Publications

2020

• An Eco-Routing Algorithm for HEVs Under Traffic Conditions
  
  • Rhun Arthur Le
  • Bonnans Frédéric
  • Nunzio Giovanni De
  • Leroy Thomas
  • Martinon Pierre
  IFAC-PapersOnLine, Elsevier, 2020, 53 (2), pp.14242 - 14247. In a previous work, a bi-level optimization approach was presented for the energy management of Hybrid Electric Vehicles (HEVs), using a statistical model for traffic conditions. The present work is an extension of this framework to the eco-routing problem. The optimal trajectory is computed as the shortest path on a weighted graph whose nodes are (position, state of charge) pairs for the vehicle. The edge costs are provided by cost maps from an offline optimization at the lower level of small road segments. The error due to the discretization of the state of charge is proven to be linear if the cost maps are Lipschitz. The classical A * algorithm is used to solve the problem, with a heuristic based on a lower bound of the energy needed to complete the travel. The eco-routing method is compared to the fastest-path strategy by numerical simulations on a simple synthetic road network. (10.1016/j.ifacol.2020.12.1158)
  DOI : 10.1016/j.ifacol.2020.12.1158
• Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions
  
  • Bansaye Vincent
  • Cloez Bertrand
  • Gabriel Pierre
  Acta Applicandae Mathematicae, Springer Verlag, 2020, 166 (1), pp.29-72. We provide quantitative estimates in total variation distance for positive semi-groups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin's type conditions for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semi-groups and new bounds in the homogeneous setting. They are are illustrated on the renewal equation. (10.1007/s10440-019-00253-5)
  DOI : 10.1007/s10440-019-00253-5
• Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions
  
  • Akimoto Youhei
  • Auger Anne
  • Hansen Nikolaus
  Theoretical Computer Science, Elsevier, 2020, 832, pp.42-67. Quality gain is the expected relative improvement of the function value in a single step of a search algorithm. Quality gain analysis reveals the dependencies of the quality gain on the parameters of a search algorithm, based on which one can derive the optimal values for the parameters. In this paper, we investigate evolution strategies with weighted recombination on general convex quadratic functions. We derive a bound for the quality gain and two limit expressions of the quality gain. From the limit expressions, we derive the optimal recombination weights and the optimal step-size, and find that the optimal recombination weights are independent of the Hessian of the objective function. Moreover, the dependencies of the optimal parameters on the dimension and the population size are revealed. Differently from previous works where the population size is implicitly assumed to be smaller than the dimension, our results cover the population size proportional to or greater than the dimension. Simulation results show the optimal parameters derived in the limit approximates the optimal values in non-asymptotic scenarios. (10.1016/j.tcs.2018.05.015)
  DOI : 10.1016/j.tcs.2018.05.015
• ADDITIVE MANUFACTURING SCANNING PATHS OPTIMIZATION USING SHAPE OPTIMIZATION TOOLS
  
  • Boissier M
  • Allaire G.
  • Tournier Christophe
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020, 61, pp.2437–2466. This paper investigates path planning strategies for additive manufacturing processes such as powder bed fusion. The state of the art mainly studies trajectories based on existing patterns. Parametric optimization on these patterns or allocating them to the object areas are the main strategies. We propose in this work a more systematic optimization approach without any a priori restriction on the trajectories. The typical optimization problem is to melt the desired structure, without overheating (to avoid thermally induced residual stresses) and possibly with a minimal path length. The state equation is the heat equation with a source term depending on the scanning path. First, in a steady-state context, shape optimization tools are applied to trajec-tories. Second, for time-dependent problems, an optimal control method is considered instead. In both cases, gradient type algorithms are deduced and tested on 2-d examples. Numerical results are discussed, leading to a better understanding of the problem and thus to short-and long-term perspectives. (10.1007/s00158-020-02614-3)
  DOI : 10.1007/s00158-020-02614-3
• Variance Reduction Methods and Multilevel Monte Carlo Strategy for Estimating Densities of Solutions to Random Second-Order Linear Differential Equations
  
  • Jornet Marc
  • Calatayud Julia
  • Le Maitre Olivier
  • Cortés Juan Carlos
  International Journal for Uncertainty Quantification, Begell House Publishers, 2020, 10 (5), pp.467-497. This paper concerns the estimation of the density function of the solution to a random nonautonomous second-order linear differential equation with analytic data processes. In a recent contribution, we proposed to express the density function as an expectation, and we used a standard Monte Carlo algorithm to approximate the expectation. Although the algorithms worked satisfactorily for most test problems, some numerical challenges emerged for others, due to large statistical errors. In these situations, the convergence of the Monte Carlo simulation slows down severely, and noisy features plague the estimates. In this paper, we focus on computational aspects and propose several variance reduction methods to remedy these issues and speed up the convergence. First, we introduce a pathwise selection of the approximating processes which aims at controlling the variance of the estimator. Second, we propose a hybrid method, combining Monte Carlo and deterministic quadrature rules, to estimate the expectation. Third, we exploit the series expansions of the solutions to design a multilevel Monte Carlo estimator. The proposed methods are implemented and tested on several numerical examples to highlight the theoretical discussions and demonstrate the significant improvements achieved.
• Hölder-logarithmic stability in Fourier synthesis
  
  • Isaev Mikhail
  • Novikov Roman G
  Inverse Problems, IOP Publishing, 2020, 36 (12), pp.125003(17 pp.). We prove a Hölder-logarithmic stability estimate for the problem of finding a sufficiently regular compactly supported function v on R^d from its Fourier transform Fv given on [−r, r]^d. This estimate relies on a Hölder stable continuation of Fv from [−r, r]^d to a larger domain. The related reconstruction procedures are based on truncated series of Chebyshev polynomials. We also give an explicit example showing optimality of our stability estimates. (10.1088/1361-6420/abb5df)
  DOI : 10.1088/1361-6420/abb5df
• SCALPEL3: a scalable open-source library for healthcare claims databases
  
  • Bacry Emmanuel
  • Gaiffas Stéphane
  • Leroy Fanny
  • Morel Maryan
  • Nguyen D.P.
  • Sebiat Youcef
  • Sun Dian
  International Journal of Medical Informatics, Elsevier, 2020.
• State-constrained control-affine parabolic problems I: First and Second order necessary optimality conditions
  
  • Aronna M Soledad
  • Bonnans J. Frederic
  • Kröner Axel
  Set-Valued and Variational Analysis, Springer, 2020. In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimen-sional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order necessary conditions relying on the concept of alternative costates and quasi-radial critical directions. The appendix provides an example illustrating the applicability of our results.
• Tropical planar networks
  
  • Gaubert Stéphane
  • Niv Adi
  Linear Algebra and its Applications, Elsevier, 2020, 595, pp.123-144. We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary Jacobi matrices. (10.1016/j.laa.2020.02.019)
  DOI : 10.1016/j.laa.2020.02.019
• The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities
  
  • Backhoff Julio
  • Conforti Giovanni
  • Gentil Ivan
  • Léonard Christian
  Probability Theory and Related Fields, Springer Verlag, 2020, 178, pp.475-530. (10.1007/s00440-020-00977-8)
  DOI : 10.1007/s00440-020-00977-8
• Optimal Hedging Under Fast-Varying Stochastic Volatility
  
  • Garnier Josselin
  • Sølna Knut
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2020, 11 (1), pp.274-325. In a market with a rough or Markovian mean-reverting stochastic volatility thereis no perfect hedge. Here it is shown how various delta-type hedging strategies perform and canbe evaluated in such markets in the case of European options.A precise characterization of thehedging cost, the replication cost caused by the volatilityfluctuations, is presented in an asymptoticregime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset basedhedging strategy in the considered regime is identified as the so-called “practitioners” delta hedgingscheme. It is moreover shown that the performances of the delta-type hedging schemes are essentiallyindependent of the regularity of the volatility paths in theconsidered regime and that the hedgingcosts are related to a Vega risk martingale whose magnitude is proportional to a new market riskparameter. It is also shown via numerical simulations that the proposed hedging schemes whichderive from option price approximations in the regime of rapid mean reversion, are robust: the“practitioners” delta hedging scheme that is identified as being optimal by our asymptotic analysiswhen the mean reversion time is small seems to be optimal witharbitrary mean reversion times. (10.1137/18M1221655)
  DOI : 10.1137/18M1221655
• Regression Monte Carlo methods for HJB-type equations: which approximation space?
  
  • Barrera David
  • Gobet Emmanuel
  • Lopez-Salas Jose
  • Turkedjiev Plamen
  • Vasquez Carlos
  • Zubelli Jorge
  , 2020.
• Kinetic derivation of diffuse-interface fluid models
  
  • Giovangigli Vincent
  Physical Review E, American Physical Society (APS), 2020, 102. We present a full derivation of capillary fluid equations from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, and Dunn and Serrin's heat flux as well as viscous and heat dissipative fluxes. Starting from macroscopic equations obtained from the kinetic theory of dense gases, we use a second-order expansion of the pair distribution function in order to derive the diffuse interface model. The capillary extra terms and the capillarity coefficient are then associated with intermolecular forces and the pair interaction potential. (10.1103/physreve.102.012110)
  DOI : 10.1103/physreve.102.012110
• Commentaires sur le rapport de surveillance de culture du MON 810 en 2018. Paris, le 25 février 2020
  
  • Du Haut Conseil Des Biotechnologies Comité Scientifique
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • de Verneuil Hubert
  • Vilotte Jean-Luc
  , 2020, pp.35 p.. Les analyses contenues dans le rapport de surveillance de Bayer Agriculture BVBA ne font apparaître aucun problème majeur associé à la culture de maïs MON 810 en 2018. Toutefois, le CS du HCB identifie encore certaines faiblesses et limites méthodologiques concernant la surveillance de la sensibilité des ravageurs ciblés à la toxine Cry1Ab, remettant en question les conclusions du rapport. Le HCB estime notamment que l’utilisation d’une dose diagnostic présente certaines limites pour la détection précoce de l’évolution de la résistance, tant dans son principe intrinsèque que dans sa mise en oeuvre par Bayer, et recommande une méthode alternative de type F2 screen permettant de déterminer la fréquence des allèles de résistance au sein d’une population de ravageurs cibles. Par ailleurs, le HCB formule des recommandations destinées à renforcer la mise en oeuvre des zones refuges pour prévenir ou retarder le développement de résistance à la toxine Cry1Ab chez les ravageurs ciblés. Concernant la surveillance générale, le CS du HCB relève un problème de pertinence méthodologique quant aux questions étudiées, avec des règles de décision arbitraires, des conclusions incorrectement justifiées et un possible biais associé au format d’enquête auprès du panel d’agriculteurs qui ont accepté de répondre au questionnaire. Enfin, le CS du HCB recommande que le rapport de surveillance considère la présence de téosinte dans des zones de culture du maïs MON 810 en Espagne et les risques potentiels associés à une éventuelle introgression de gènes de maïs MON 810 chez le téosinte.
• A quantitative McDiarmid’s inequality for geometrically ergodic Markov chains
  
  • Havet Antoine
  • Lerasle Matthieu
  • Moulines Éric
  • Vernet Elodie
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2020, 25. (10.1214/20-ECP286)
  DOI : 10.1214/20-ECP286
• A metric interpretation of the geodesic curvature in the Heisenberg group
  
  • Kohli Mathieu
  Journal of Dynamical and Control Systems, Springer Verlag, 2020, 26 (1), pp.159–174. In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve. (10.1007/s10883-019-09444-7)
  DOI : 10.1007/s10883-019-09444-7
• Approximate and exact controllability of linear difference equations
  
  • Chitour Yacine
  • Mazanti Guilherme
  • Sigalotti Mario
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2020, 7, pp.93--142. In this paper, we study approximate and exact controllability of the linear difference equation $x(t) = \sum_{j=1}^N A_j x(t - \Lambda_j) + B u(t)$ in $L^2$, with $x(t) \in \mathbb C^d$ and $u(t) \in \mathbb C^m$, using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$, and some suitable matrix coefficients. When $\Lambda_1, \dotsc, \Lambda_N$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^2$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays. (10.5802/jep.112)
  DOI : 10.5802/jep.112
• A MOMENT CLOSURE BASED ON A PROJECTION ON THE BOUNDARY OF THE REALIZABILITY DOMAIN: 1D CASE
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2020, 13 (6), pp.1243-1280. This work aims to develop and test a projection technique for the construction of closing equations of moment systems. One possibility to define such a closure consists in reconstructing an underlying kinetic distribution from a vector of moments, then expressing the closure based on this reconstructed function. Exploiting the geometry of the realizability domain, i.e. the set of moments of positive distribution function, we decompose any realizable vectors into two parts, one corresponding to the moments of a chosen equilibrium function, and one obtain by a projection onto the boundary of the realizability domain in the direction of equilibrium function. A realizable closure of both of these parts are computed with standard techniques providing a realizable closure for the full system. This technique is tested for the reduction of a radiative transfer equation in slab geometry. (10.3934/xx.xx.xx.xx)
  DOI : 10.3934/xx.xx.xx.xx
• A second order analysis of McKean-Vlasov semigroups
  
  • Arnaudon Marc
  • del Moral Pierre
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2020. We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gröbner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities. (10.1214/20-AAP1568)
  DOI : 10.1214/20-AAP1568
• Surface waves in a channel with thin tunnels and wells at the bottom: non-reflecting underwater tomography
  
  • Chesnel Lucas
  • Nazarov Sergei
  • Taskinen Jari
  Asymptotic Analysis, IOS Press, 2020. We consider the propagation of surface water waves in a straight planar channel perturbed at the bottom by several thin curved tunnels and wells. We propose a method to construct non reflecting underwater topographies of this type at an arbitrary prescribed wave number. To proceed, we compute asymptotic expansions of the diffraction solutions with respect to the small parameter of the geometry taking into account the existence of boundary layer phenomena. We establish error estimates to validate the expansions using advances techniques of weighted spaces with detached asymptotics. In the process, we show the absence of trapped surface waves for perturbations small enough. This analysis furnishes asymptotic formulas for the scattering matrix and we use them to determine underwater topographies which are non-reflecting. Theoretical and numerical examples are given.
• Support optimization in additive manufacturing for geometric and thermo-mechanical constraints
  
  • Allaire Grégoire
  • Bihr Martin
  • Bogosel Beniamin
  Structural and Multidisciplinary Optimization, Springer Verlag, 2020, 61, pp.2377-2399. Supports are often required to safely complete the building of complicated structures by additive manufacturing technologies. In particular, supports are used as scaffoldings to reinforce overhanging regions of the structure and/or are necessary to mitigate the thermal deformations and residual stresses created by the intense heat flux produced by the source term (typically a laser beam). However, including supports increase the fabrication cost and their removal is not an easy matter. Therefore, it is crucial to minimize their volume while maintaining their efficiency. Based on earlier works, we propose here some new optimization criteria. First, simple geometric criteria are considered like the projected area and the volume of supports required for overhangs: they are minimized by varying the structure orientation with respect to the baseplate. In addition, an accessibility criterion is suggested for the removal of supports, which can be used to forbid some parts of the structure to be supported. Second, shape and topology optimization of supports for compliance minimization is performed. The novelty comes from the applied surface loads which are coming either from pseudo gravity loads on overhanging parts or from equivalent thermal loads arising from the layer by layer building process. Here, only the supports are optimized, with a given non-optimizable structure, but of course many generalizations are possible, including optimizing both the structure and its supports. Our optimization algorithm relies on the level set method and shape derivatives computed by the Hadamard method. Numerical examples are given in 2-d and 3-d.
• Uncertainty Quantification for Stochastic Approximation Limits Using Chaos Expansion
  
  • Crépey Stéphane
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2020, 8 (3), pp.1061–1089. The uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm is analyzed. In our setup, this limit $f^*$ is defined as a zero of an intractable function and is modeled as uncertain through a parameter $\theta$. We aim at deriving the function $f^*$, as well as the probabilistic distribution of $f^*(\theta)$ given a probability distribution $\pi$ for $\theta$. We introduce the so-called Uncertainty Quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of $\theta \mapsto f^*(\theta)$ on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations $K$, returns a finite set of coefficients, providing an approximation $\widehat{f^*_K}(\cdot)$ of $f^*$. We establish the almost-sure and $L^p$-convergences in the Hilbert space of the sequence of functions $\widehat{f^*_K}(\cdot)$ when the number of iterations $K$ tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix and of higher order moments of the quantity $\widehat{f^*_K}(\theta)$ when $\theta$ is random with distribution $\pi$. UQSA is illustrated and the role of its design parameters is discussed numerically. (10.1137/18M1178517)
  DOI : 10.1137/18M1178517
• Multipoint formulas for scattered far field in multidimensions
  
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2020, 36 (9), pp.095001(12 pp.). We give asymptotic formulas for finding the scattering amplitude at fixed frequency and angles (scattered far field) from the scattering wave function given at $n$ points in dimension $d\geq 2$. These formulas are explicit and their precision is proportional to $n$. To our knowledge these formulas are new already for $n\geq 2$. (10.1088/1361-6420/aba891)
  DOI : 10.1088/1361-6420/aba891
• A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices
  
  • Gaubert Stéphane
  • Stott Nikolas
  Mathematical Control and Related Fields, AIMS, 2020, 10 (3), pp.573-590. We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques. (10.3934/mcrf.2020011)
  DOI : 10.3934/mcrf.2020011
• Model Reduction for Large-Scale Earthquake Simulation in an Uncertain 3D Medium
  
  • Sochala Pierre
  • de Martin Florent
  • Le Maitre Olivier
  International Journal for Uncertainty Quantification, Begell House Publishers, 2020, 10 (2), pp.101-127. In this paper, we are interested in the seismic wave propagation into an uncertain medium. To this end, we performed an ensemble of 400 large-scale simulations that requires 4 million core-hours of CPU time. In addition to the large computational load of these simulations, solving the uncertainty propagation problem requires dedicated procedures to handle the complexities inherent to large data set size and the low number of samples. We focus on the peak ground motion at the free surface of the 3D domain, and our analysis utilizes a surrogate model combining two key ingredients for complexity mitigation: i) a dimension reduction technique using empirical orthogonal basis functions and ii) a functional approximation of the uncertain reduced coordinates by polynomial chaos expansions. We carefully validate the resulting surrogate model by estimating its predictive error using bootstrap, truncation, and cross-validation procedures. The surrogate model allows us to compute various statistical information of the uncertain prediction, including marginal and joint probability distributions, interval probability maps, and 2D fields of global sensitivity indices. (10.1615/Int.J.UncertaintyQuantification.2020031165)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2020031165