Publications

2018

• Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
  
  • Agrachev Andrei
  • Boscain Ugo
  • Neel Robert
  • Rizzi Luca
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2018, 24 (3), pp.1075–1105. We relate some basic constructions of stochastic analysis to differential geometry , via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators , on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian mani-fold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume. (10.1051/cocv/2017037)
  DOI : 10.1051/cocv/2017037
• Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost
  
  • Kröner Axel
  • Picarelli Athena
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2018, 56 (5), pp.3296-3319. An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order Hamilton-Jacobi-Bellman (HJB) equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. These properties are verified for a specific semi-Lagrangian scheme. (10.1137/17M115253X)
  DOI : 10.1137/17M115253X
• Fluctuations and Temperature Effects in Bose-Einstein Condensation
  
  • de Bouard Anne
  • Debussche Arnaud
  • Fukuizumi Reika
  • Poncet Romain
  ESAIM: Proceedings and Surveys, EDP Sciences, 2018, 61, pp.55-67. The modeling of cold atoms systems has known an increasing interest in the theoretical physics community, after the first experimental realizations of Bose Einstein condensates, some twenty years ago. We here review some analytical and numerical results concerning the influence of fluctuations , either arising from fluctuations of the confining parameters, or due to temperature effects, in the models describing the dynamics of such condensates. (10.1051/proc/201861055)
  DOI : 10.1051/proc/201861055
• Random planar maps and growth-fragmentations
  
  • Bertoin Jean
  • Curien Nicolas
  • Kortchemski Igor
  The Annals of Probability, Institute of Mathematical Statistics, 2018, 46 (1), pp.207-260. (10.1214/17-AOP1183)
  DOI : 10.1214/17-AOP1183
• Characteristic and Universal Tensor Product Kernels
  
  • Szabó Zoltán
  • Sriperumbudur Bharath K
  Journal of Machine Learning Research, Microtome Publishing, 2018, 18, pp.233. Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of characteristic property of the tensor product kernel.
• A NON-INTRUSIVE STRATIFIED RESAMPLER FOR REGRESSION MONTE CARLO: APPLICATION TO SOLVING NON-LINEAR EQUATIONS
  
  • Gobet Emmanuel
  • Liu Gang
  • Zubelli Jorge
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2018, 56 (1), pp.50-77. Our goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that the model for X is not known in full detail and only a root sample X1, . . . , XM of such process is available. By a stratification of the space and a suitable choice of a probability measure ν, we design a new resampling scheme that allows to compute local regressions (on basis functions) in each stratum. The combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation (possibly in large dimensions) using only a relatively small set of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates in L2(ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root paths. (10.1137/16M1066865)
  DOI : 10.1137/16M1066865
• Laser Beam Imaging from the Speckle Pattern of the Off-Axis Scattered Intensity
  
  • Borcea Liliana
  • Garnier Josselin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (2), pp.677-704. (10.1137/17M1139059)
  DOI : 10.1137/17M1139059
• Inverse scattering for the Bethe-Peierls model
  
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2018, 6 (1), pp.52-55. We consider the phased and phaseless inverse scattering problems for the Bethe-Peierls model. We give complete solutions of these problems including questions of uniqueness, nonuniqueness, reconstruction and characterization.
• Avis en réponse à la saisine HCB - dossier 2017-143. Paris, le 15 mai 2018
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • 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
  • Vilotte Jean-Luc
  , 2018.
• Avis en réponse à la saisine HCB - dossier C/NL/06/01_001. Paris, le 17 octobre 2018
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie Anne M. A.
  • Bellivier Florence
  • Berny Philippe
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Coléno François
  • Couvet Denis
  • Dassa Elie
  • 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
  • de Verneuil Hubert
  • Vilotte Jean-Luc
  , 2018, pp.25 p.. Le Haut Conseil des biotechnologies (HCB) a été saisi le 14 août 2018 par les autorités compétentes françaises (le ministère de l’Agriculture et de l’Alimentation) d’une demande d’avis relative au dossier C/NL/06/01_001 de demande de renouvellement d’autorisation de mise sur le marché de la lignée d’oeillets génétiquement modifiés 123.8.12 (identificateur unique FLO- 40689-6) à des fins d’importation et de commercialisation de fleurs coupées. Ce dossier a été déposé par la société Suntory Flowers Limited auprès des autorités compétentes néerlandaises sur le fondement 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, qui disposent de 60 jours pour faire des commentaires, demander des informations complémentaires ou émettre des objections à la mise sur le marché. Par cette saisine, les autorités compétentes françaises consultent le HCB dans cette perspective, en amont du vote des Etats membres à la Commission européenne. Le Comité scientifique (CS)2 du HCB a examiné le dossier en séance du 17 octobre 2018 sous la présidence de Jean-Christophe Pagès. Le présent avis a été adopté en séance et publié le 23 octobre 2018.
• Minimization of the eigenvalues of the Dirichlet-LapIacian with a diameter constraint
  
  • Bogosel B
  • Henrot A
  • Lucardesi I
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2018, 50 (5), pp.5337-5361. In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian λ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane. (10.1137/17M1162147)
  DOI : 10.1137/17M1162147
• Global acoustic daylight imaging in a stratified Earth-like model
  
  • Garnier Josselin
  • de Hoop Maarten V.
  • Sølna Knut
  Inverse Problems, IOP Publishing, 2018, 34 (1). (10.1088/1361-6420/aa9ad7)
  DOI : 10.1088/1361-6420/aa9ad7
• Long time behavior of Gross-Pitaevskii equation at positive temperature
  
  • de Bouard Anne
  • Debussche Arnaud
  • Fukuizumi Reika
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2018, 50 (6), pp.5887–5920. The stochastic Gross-Pitaevskii equation is used as a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg Landau equation with a trapping potential and an additive space-time white noise. Two important questions for this system are the global existence of solutions in the support of the Gibbs measure, and the convergence of those solutions to the equilibrium for large time. In this paper, we give a proof of these two results in one space dimension. In order to prove the convergence to equilibrium, we use the associated purely dissipative equation as an auxiliary equation, for which the convergence may be obtained using standard techniques. (10.1137/17M1149195)
  DOI : 10.1137/17M1149195
• High order moment model for polydisperse evaporating sprays towards interfacial geometry
  
  • Essadki Mohamed
  • de Chaisemartin Stephane
  • Laurent Frédérique
  • Massot Marc
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (4), pp.2003-2027. In this paper we propose a new Eulerian modeling and related accurate and robust numerical methods, describing polydisperse evaporating sprays, based on high order moment methods in size. The main novelty of this model is its capacity to describe some geometrical variables of the droplet-gas interface, by analogy with the liquid-gas interface in interfacial flows. For this purpose, we use fractional size-moments, where the size variable is taken as the droplet surface. In order to evaluate the evaporation of the polydisperse spray, we use a smooth reconstruction which maximizes the Shannon entropy. However, the use of fractional moments introduces some theoretical and numerical difficulties, which need to be tackled. First, relying on a study of the moment space, we extend the Maximum Entropy (ME) reconstruction of the size distribution to the case of fractional moments. Then, we propose a new accurate and realizable algorithm to solve the moment evolution due to evaporation, which preserves the structure of the moment space. This algorithm is based on a mathematical analysis of the kinetic evolution due to evaporation, where it shown that the evolution of some negative order fractional moments have to be properly predicted, a peculiarity related to the use of fractional moments. The present model and numerical schemes yield an accurate and stable evaluation of the moment dynamics with minimal number of variables, as well as a minimal computational cost as with the EMSM model, but with the very interesting additional capacity of coupling with diffuse interface model and transport equation of averaged geometrical interface variables, which are essential in oder to describe atomization. (10.1137/16M1108364)
  DOI : 10.1137/16M1108364
• Optimization of classification and regression analysis of four monoclonal antibodies from Raman spectra using collaborative machine learning approach
  
  • Le Laetitia Minh Maï
  • Kégl Balázs
  • Gramfort Alexandre
  • Marini Camille
  • Nguyen David
  • Cherti Mehdi
  • Tfaili Sana
  • Tfayli Ali
  • Baillet-Guffroy Arlette
  • Prognon Patrice
  • Chaminade Pierre
  • Caudron Eric
  Talanta, Elsevier, 2018, 184, pp.260-265. The use of monoclonal antibodies (mAbs) constitutes one of the most important strategies to treat patients suffering from cancers such as hematological malignancies and solid tumors. These antibodies are prescribed by the physician and prepared by hospital pharmacists. An analytical control enables the quality of the preparations to be ensured. The aim of this study was to explore the development of a rapid analytical method for quality control. The method used four mAbs (Infliximab, Bevacizumab, Rituximab and Ramucirumab) at various concentrations and was based on recording Raman data and coupling them to a traditional chemometric and machine learning approach for data analysis. Compared to conventional linear approach, prediction errors are reduced with a data-driven approach using statistical machine learning methods. In the latter, preprocessing and predictive models are jointly optimized. An additional original aspect of the work involved on submitting the problem to a collaborative data challenge platform called Rapid Analytics and Model Prototyping (RAMP). This allowed using solutions from about 300 data scientists in collaborative work. Using machine learning, the prediction of the four mAbs samples was considerably improved. The best predictive model showed a combined error of 2.4% versus 14.6% using linear approach. The concentration and classification errors were 5.8% and 0.7%, only three spectra were misclassified over the 429 spectra of the test set. This large improvement obtained with machine learning techniques was uniform for all molecules but maximal for Bevacizumab with an 88.3% reduction on combined errors (2.1% versus 17.9%). (10.1016/j.talanta.2018.02.109)
  DOI : 10.1016/j.talanta.2018.02.109
• Parking 3-sphere swimmer I. Energy minimizing strokes
  
  • Alouges François
  • Di Fratta Giovanni
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2018, 23 (4), pp.1797-1817. The paper is about the parking 3-sphere swimmer (sPr$_3$), a low-Reynolds number model swimmer composed of three balls of equal radii. The three balls can move along three horizontal axes (supported in the same plane) that mutually meet at the center of sPr$_3$ with angles of 120°. The governing dynamical system is introduced and the implications of its geometric symmetries revealed. It is then shown that, in the first order range of small strokes, optimal periodic strokes are ellipses embedded in 3d space, i.e., closed curves of the form t ∈ [0,2π ] → (cos t)${u}$ +(sin t)${v}$ for suitable vectors ${u}$ and ${v}$ of $\mathbb{R}^3$. A simple analytic expression for the vectors ${u}$ and ${v}$ is derived (10.3934/dcdsb.2018085)
  DOI : 10.3934/dcdsb.2018085
• Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding
  
  • Gautier Eric
  • Le Pennec Erwan
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2018, 12 (1), pp.277-320. (10.1214/17-EJS1383)
  DOI : 10.1214/17-EJS1383
• Quadratic backward stochastic differential equations driven by $G$-Brownian motion: discrete solutions and approximation
  
  • Hu Ying
  • Lin Yiqing
  • Soumana-Hima Abdoulaye
  Stochastic Processes and their Applications, Elsevier, 2018, 128 (11), pp.3724-3750. In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces. (10.1016/j.spa.2017.12.004)
  DOI : 10.1016/j.spa.2017.12.004
• Adiabatic ensemble control of a continuum of quantum systems
  
  • Augier Nicolas
  • Boscain Ugo
  • Sigalotti Mario
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2018, 56, pp.4045-4068. In this article we discuss how to control a parameter-dependent family of quantum systems. Our technique is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. As particular cases, we recover chirped pulses for two-level quantum systems and counter-intuitive solutions for three-level stimulated Raman adiabatic passage (STIRAP). The proposed technique works for systems evolving both in finite-dimensional and infinite-dimensional Hilbert spaces. We show that the assumptions guaranteeing ensemble controllability are structurally stable with respect to perturbations of the parametrized family of systems.
• Transport et diffusion
  
  • Allaire Grégoire
  • Blanc Xavier
  • Després Bruno
  • Golse François
  , 2018, pp.328. Ce livre est issu d'un cours enseigné par les auteurs en troisième année de l'Ecole Polytechnique, ce qui correspond à un niveau de première année de Master. Le sujet en est l'étude mathématique et numérique de modèles d'équations aux dérivées partielles, dits de transport et diffusion. Ces équations modélisent l'évolution d'une densité de particules ou d'individus en interaction avec leur milieu. L'origine de ces modèles est très variée. Ils proviennent classiquement de la physique et servent à décrire des particules neutres comme les neutrons et les photons. Dans le premier cas on parle aussi de neutronique alors que le dernier cas est appelé transfert radiatif. La biologie fait aussi appel aux équations de transport pour modéliser la dynamique de populations structurées. Citons aussi pour mémoire la dynamique des gaz raréfiés, le transport d'électrons dans les semi-conducteurs ou encore la physique des plasmas qui sont des phénomènes modélisés par des équations où l'opérateur de transport est une brique de base essentielle.
• The derivation of homogenized diffusion kurtosis models for diffusion MRI
  
  • Haddar Houssem
  • Kchaou Marwa
  • Moakher Maher
  Journal of Magnetic Resonance, Elsevier, 2018, 298, pp.48-57.
• Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs
  
  • Czichowsky Christoph
  • Peyre Rémi
  • Schachermayer Walter
  • Yang Junjian
  Finance and Stochastics, Springer Verlag (Germany), 2018, 22 (1), pp.161-180. The present paper accomplishes a major step towards a reconciliation of two conflicting approaches in mathematical finance: on the one hand, the mainstream approach based on the notion of no arbitrage (Black, Merton & Scholes); and on the other hand, the consideration of non-semimartingale price processes, the archetype of which being fractional Brownian motion (Mandelbrot). Imposing (arbitrarily small) proportional transaction costs and considering logarithmic utility optimisers, we are able to show the existence of a semimartingale, frictionless shadow price process for an exponential fractional Brownian financial market. (10.1007/s00780-017-0351-5)
  DOI : 10.1007/s00780-017-0351-5
• Non reflection and perfect reflection via Fano resonance in waveguides
  
  • Chesnel Lucas
  • Nazarov Sergei A
  Communications in Mathematical Sciences, International Press, 2018. We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter ε and of the frequency λ is in general not continuous at a point $(ε, λ) = (0,λ_0$) where trapped modes exist. In particular, we prove that for a given $ε = 0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $λ_0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.
• Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule
  
  • Akimoto Youhei
  • Auger Anne
  • Glasmachers Tobias
  , 2018. This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the sphere function. To deal with potential functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension $d$. The bounds are akin to linear convergence. We then study the dependency of the different terms on $d$ proving a convergence rate dependency of $\Theta(1/d)$. Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.
• Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau.
  
  • Durmus Alain
  • Moulines Éric
  • Pereyra Marcelo
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2018, 11 (1). In this paper, two new algorithms to sample from possibly non-smooth log-concave probability measures are introduced. These algorithms use Moreau-Yosida envelope combined with the Euler-Maruyama discretization of Langevin diffusions. They are applied to a de-convolution problem in image processing, which shows that they can be practically used in a high dimensional setting. Finally, non-asymptotic bounds for one of the proposed methods are derived. These bounds follow from non-asymptotic results for ULA applied to probability measures with a convex continuously differentiable log-density with respect to the Lebesgue measure. (10.1137/16M110834)
  DOI : 10.1137/16M110834