valued Gaussian white noise. We first perform the numerical analysis of a semi-discrete Crank-Nicolson scheme in the case when the continuous equation possesses a unique global solution. We prove that the strong order of convergence in probability is equal to one in this case. In a second step, we numerically investigate, in space dimension one, the behavior of the solutions of the equation for different power nonlinearities, corresponding to subcritical, critical or supercritical nonlinearities in the deterministic case. Numerical evidence of a change in the critical power due to the presence of the noise is pointed out. (10.1007/s40072-015-0044-z)
  DOI : 10.1007/s40072-015-0044-z
• On the scaling limits of Galton–Watson processes in varying environments
  
  • Bansaye Vincent
  • Simatos Florian
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20 (75), pp.1-36. We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes. (10.1214/EJP.v20-3812)
  DOI : 10.1214/EJP.v20-3812
• Converse Lyapunov-Krasovskii theorems for uncertain retarded differential equations
  
  • Haidar Ihab
  • Mason Paolo
  • Sigalotti Mario
  Automatica, Elsevier, 2015, 62, pp.263-273. In this article we give a collection of converse Lyapunov–Krasovskii theorems for uncertain retarded differential equations. We show that the existence of a weakly-degenerate Lyapunov–Krasovskii functional is a necessary and sufficient condition for the global exponential stability of linear retarded functional differential equations. This is carried out using a switched system representation approach. (10.1016/j.automatica.2015.09.034)
  DOI : 10.1016/j.automatica.2015.09.034
• A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions.
  
  • Chambolle Antonin
  • Pock Thomas
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2015, 1, pp.29 - 54. We analyze alternating descent algorithms for minimizing the sum of a quadratic function and block separable non-smooth functions. In case the quadratic interactions between the blocks are pairwise, we show that the schemes can be accelerated, leading to improved convergence rates with respect to related accelerated parallel proximal descent. As an application we obtain very fast algorithms for computing the proximity operator of the 2D and 3D total variation. (10.5802/smai-jcm.3)
  DOI : 10.5802/smai-jcm.3
• Avis en réponse à la saisine 150519 - dossier C-NL-13-02. Paris, le 7 septembre 2015
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie Anne M. A.
  • Bellivier Florence
  • Berny Philippe
  • Bertheau Yves
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Coléno François
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Eychenne Nathalie
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Jestin André
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie V.
  • Lemaire Olivier O.
  • Lereclus Didier
  • Maximilien Rémi
  • Meurs Eliane
  • Moreau de Bellaing Cédric
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Parzy Daniel
  • Regnault-Roger Catherine
  • Renard Michel
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2015, pp.21 p.. Le Haut Conseil des biotechnologies (HCB) a été saisi le 20 mai 2015 par les autorités compétentes françaises (le ministère de l’Agriculture, de l’Agroalimentaire et de la Forêt) d’une demande d’avis relative au dossier C/NL/13/02 de demande de mise sur le marché de la lignée d’oeillets génétiquement modifiés FLO-40685-1 à des fins d’importation et de commercialisation de fleurs coupées. Ce dossier a été déposé par la société Suntory Holdings Limited auprès des autorités compétentes néerlandaises dans le cadre de la directive 2001/18/CE. Conformément à cette directive, la Commission européenne a adressé le rapport d’évaluation des Pays-Bas ainsi que le dossier du pétitionnaire à l’ensemble des Etats membres. Par cette saisine, les autorités compétentes françaises consultent le HCB au stade ultime de la préparation au vote des Etats membres à la Commission européenne. Le Comité scientifique (CS)2 du HCB a examiné le dossier en séance du 25 juin 2015 sous la présidence de Jean-Christophe Pagès. Le présent avis a été adopté par voie électronique le 7 septembre 2015 et publié le 10 septembre 2015.
• Phaseless inverse scattering in the one-dimensional case
  
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2015, 3 (1), pp.64-70. We consider the one-dimensional Schrödinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line x ≥ 0. For this equation at fixed positive energy we give explicit formulas for finding the full complex valued reflection coefficient to the left from appropriate phaseless scattering data measured on the left, i.e. for x < 0. Using these formulas and known inverse scattering results we obtain global uniqueness and reconstruction results for phaseless inverse scattering in dimension d = 1.
• Artificial boundary conditions for axisymmetric eddy current probe problems
  
  • Haddar Houssem
  • Jiang Zixian
  • Lechleiter Armin
  Computers & Mathematics with Applications, Elsevier, 2015, 68 (12, Part A,), pp.1844–1870. We study different strategies for the truncation of computational domains in the simulation of eddy current probes of elongated axisymmetric tubes. For axial fictitious boundaries, an exact Dirichlet-to-Neumann map is proposed and mathematically analyzed via a non-selfadjoint spectral problem: under general assumptions we show convergence of the solution to an eddy current problem involving a truncated Dirichlet-to-Neumann map to the solution on the entire, unbounded axisymmetric domain as the truncation parameter tends to infinity. Under stronger assumptions on the physical parameters of the eddy current problem, convergence rates are shown. We further validate our theoretical results through numerical experiments for a realistic physical setting inspired by eddy current probes of nuclear reactor core tubes. (10.1016/j.camwa.2014.10.008)
  DOI : 10.1016/j.camwa.2014.10.008
• The topological derivative of stress-based cost functionals in anisotropic elasticity
  
  • Delgado Gabriel
  • Bonnet Marc
  Computers & Mathematics with Applications, Elsevier, 2015, 69, pp.1144-1166. The topological derivative of cost functionals J that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative dJ(z) of J quantifies the asymptotic behavior of J under the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius a at a specified location z. The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small a makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals. The asymptotic perturbation of J is shown to be of order O(a^3) for a wide class of stress-based cost functionals having smooth densities. The topological derivative of J, i.e. the coefficient of the O(a^3) perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of J is mathematically justified (i.e. its remainder is proved to be of order o(a^3)). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of \dJ on cases involving anisotropic elasticity and non-quadratic cost functionals. (10.1016/j.camwa.2015.03.010)
  DOI : 10.1016/j.camwa.2015.03.010
• Energy release rate for non smooth cracks in planar elasticity
  
  • Babadjian Jean-François
  • Chambolle Antonin
  • Lemenant Antoine
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2015, 2, pp.117-152. This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.