Publications

2022

• A class of short-term models for the oil industry addressing speculative storage
  
  • Achdou Yves
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  • Rostand Antoine
  • Scheinkman Jose
  Finance and Stochastics, Springer Verlag (Germany), 2022, 26 (3), pp.631-669. This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon. (10.1007/s00780-022-00481-y)
  DOI : 10.1007/s00780-022-00481-y
• A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2022, 15 (5), pp.793. A closure relation for moments equation in kinetic theory was recently introduced in [38], based on the study of the geometry of the set of moments. This relation was constructed from a projection of a moment vector toward the boundary of the set of moments and corresponds to approximating the underlying kinetic distribution as a sum of a chosen equilibrium distribution plus a sum of purely anisotropic Dirac distributions. The present work generalizes this construction for kinetic equations involving unbounded velocities, i.e. to the Hamburger problem, and provides a deeper analysis of the resulting moment system. Especially, we provide representation results for moment vectors along the boundary of the moment set that implies the well-definition of the model. And the resulting moment model is shown to be weakly hyperbolic with peculiar properties of hyperbolicity and entropy of two subsystems, corresponding respectively to the equilibrium and to the purely anisotropic parts of the underlying kinetic distribution. (10.3934/krm.2022014)
  DOI : 10.3934/krm.2022014
• Differentially Private Federated Learning on Heterogeneous Data
  
  • Noble Maxence
  • Bellet Aurélien
  • Dieuleveut Aymeric
  , 2022. Federated Learning (FL) is a paradigm for large-scale distributed learning which faces two key challenges: (i) training efficiently from highly heterogeneous user data, and (ii) protecting the privacy of participating users. In this work, we propose a novel FL approach (DP-SCAFFOLD) to tackle these two challenges together by incorporating Differential Privacy (DP) constraints into the popular SCAFFOLD algorithm. We focus on the challenging setting where users communicate with a "honest-but-curious" server without any trusted intermediary, which requires to ensure privacy not only towards a third party observing the final model but also towards the server itself. Using advanced results from DP theory and optimization, we establish the convergence of our algorithm for convex and non-convex objectives. Our paper clearly highlights the trade-off between utility and privacy and demonstrates the superiority of DP-SCAFFOLD over the state-ofthe-art algorithm DP-FedAvg when the number of local updates and the level of heterogeneity grows. Our numerical results confirm our analysis and show that DP-SCAFFOLD provides significant gains in practice.
• An ODE Method to Prove the Geometric Convergence of Adaptive Stochastic Algorithms
  
  • Akimoto Youhei
  • Auger Anne
  • Hansen Nikolaus
  Stochastic Processes and their Applications, Elsevier, 2022, 145, pp.269-307. We consider stochastic algorithms derived from methods for solving deterministic optimization problems, especially comparison-based algorithms derived from stochastic approximation algorithms with a constant step-size. We develop a methodology for proving geometric convergence of the parameter sequence {θn}n≥0 of such algorithms. We employ the ordinary differential equation (ODE) method, which relates a stochastic algorithm to its mean ODE, along with a Lyapunov-like function Ψ such that the geometric convergence of Ψ(θn) implies -- in the case of an optimization algorithm -- the geometric convergence of the expected distance between the optimum and the search point generated by the algorithm. We provide two sufficient conditions for Ψ(θn) to decrease at a geometric rate: Ψ should decrease "exponentially" along the solution to the mean ODE, and the deviation between the stochastic algorithm and the ODE solution (measured by Ψ) should be bounded by Ψ(θn) times a constant. We also provide practical conditions under which the two sufficient conditions may be verified easily without knowing the solution of the mean ODE. Our results are any-time bounds on Ψ(θn), so we can deduce not only the asymptotic upper bound on the convergence rate, but also the first hitting time of the algorithm. The main results are applied to a comparison-based stochastic algorithm with a constant step-size for optimization on continuous domains. (10.1016/j.spa.2021.12.005)
  DOI : 10.1016/j.spa.2021.12.005
• Extended mean field control problem: a propagation of chaos result
  
  • Djete Mao Fabrice
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2022, 27 (none). (10.1214/21-EJP726)
  DOI : 10.1214/21-EJP726
• Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field
  
  • Alouges François
  • Chambolle Antonin
  • Stantejsky Dominik
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2022, 63 (5), pp.129. We use the Landau-de Gennes energy to describe a particle immersed into nematic liquid crystals with a constant applied magnetic field. We derive a limit energy in a regime where both line and point defects are present, showing quantitatively that the close-to-minimal energy is asymptotically concentrated on lines and surfaces nearby or on the particle. We also discuss regularity of minimizers and optimality conditions for the limit energy. (10.1007/s00526-024-02717-5)
  DOI : 10.1007/s00526-024-02717-5
• ON THE DISCRETIZATION OF DISCONTINUOUS SOURCES OF HYPERBOLIC BALANCE LAWS
  
  • Pichard Teddy
  , 2022. We focus on a toy problem which corresponds to a simplification of a boiling twophase flow model. This model is a hyperbolic system of balance laws with a source term defined as a discontinuous function of the unknown. Several discretizations of this source terms are studied, and we illustrate their capacity to capture steady states. (10.23967/eccomas.2022.172)
  DOI : 10.23967/eccomas.2022.172
• Local-Global MCMC kernels: the best of both worlds
  
  • Samsonov Sergey
  • Lagutin Evgeny
  • Gabrié Marylou
  • Durmus Alain
  • Naumov Alexey
  • Moulines Eric
  , 2022. Recent works leveraging learning to enhance sampling have shown promising results, in particular by designing effective non-local moves and global proposals. However, learning accuracy is inevitably limited in regions where little data is available such as in the tails of distributions as well as in high-dimensional problems. In the present paper we study an Explore-Exploit Markov chain Monte Carlo strategy ($Ex^2MCMC$) that combines local and global samplers showing that it enjoys the advantages of both approaches. We prove $V$-uniform geometric ergodicity of $Ex^2MCMC$ without requiring a uniform adaptation of the global sampler to the target distribution. We also compute explicit bounds on the mixing rate of the Explore-Exploit strategy under realistic conditions. Moreover, we also analyze an adaptive version of the strategy ($FlEx^2MCMC$) where a normalizing flow is trained while sampling to serve as a proposal for global moves. We illustrate the efficiency of $Ex^2MCMC$ and its adaptive version on classical sampling benchmarks as well as in sampling high-dimensional distributions defined by Generative Adversarial Networks seen as Energy Based Models. We provide the code to reproduce the experiments at the link: https://github.com/svsamsonov/ex2mcmc_new.
• The mesoscopic geometry of sparse random maps
  
  • Curien Nicolas
  • Kortchemski Igor
  • Marzouk Cyril
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2022, 9, pp.1305-1345. (10.5802/jep.207)
  DOI : 10.5802/jep.207
• SAMBA: a Novel Method for Fast Automatic Model Building in Nonlinear Mixed-Effects Models
  
  • Prague Mélanie
  • Lavielle Marc
  CPT: Pharmacometrics and Systems Pharmacology, American Society for Clinical Pharmacology and Therapeutics ; International Society of Pharmacometrics, 2022, 11 (2). The success of correctly identifying all the components of a nonlinear mixed-effects model is far from straightforward: it is a question of finding the best structural model, determining the type of relationship between covariates and individual parameters, detecting possible correlations between random effects, or also modeling residual errors. We present the SAMBA (Stochastic Approximation for Model Building Algorithm) procedure and show how this algorithm can be used to speed up this process of model building by identifying at each step how best to improve some of the model components. The principle of this algorithm basically consists in 'learning something' about the 'best model', even when a 'poor model' is used to fit the data. A comparison study of the SAMBA procedure with SCM and COSSAC show similar performances on several real data examples but with a much-reduced computing time. This algorithm is now implemented in Monolix and in the R package Rsmlx. (10.1002/psp4.12742)
  DOI : 10.1002/psp4.12742
• Long-time behaviour of entropic interpolations
  
  • Clerc Gauthier
  • Conforti Giovanni
  • Gentil Ivan
  Potential Analysis, Springer Verlag, 2022. In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schrödinger problem [Sch31], which is the problem of finding the most likely evolution of a system of independent Brownian particles conditionally to observations. It is well known that in the short time limit entropic interpolations converge to the McCann-geodesics of optimal transport. Here we focus on the long-time behaviour, proving in particular asymptotic results for the entropic cost and establishing the convergence of entropic interpolations towards the heat equation, which is the gradient flow of the entropy according to the Otto calculus interpretation. Explicit rates are also given assuming the Bakry-Émery curvature-dimension condition. In this respect, one of the main novelties of our work is that we are able to control the long time behavior of entropic interpolations assuming the CD(0, n) condition only. (10.1007/s11118-021-09961-w)
  DOI : 10.1007/s11118-021-09961-w
• Two dimensional Gross-Pitaevskii equation with space-time white noise
  
  • de Bouard Anne
  • Debussche Arnaud
  • Fukuizumi Reika
  International Mathematics Research Notices, Oxford University Press (OUP), 2022. In this paper we consider the two-dimensional stochastic Gross-Pitaevskii equation, which is a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg-Landau equation with a harmonic potential and an additive space-time white noise. We study the well-posedness of the model using an inhomogeneous Wick renormalization due to the potential, and prove the existence of an invariant measure and of stationary martingale solutions. (10.1093/imrn/rnac137)
  DOI : 10.1093/imrn/rnac137
• Points and lines configurations for perpendicular bisectors of convex cyclic polygons
  
  • Melotti Paul
  • Ramassamy Sanjay
  • Thévenin Paul
  The Electronic Journal of Combinatorics, Open Journal Systems, 2022, 29 (1), pp.P1.59. We characterize the topological configurations of points and lines that may arise when placing n points on a circle and drawing the n perpendicular bisectors of the sides of the corresponding convex cyclic n-gon. We also provide a functional central limit theorem describing the shape of a large realizable configuration of points and lines taken uniformly at random among realizable configurations.
• Multiresolution-based mesh adaptation and error control for lattice Boltzmann methods with applications to hyperbolic conservation laws
  
  • Bellotti Thomas
  • Gouarin Loïc
  • Graille Benjamin
  • Massot Marc
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2022, 44 (4), pp.A2599-A2627. Lattice Boltzmann Methods (LBM) stand out for their simplicity and computational efficiency while offering the possibility of simulating complex phenomena. While they are optimal for Cartesian meshes, adapted meshes have traditionally been a stumbling block since it is difficult to predict the right physics through various levels of meshes. In this work, we design a class of fully adaptive LBM methods with dynamic mesh adaptation and error control relying on multiresolution analysis. This wavelet-based approach allows to adapt the mesh based on the regularity of the solution and leads to a very efficient compression of the solution without loosing its quality and with the preservation of the properties of the original LBM method on the finest grid. This yields a general approach for a large spectrum of schemes and allows precise error bounds, without the need for deep modifications on the reference scheme. An error analysis is proposed. For the purpose of assessing the approach, we conduct a series of test-cases for various schemes and scalar and systems of conservation laws, where solutions with shocks are to be found and local mesh adaptation is especially relevant. Theoretical estimates are retrieved while a reduced memory footprint is observed. It paves the way to an implementation in a multi-dimensional framework and high computational efficiency of the method for both parabolic and hyperbolic equations, which is the subject of a companion paper. (10.1137/21M140256X)
  DOI : 10.1137/21M140256X
• Docent: A content-based recommendation system to discover contemporary art
  
  • Fosset Antoine
  • El-Mennaoui Mohamed
  • Rebei Amine
  • Calligaro Paul
  • Di Maria Elise Farge
  • Nguyen-Ban Hélène
  • Rea Francesca
  • Vallade Marie-Charlotte
  • Vitullo Elisabetta
  • Zhang Christophe
  • Charpiat Guillaume
  • Rosenbaum Mathieu
  , 2022. Recommendation systems have been widely used in various domains such as music, films, e-shopping etc. After mostly avoiding digitization, the art world has recently reached a technological turning point due to the pandemic, making online sales grow significantly as well as providing quantitative online data about artists and artworks. In this work, we present a content-based recommendation system on contemporary art relying on images of artworks and contextual metadata of artists. We gathered and annotated artworks with advanced and art-specific information to create a completely unique database that was used to train our models. With this information, we built a proximity graph between artworks. Similarly, we used NLP techniques to characterize the practices of the artists and we extracted information from exhibitions and other event history to create a proximity graph between artists. The power of graph analysis enables us to provide an artwork recommendation system based on a combination of visual and contextual information from artworks and artists. After an assessment by a team of art specialists, we get an average final rating of 75% of meaningful artworks when compared to their professional evaluations.
• Algorithmic market making in dealer markets with hedging and market impact
  
  • Barzykin Alexander
  • Bergault Philippe
  • Guéant Olivier
  Mathematical Finance, Wiley, 2022. In dealer markets, dealers provide prices at which they agree to buy and sell the assets and securities they have in their scope. With ever increasing trading volume, this quoting task has to be done algorithmically in most markets such as foreign exchange markets or corporate bond markets. Over the last ten years, many mathematical models have been designed that can be the basis of quoting algorithms in dealer markets. Nevertheless, in most (if not all) models, the dealer is a pure internalizer, setting quotes and waiting for clients. However, on many dealer markets, dealers also have access to an inter-dealer market or even public trading venues where they can hedge part of their inventory. In this paper, we propose a model taking this possibility into account, therefore allowing dealers to externalize part of their risk. The model displays an important feature well known to practitioners that within a certain inventory range the dealer internalizes the flow by appropriately adjusting the quotes and starts externalizing outside of that range. The larger the franchise, the wider is the inventory range suitable for pure internalization. The model is illustrated numerically with realistic parameters for USDCNH spot market.
• Topology optimization of supports with imperfect bonding in additive manufacturing
  
  • Allaire Grégoire
  • Bogosel Beniamin
  • Godoy Matías
  Structural and Multidisciplinary Optimization, Springer Verlag, 2022. Supports are an important ingredient of the building process of structures by additive manufacturing technologies. They are used to reinforce overhanging regions of the desired structure and/or to facilitate the mitigation of residual thermal stresses due to the extreme heat flux produced by the source term (laser beam). Very often, supports are, on purpose, weakly connected to the built structure for easing their removal. In this work, we consider an imperfect interface model for which the interaction between supports and the built structure is not ideal, meaning that the displacement is discontinuous at the interface while the normal stress is continuous and proportional to the jump of the displacement. The optimization process is based on the level set method, body-fitted meshes and the notion of shape derivative using the adjoint method. We provide 2-d and 3-d numerical examples, as well as a comparison with the usual perfect interface model. Completely different designs of supports are obtained with perfect or imperfect interfaces.
• Spatio-temporal mixture process estimation to detect population dynamical changes
  
  • Pruilh Solange
  • Jannot Anne-Sophie
  • Allassonnière Stéphanie
  Artificial Intelligence in Medicine, Elsevier, 2022, 126, pp.102258. Population monitoring is a challenge in many areas such as public health or ecology. We propose a method to model and monitor population distributions over space and time, in order to build an alert system for spatio-temporal data evolution. Assuming that mixture models can correctly model populations, we propose new versions of the Expectation-Maximization algorithm to better estimate both the number of clusters together with their parameters. We then combine these algorithms with a temporal statistical model, allowing to detect dynamical changes in population distributions, and name it a spatio-temporal mixture process (STMP). We test STMP on synthetic data, and consider several different behaviors of the distributions, to adjust this process. Finally, we validate STMP on a real data set of positive diagnosed patients to corona virus disease 2019. We show that our pipeline correctly models evolving real data and detects epidemic changes. (10.1016/j.artmed.2022.102258)
  DOI : 10.1016/j.artmed.2022.102258
• Gâteaux type path-dependent PDEs and BSDEs with Gaussian forward processes
  
  • Barrasso Adrien
  • Russo Francesco
  Stochastics and Dynamics, World Scientific Publishing, 2022, 22, pp.2250007,. We are interested in path-dependent semilinear PDEs, where the derivatives are of Gâteaux type in specific directions k and b, being the kernel functions of a Volterra Gaussian process X. Under some conditions on k, b and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is X. (10.1142/S0219493722500071)
  DOI : 10.1142/S0219493722500071
• Fixed-distance multipoint formulas for the scattering amplitude from phaseless measurements
  
  • Novikov Roman
  • Sivkin Vladimir
  Inverse Problems, IOP Publishing, 2022, 38 (2), pp.025012. We give new formulas for finding the complex (phased) scattering amplitude at fixed frequency and angles from absolute values of the scattering wave function at several points $x_1,..., x_m$. In dimension $d\geq 2$, for $m>2$, we significantly improve previous results in the following two respects. First, geometrical constraints on the points needed in previous results are significantly simplified. Essentially, the measurement points $x_j$ are assumed to be on a ray from the origin with fixed distance $\tau=|x_{j+1}- x_j|$, and high order convergence (linearly related to $m$) is achieved as the points move to infinity with fixed $\tau$. Second, our new asymptotic reconstruction formulas are significantly simpler than previous ones. In particular, we continue studies going back to [Novikov, Bull. Sci. Math. 139(8), 923-936, 2015]. (10.1088/1361-6420/ac44db)
  DOI : 10.1088/1361-6420/ac44db
• A Non-Conservative Harris Ergodic Theorem
  
  • Bansaye Vincent
  • Cloez Bertrand
  • Gabriel Pierre
  • Marguet Aline
  Journal of the London Mathematical Society, London Mathematical Society ; Wiley, 2022, 106 (3), pp.2459-2510. We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides quantitative estimates of spectral gaps, complementing Krein-Rutman theorems and generalizing recent results relying on probabilistic approaches. The proof is based on a non-homogenous h-transform of the semi-group and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris' theorem for conservative semigroups. We apply these results and obtain exponential convergence of birth and death processes conditioned on survival to their quasi-stationary distribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs.We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides quantitative estimates of spectral gaps, complementing Krein-Rutman theorems and generalizing probabilistic approaches. The proof is based on a non-homogenous h-transform of the semigroup and the construction of Lyapunov functions for this latter. It exploits then the classical necessary and sufficient conditions of Harris’s theorem for conservative semigroups and recent techniques developed for the study for absorbed Markov process. We apply these results to population dynamics. We obtain exponential convergence of birth and death processes conditioned on survival to their quasi-stationary distribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs. (10.1112/jlms.12639)
  DOI : 10.1112/jlms.12639
• Firm non-expansive mappings in weak metric spaces
  
  • Gutiérrez Armando W.
  • Walsh Cormac
  Archiv der Mathematik, Springer Verlag, 2022,  119, pp.389-400. We introduce the notion of firm non-expansive mapping in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces. We prove that, for firm non-expansive mappings, the minimal displacement, the linear rate of escape, and the asymptotic step size are all equal. This generalises a theorem by Reich and Shafrir.
• Leveraging Deep Learning for Efficient Explicit MPC of High-Dimensional and Non-linear Chemical Processes
  
  • Shokry Ahmed
  • El Qassime Mehdi Abou
  • Moulines Eric
  , 2022, 51, pp.1171-1176. (10.1016/B978-0-323-95879-0.50196-X)
  DOI : 10.1016/B978-0-323-95879-0.50196-X
• Weak Langmuir turbulence in disordered multimode optical fibers
  
  • Baudin Kilian
  • Garnier Josselin
  • Fusaro Adrien
  • Berti Nicolas
  • Millot Guy
  • Picozzi Antonio
  Physical Review A, American Physical Society, 2022, 105 (1), pp.013528. We consider the propagation of temporally incoherent waves in multimode optical fibers (MMFs) in the framework of the multimode nonlinear Schrödinger (NLS) equation accounting for the impact of the natural structural disorder that affects light propagation in standard MMFs (random mode coupling and polarization fluctuations). By averaging the dynamics over the fast disordered fluctuations, we derive a Manakov equation from the multimode NLS equation, which reveals that the Raman effect introduces a previously unrecognized nonlinear coupling among the modes. Applying the wave turbulence theory on the Manakov equation, we derive a very simple scalar kinetic equation describing the evolution of the multimode incoherent waves. The structure of the kinetic equation is analogous to that developed in plasma physics to describe weak Langmuir turbulence. The extreme simplicity of the derived kinetic equation provides physical insight into the multimode incoherent wave dynamics. It reveals the existence of different collective behaviors where all modes self-consistently form a multimode spectral incoherent soliton state. Such an incoherent soliton can exhibit a discrete behavior characterized by collective synchronized spectral oscillations in frequency space. The theory is validated by accurate numerical simulations: The simulations of the generalized multimode NLS equation are found in quantitative agreement with those of the derived scalar kinetic equation without using adjustable parameters. (10.1103/PhysRevA.105.013528)
  DOI : 10.1103/PhysRevA.105.013528
• Asymptotic Analysis of a Matrix Latent Decomposition Model
  
  • Mantoux Clément
  • Durrleman​ Stanley
  • Allassonnière Stéphanie
  ESAIM: Probability and Statistics, EDP Sciences, 2022, 26, pp.208-242. Matrix data sets arise in network analysis for medical applications, where each network belongs to a subject and represents a measurable phenotype. These large dimensional data are often modeled using lower-dimensional latent variables, which explain most of the observed variability and can be used for predictive purposes. In this paper, we provide asymptotic convergence guarantees for the estimation of a hierarchical statistical model for matrix data sets. It captures the variability of matrices by modeling a truncation of their eigendecomposition. We show that this model is identifiable, and that consistent Maximum A Posteriori (MAP) estimation can be performed to estimate the distribution of eigenvalues and eigenvectors. The MAP estimator is shown to be asymptotically normal for a restricted version of the model. (10.1051/ps/2022004)
  DOI : 10.1051/ps/2022004