Publications

2013

• First and second order optimality conditions for optimal control problems of state constrained integral equations
  
  • Bonnans J. Frédéric
  • de La Vega Constanza
  • Dupuis Xavier
  Journal of Optimization Theory and Applications, Springer Verlag, 2013, 159 (1), pp.1-40. This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient conditions. (10.1007/s10957-013-0299-3)
  DOI : 10.1007/s10957-013-0299-3
• 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.
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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.
• 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
• 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
• 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.
• 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
• 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.
• 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
• 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
• 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 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.
• 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.
• 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