Publications

2013

• 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.
• Stability estimates for determination of potential from the impedance boundary map
  
  • Isaev Mikhail
  • Novikov Roman
  Algebra and Analysis, 2013, 25 (1), pp.37-63. We study the impedance boundary map (or Robin-to-Robin map) for the Schrodinger equation in open bounded demain at fixed energy in multidimensions. We give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map.
• Stabilization of two-dimensional persistently excited linear control systems with arbitrary rate of convergence
  
  • Chitour Yacine
  • Mazanti Guilherme
  • Sigalotti Mario
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2013, 51 (2), pp.801-823. We study the control system $\dot x = A x + \alpha(t) b u$ where the pair $(A, b)$ is controllable, $x \in \mathbb R^2$, $u \in \mathbb R$ is a scalar control and the unknown signal $\alpha: \mathbb R_+ \to [0, 1]$ is $(T,\mu)$-persistently exciting (PE), i.e., there exists $T \geq \mu > 0$ such that, for all $t \in \mathbb R_+$, $\int_t^{t + T} \alpha(s) ds \geq \mu$. We are interested in the stabilization problem of this system by a linear state feedback $u = - K x$. In this paper, we positively answer a question asked in \cite{YacineMario} and prove the following: Assume that the class of $(T,\mu)$-PE signals is restricted to those which are $M$\nobreakdash-Lipschitzian, where $M>0$ is a positive constant. Then, given any $C>0$, there exists a linear state feedback $u = - K x$ where $K$ only depends on $(A,b)$ and $T,\mu,M$ so that, for every $M$-Lipschitzian $(T,\mu)$-PE signal, the rate of exponential decay of the time-varying system $\dot x = (A -\alpha(t) bK)x $ is greater than $C$. (10.1137/110848153)
  DOI : 10.1137/110848153
• A general Hamilton-Jacobi framework for nonlinear state-constrained control problems
  
  • Altarovici Albert
  • Bokanowski Olivier
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (2), pp.337--357. The paper deals with deterministic optimal control problem with state constraints and non-linear dynamics. It is known for such a problem that the value function is in general discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumption, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypass the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach. (10.1051/cocv/2012011)
  DOI : 10.1051/cocv/2012011
• New approximations in local volatility models
  
  • Gobet Emmanuel
  • Suleiman Ali
  , 2013, pp.305--330. For general time-dependent local volatility models, we propose new approximation formulas for the price of call options. This extends previous results of [BGM10b] where stochastic expansions combined with Malliavin calculus were performed to obtain approximation formulas based on the local volatility At The Money. Here, we derive alternative expansions involving the local volatility at strike. Averaging both expansions give even more accurate results. Approximations of the implied volatility are provided as well.
• A mesh evolution algorithm based on the level set method for geometry and topology optimization
  
  • Allaire Grégoire
  • Dapogny Charles
  • Frey Pascal
  Structural and Multidisciplinary Optimization, Springer Verlag, 2013, 48 (4), pp.711-715. An approach for structural optimization is proposed, which combines the versatility of the level set method for handling large deformations and topology changes with the accurate description of the geometry provided by an exact mesh of the shape. The key ingredients of this method are efficient algorithms for (i) moving a level set function on an unstructured mesh, (ii) remeshing the surface corresponding to the zero level set and (iii) simultaneously adaptating the volumic mesh which fits to this surfacic mesh. (10.1007/s00158-013-0929-2)
  DOI : 10.1007/s00158-013-0929-2
• Numerical approximation of Nash equilibria for a class of non-cooperative differential games
  
  • Cacace Simone
  • Cristiani Emiliano
  • Falcone Maurizio
  , 2013, 16, pp.45-58. In this paper we propose a numerical method to obtain an approximation of Nash equilibria for multi-player non-cooperative games with a special structure. We consider the infinite horizon problem in a case which leads to a system of Hamilton-Jacobi equations. The numerical method is based on the Dynamic Programming Principle for every equation and on a global fixed point iteration. We present the numerical solutions of some two-player games in one and two dimensions. The paper has an experimental nature, but some features and properties of the approximation scheme are discussed.
• On the Well-Posedness , Stability And Accuracy Of An Asymptotic Model For Thin Periodic Interfaces In Electromagnetic Scattering Problems
  
  • Delourme Bérangère
  • Haddar Houssem
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We analyze the well-posedness and stability properties of a parameter dependent problem that models the reflection and transmission of electromagnetic waves at a thin and rapidly oscillating interface. The latter is modeled using approximate interface conditions that can be derived using asymptotic expansion of the exact solution with respect to the small parameter (proportional to the periodicity length of oscillations and the width of the interface). The obtained uniform stability results are then used to analyze the accuracy (with respect to the small parameter) of the proposed model.
• Coupling techniques for nonlinear hyperbolic equations. III. Well-balanced approximation of thick interfaces
  
  • Boutin Benjamin
  • Coquel Frédéric
  • LeFloch Philippe G.
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2013, 51 (2), pp.1108-1133. We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems, based on the so-called thin interface model, which uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation but introduce a new framework for coupling problems, now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe first that the Cauchy problem is well-posed. Our main achievements in the present paper are, on one hand, the design of a new well-balanced finite volume scheme which is adapted to the thick interface model and, on the other hand, a proof of the convergence of this scheme toward the unique entropy solution to the Cauchy problem for a large class of nonlinear hyperbolic equations. Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used instead. Our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the entropy dissipation of the proposed discrete scheme. (10.1137/120865768)
  DOI : 10.1137/120865768
• On the control of spin-boson systems
  
  • Boscain Ugo
  • Mason Paolo
  • Panati Gianluca
  • Sigalotti Mario
  , 2013, pp.2110-2115. (10.23919/ecc.2013.6669621)
  DOI : 10.23919/ecc.2013.6669621
• Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations
  
  • Amirat Youcef
  • Hamdache Kamel
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 33 (8), pp.3289-3320. In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field. (10.3934/dcds.2013.33.3289)
  DOI : 10.3934/dcds.2013.33.3289
• Log-normal continuous cascade model of asset returns: aggregation properties and estimation
  
  • Bacry Emmanuel
  • Kozhemyak Alexey
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2013, 13 (5), pp.795-818. no abstract (10.1080/14697688.2011.647411)
  DOI : 10.1080/14697688.2011.647411
• Some limit theorems for Hawkes processes and application to financial statistics
  
  • Bacry Emmanuel
  • Delattre Sylvain
  • Hoffmann Marc
  • Muzy Jean-François
  Stochastic Processes and their Applications, Elsevier, 2013, 123 (7), pp.2475 - 2499. Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms. (10.1016/j.spa.2013.04.007)
  DOI : 10.1016/j.spa.2013.04.007
• The 2d-Directed Spanning Forest is almost surely a tree
  
  • Coupier David
  • Tran Viet Chi
  Random Structures and Algorithms, Wiley, 2013, 42 (1), pp.59-72. We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF. (10.1002/rsa.20400)
  DOI : 10.1002/rsa.20400
• A geometric approach for convexity in some variational problem in the Gauss space
  
  • Goldman Michael
  Rendiconti del Seminario Matematico della Università di Padova, University of Padua / European Mathematical Society, 2013. In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
• Quantifying the Mutational Meltdown in Diploid Populations
  
  • Coron Camille
  • Méléard Sylvie
  • Porcher Emmanuelle
  • Robert Alexandre
  The American Naturalist, University of Chicago Press, 2013, 181 (5), pp.623-636. Mutational meltdown, in which demographic and genetic processes mutually reinforce one another to accelerate the extinction of small populations, has been poorly quantified despite its potential importance in conservation biology. Here we present a model-based framework to study and quantify the mutational meltdown in a finite diploid population that is evolving continuously in time and subject to resource competition. We model slightly deleterious mutations affecting the population demographic parameters and study how the rate of mutation fixation increases as the genetic load increases, a process that we investigate at two timescales: an ecological scale and a mutational scale. Unlike most previous studies, we treat population size as a random process in continuous time. We show that as deleterious mutations accumulate, the decrease in mean population size accelerates with time relative to a null model with a constant mean fixation time. We quantify this mutational meltdown via the change in the mean fixation time after each new mutation fixation, and we show that the meltdown appears less severe than predicted by earlier theoretical work. We also emphasize that mean population size alone can be a misleading index of the risk of population extinction, which could be better evaluated with additional information on demographic parameters. (10.1086/670022)
  DOI : 10.1086/670022
• How pilots fly: An inverse optimal control problem approach
  
  • Maillot Thibault
  • Serres Ulysse
  • Gauthier Jean-Paul
  • Ajami Alain
  , 2013, pp.1792-1797. (10.1109/CDC.2013.6760142)
  DOI : 10.1109/CDC.2013.6760142
• Statistics of animal movement
  
  • Berthelot Geoffroy C.B.
  • Bansaye Vincent
  • Calenge C.
  , 2013.
• Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Jing Wenjia
  • Kang Hyeonbae
  • Wahab Abdul
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2013, 6 (4), pp.2174 - 2212. (10.1137/120899303)
  DOI : 10.1137/120899303
• Lineage selection and the maintenance of sex
  
  • de Vienne Damien M.
  • Giraud Tatiana
  • Gouyon Pierre-Henri
  PLoS ONE, Public Library of Science, 2013, 8 (6), pp.e66906. Sex predominates in eukaryotes, despite its short-term disadvantage when compared to asexuality. Myriad models have suggested that short-term advantages of sex may be sufficient to counterbalance its twofold costs. However, despite decades of experimental work seeking such evidence, no evolutionary mechanism has yet achieved broad recognition as explanation for the maintenance of sex. We explore here, through lineage-selection models, the conditions favouring the maintenance of sex. In the first model, we allowed the rate of transition to asexuality to evolve, to determine whether lineage selection favoured species with the strongest constraints preventing the loss of sex. In the second model, we simulated more explicitly the mechanisms underlying the higher extinction rates of asexual lineages than of their sexual counterparts. We linked extinction rates to the ecological and/ or genetic features of lineages, thereby providing a formalisation of the only figure included in Darwin's "The origin of species". Our results reinforce the view that the long-term advantages of sex and lineage selection may provide the most satisfactory explanations for the maintenance of sex in eukaryotes, which is still poorly recognized, and provide figures and a simulation website for training and educational purposes. Short-term benefits may play a role, but it is also essential to take into account the selection of lineages for a thorough understanding of the maintenance of sex. (10.1371/journal.pone.0066906)
  DOI : 10.1371/journal.pone.0066906
• Diffraction of Bloch Wave Packets for Maxwell's Equations
  
  • Allaire Grégoire
  • Palombaro Mariapia
  • Rauch Jeffrey
  Communications in Contemporary Mathematics, World Scientific Publishing, 2013, 15 (6), pp.1350040. We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations. (10.1142/S0219199713500405)
  DOI : 10.1142/S0219199713500405
• Ergodic Control and Polyhedral approaches to PageRank Optimization
  
  • Fercoq Olivier
  • Akian Marianne
  • Bouhtou Mustapha
  • Gaubert Stéphane
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2013, 58 (1), pp.134--148. We study a general class of PageRank optimization problems which involve finding an optimal outlink strategy for a web site subject to design constraints. We consider both a continuous problem, in which one can choose the intensity of a link, and a discrete one, in which in each page, there are obligatory links, facultative links and forbidden links. We show that the continuous problem, as well as its discrete variant when there are no constraints coupling different pages, can both be modeled by constrained Markov decision processes with ergodic reward, in which the webmaster determines the transition probabilities of websurfers. Although the number of actions turns out to be exponential, we show that an associated polytope of transition measures has a concise representation, from which we deduce that the continuous problem is solvable in polynomial time, and that the same is true for the discrete problem when there are no coupling constraints. We also provide efficient algorithms, adapted to very large networks. Then, we investigate the qualitative features of optimal outlink strategies, and identify in particular assumptions under which there exists a "master" page to which all controlled pages should point. We report numerical results on fragments of the real web graph. (10.1109/TAC.2012.2226103)
  DOI : 10.1109/TAC.2012.2226103
• The Moutard transformation and two-dimensional multi-point delta-type potentials
  
  • Novikov Roman
  • Taimanov Iskander
  Russian Mathematical Surveys, Turpion, 2013, 68 (5), pp.957–959. In the framework of the Moutard transformation formalism we find multi-point delta-type potentials for two-dimensional Schrodinger operators and their isospectral deformations on the zero energy level. In particular, these potentials are "reflectionless" in the sense of the Faddeev generalized "scattering" data.
• Statistical models for deformable templates in image and shape analysis (Modèles statistiques d'atlas déformables pour l'analyse d'images et de formes)
  
  • Allassonnière Stéphanie
  • Bigot Jérémie
  • Glaunès Joan Alexis
  • Maire Florian
  • Richard Frederic J.P.
  Annales Mathématiques Blaise Pascal, Université Blaise-Pascal - Clermont-Ferrand, 2013, 20 (1), pp.1-35. Les données de grande dimensions sont de plus en plus fréquemment collectées dans de nombreux domaines d'application. Il devient alors particulièrement important d'être capable d'extraire des caractéristiques significatives de ces bases de données. Le modèle d'atlas déformable (Deformable template model) est un outil maintenant répandu pour atteindre ce but. Cet article présente un panorama des aspects statistiques de ce modèle ainsi que ses généralisations. Nous décrivons les différents cadres mathématiques permettant de prendre en compte des types variés de données et de déformations. Nous rappelons les propriétés théoriques de convergence des estimateurs et des algorithmes permettant l'estimation de ces caractéristiques. Nous terminons cet article par la présentation de quelques résultats publiés utilisant des données réelles. (10.5802/ambp.320)
  DOI : 10.5802/ambp.320