Publications

2024

• Learning of extreme Expected Shortfall with neural networks. Application to cryptocurrency data
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2024. We propose new parametrizations for neural networks in order to estimate extreme Value-at-Risk and Expected-Shortfall in heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order.The convergence rate of the uniform error between extreme log quantities and their neural network approximation is established. The finite sample performances of the neural network estimator are compared to other bias-reduced extreme-value competitors on both real and simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail.
• Quantitative modelling and analysis of the Automated Market Maker Uniswap
  
  • Echenim Mnacho
  • Gobet Emmanuel
  • Maurice Anne-Claire
  , 2024.
• Automated approach for recovering modal components in shallow waters
  
  • Niclas Angèle
  • Garnier Josselin
  Journal of the Acoustical Society of America, Acoustical Society of America, 2024, 155 (4), pp.2347-2358. This paper proposes a fully automated method for recovering modal components from a signal in shallow waters. The scenario involves an unknown source emitting low-frequency sound waves in a shallow water environment, and a single hydrophone recording the signal. The proposed automated algorithm is based on the warping method to separate each modal component in the time-frequency space. However, instead of manually choosing a single arrival time for extraction, the method performs successive extractions with automated time selection based on an explicit quality factor. Modal component separation is achieved through a watershed algorithm, streamlining the process and eliminating the need for manual intervention. The proposed method is tested on experimental data of a right whale gunshot, a combustive sound source, and a bowhead whale upsweep, demonstrating its effectiveness in real-world scenarios. (10.1121/10.0025471)
  DOI : 10.1121/10.0025471
• Reference prior for Bayesian estimation of seismic fragility curves
  
  • Van Biesbroeck Antoine
  • Gauchy Clément
  • Feau Cyril
  • Garnier Josselin
  Probabilistic Engineering Mechanics, Elsevier, 2024, 76, pp.103622. One of the key elements of probabilistic seismic risk assessment studies is the fragility curve, which represents the conditional probability of failure of a mechanical structure for a given scalar measure derived from seismic ground motion. For many structures of interest, estimating these curves is a daunting task because of the limited amount of data available; data which is only binary in our framework, i.e., only describing the structure as being in a failure or non-failure state. A large number of methods described in the literature tackle this challenging framework through parametric log-normal models. Bayesian approaches, on the other hand, allow model parameters to be learned more efficiently. However, the impact of the choice of the prior distribution on the posterior distribution cannot be readily neglected and, consequently, neither can its impact on any resulting estimation. This paper proposes a comprehensive study of this parametric Bayesian estimation problem for limited and binary data. Using the reference prior theory as a cornerstone, this study develops an objective approach to choosing the prior. This approach leads to the Jeffreys prior, which is derived for this problem for the first time. The posterior distribution is proven to be proper (i.e., it integrates to unity) with the Jeffreys prior but improper with some traditional priors found in the literature. With the Jeffreys prior, the posterior distribution is also shown to vanish at the boundaries of the parameters’ domain, which means that sampling the posterior distribution of the parameters does not result in anomalously small or large values. Therefore, the use of the Jeffreys prior does not result in degenerate fragility curves such as unit-step functions, and leads to more robust credibility intervals. The numerical results obtained from two different case studies—including an industrial example—illustrate the theoretical predictions. (10.1016/j.probengmech.2024.103622)
  DOI : 10.1016/j.probengmech.2024.103622
• High order weak approximation of Stochastic Differential Equations for bounded and measurable test functions
  
  • Rey Clément
  , 2024. We present a method for approximating solutions of Stochastic Differential Equations (SDEs) with arbitrary rates. This approximation is derived for bounded and measurable test functions. Specifically, we demonstrate that, leveraging the standard weak approximation properties of numerical schemes for smooth test functions (such as first-order weak convergence for the Euler scheme) we can achieve convergence for simply bounded and measurable test functions at any desired rate by constructing a tailored approximation for the semigroup of the SDE. This is achieved by evaluating the scheme (e.g., Euler) on a random time grid. To establish convergence, we exploit the regularization properties of the scheme, which hold under a weak uniform Hörmander condition. (10.48550/arXiv.2403.17596)
  DOI : 10.48550/arXiv.2403.17596
• Projet ICI — Utilisation d’un jumeau numérique pour des simulations épidémiologiques
  
  • Colomb Maxime
  • Cormier Quentin
  • Graham Carl
  • Perret Julien
  • Talay Denis
  , 2024. La plateforme ICI à pour objectif de simuler la diffusion d’épidémie au sein de modélisations détaillées de populations au sein d’un espace géographique. La modélisation de l’intérieur des bâtiments, la génération de populations synthétiques et la composition de flux de populations ont été présentés lors des journées de la recherche de l'IGN 2023. Nous proposons de focaliser cette présentation sur les points suivants : l’attribution de lieux d’activités aux individus en fonction de leurs emplois du temps ; la description du module épidémiologique ; l’application de différents scénarios sanitaires modifiant les habitudes des individus ainsi que l’espace urbain ; la simulation systématique permettant l’analyse de sensibilité de divers groupes de paramètres ainsi que l’exploration et l’optimisation des mesures sanitaires.
• Unified two-scale Eulerian multi-fluid modeling of separated and dispersed two-phase flows
  
  • Loison Arthur
  , 2024. Liquid-gas two-phase flows are present in numerous industrial applications such as aerospace propulsion, nuclear hydraulics or bubble column reactors in the chemical industry.The simulation of such flows is of primary interest for their understanding and optimization.However, the dynamics of the interface separating the gas from the liquid can present a multiscale dynamics and thus makes simulations of industrial processes computationally too expensive.Some modelling efforts have been conducted on the development of cheaper multi-fluid models adapted to particular interface dynamics regime, e.g. in the separated regime where the fluids are separated by a single smooth surface or in the disperse regime where there are inclusions of one fluid carried by the other.Attempts of coupling between these models have showed some progress to simulate multiscale flows like atomization, but usually have physical or mathematical drawbacks.This thesis then pursues the goal of proposing a unified two-scale modelling framework with appropriate numerical methods adapted to this multiscale interface dynamics which goes from a separated to a disperse regime.The main contributions related to this modelling effort are :1- The combination of compressible multi-fluid models of the literature adapted to either the separated or the disperse regime into a unified two-scale multi-fluid model relying on Hamilton’s Stationary Action Principle;2- The local coupling of the models with an inter-scale mass transfer both regularizing the large-scale inter face and modelling mixed regime phenomena such as in primary break-up;3- Enhancing the small-scale models for the disperse regimes by adding the dynamics of geometrical quantities for oscillating droplets and pulsating bubbles, built as moments of a kinetic description.From the numerical perspective, finite-volume schemes and relaxation methods are used to solve the system of conservative laws of the models.Eventually, simulations with the open-source finite solver Josiepy demonstrates the regularization properties of the model on a set of well-chosen numerical setups leading to multi-scale interface dynamics.
• Mean field games master equations: from discrete to continuous state space
  
  • Bertucci Charles
  • Cecchin Alekos
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2024, 56 (2), pp.2569-2610. This paper studies the convergence of mean field games with finite state space to mean field games with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochasctic control, but also of finite difference numerical schemes. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both in case there is a smooth solution to the limit master equation and in case there is not. The second approach relies on the notion of monotone solutions introduced by [8, 9]. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, otherwise by compactness arguments. (10.1137/23M1552528)
  DOI : 10.1137/23M1552528
• Phenotypic plasticity trade-offs in an age-structured model of bacterial growth under stress
  
  • El Karoui Meriem
  • Madrid Ignacio
  • Méléard Sylvie
  , 2024. Under low concentrations of antibiotics causing DNA damage, \textit{Escherichia coli} bacteria can trigger stochastically a stress response known as the SOS response. While the expression of this stress response can make individual cells transiently able to overcome antibiotic treatment, it can also delay cell division, thus impacting the whole population's ability to grow and survive. In order to study the trade-offs that emerge from this phenomenon, we propose a bi-type age-structured population model that captures the phenotypic plasticity observed in the stress response. Individuals can belong to two types: either a fast-dividing but prone to death ``vulnerable" type, or a slow-dividing but ``tolerant" type. We study the survival probability of the population issued from a single cell as well as the population growth rate in constant and periodic environments. We show that the sensitivity of these two different notions of fitness with respect to the parameters describing the phenotypic plasticity differs between the stochastic approach (survival probability) and the deterministic approach (population growth rate). Moreover, under a more realistic configuration of periodic stress, our results indicate that optimal population growth can only be achieved through fine-tuning simultaneously both the induction of the stress response and the repair efficiency of the damage caused by the antibiotic.
• Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning
  
  • Dieuleveut Aymeric
  • Philippenko Constantin
  , 2024. In this paper, we investigate the impact of compression on stochastic gradient algorithms for machine learning, a technique widely used in distributed and federated learning. We underline differences in terms of convergence rates between several unbiased compression operators, that all satisfy the same condition on their variance, thus going beyond the classical worst-case analysis. To do so, we focus on the case of least-squares regression (LSR) and analyze a general stochastic approximation algorithm for minimizing quadratic functions relying on a random field. We consider weak assumptions on the random field, tailored to the analysis (specifically, expected H\"older regularity), and on the noise covariance, enabling the analysis of various randomizing mechanisms, including compression. We then extend our results to the case of federated learning. More formally, we highlight the impact on the convergence of the covariance of the additive noise induced by the algorithm. We demonstrate despite the non-regularity of the stochastic field, that the limit variance term scales with (where is the Hessian of the optimization problem and the number of iterations) generalizing the rate for the vanilla LSR case where it is (Bach and Moulines, 2013). Then, we analyze the dependency of on the compression strategy and ultimately its impact on convergence, first in the centralized case, then in two heterogeneous FL frameworks.
• Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning
  
  • Dieuleveut Aymeric
  • Philippenko Constantin
  , 2024. In this paper, we investigate the impact of compression on stochastic gradient algorithms for machine learning, a technique widely used in distributed and federated learning. We underline differences in terms of convergence rates between several unbiased compression operators, that all satisfy the same condition on their variance, thus going beyond the classical worst-case analysis. To do so, we focus on the case of least-squares regression (LSR) and analyze a general stochastic approximation algorithm for minimizing quadratic functions relying on a random field. We consider weak assumptions on the random field, tailored to the analysis (specifically, expected H\"older regularity), and on the noise covariance, enabling the analysis of various randomizing mechanisms, including compression. We then extend our results to the case of federated learning. More formally, we highlight the impact on the convergence of the covariance of the additive noise induced by the algorithm. We demonstrate despite the non-regularity of the stochastic field, that the limit variance term scales with (where is the Hessian of the optimization problem and the number of iterations) generalizing the rate for the vanilla LSR case where it is (Bach and Moulines, 2013). Then, we analyze the dependency of on the compression strategy and ultimately its impact on convergence, first in the centralized case, then in two heterogeneous FL frameworks.
• Adaptive Importance Sampling Based on Fault Tree Analysis for Piecewise Deterministic Markov Process
  
  • Chennetier Guillaume
  • Chraibi Hassane
  • Dutfoy Anne
  • Garnier Josselin
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2024, 12 (1), pp.128-156. Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling method based on a cross-entropy procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. They are well adapted to high-dimensional industrial systems. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry. (10.1137/22M1522838)
  DOI : 10.1137/22M1522838
• Inverse problem for Love waves in a layered, elastic half-space
  
  • de Hoop Maarten V.
  • Garnier Josselin
  • Iantchenko Alexei
  • Ricaud Julien
  Inverse Problems, IOP Publishing, 2024, 40 (4), pp.045013. In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves. (10.1088/1361-6420/ad2781)
  DOI : 10.1088/1361-6420/ad2781
• Nouveaux outils d'apprentissage statistiques pour l'imputation et le pronostic en conservation
  
  • ALAO AFOLABI Shadé
  • Hélou-de la Grandière Pauline
  • Thoury Mathieu
  • Perruchini Elsa
  • Le Pennec Erwan
  • Cohen Serge X.
  , 2024.
• Volterra processes in finance
  
  • Abi Jaber Eduardo
  , 2024. Empirical studies indicate the presence of memory and strong inter-temporal dependence across various phenomena in the fields of finance and economics. The Brownian motion and Poisson processes, characterized by independent increments, are not suitable for modeling such phenomena. We will consider Stochastic Volterra processes: a class of processes which extends the standard Brownian motion and Poisson processes to include memory; the fractional Brownian motion and Hawkes processes constitute a special case. First, we develop the mathematical tools needed to deal with these stochastic Volterra integral equations that go beyond the standard stochastic calculus theory of Markovian processes and semimartingales. Second, we explore the modeling flexibility of such equations in introducing memory in a broad range of problem in finance and economy including: volatilit modeling, portfolio allocation, optimal execution, principal agency, mean-field games...
• Noise through an additional variable for mean field games master equation on finite state space
  
  • Bertucci Charles
  • Meynard Charles
  , 2024. This paper provides a mathematical study of the well-posedness of master equation on finite state space involving terms modelling common noise. In this setting, the solution of the master equation depends on an additional variable modelling the value of a stochastic process impacting all players. Using technique from viscosity solutions, we give sufficient conditions for the existence of a Lipschitz continuous solution on any time interval. Under some structural assumptions, we are even able to treat cases in which the dynamics of this stochastic process depend on the state of the game.
• Determining efficient methods for solving Markov Decision Processes
  
  • Forghieri Orso
  • Hyon Emmanuel
  , 2024.
• On the consistency of supervised learning with missing values
  
  • Josse Julie
  • Chen Jacob M.
  • Prost Nicolas
  • Varoquaux Gaël
  • Scornet Erwan
  Statistical Papers, Springer Verlag, 2024, 65 (9), pp.5447-5479. In many application settings, the data have missing entries which make analysis challenging. An abundant literature addresses missing values in an inferential framework: estimating parameters and their variance from incomplete tables. Here, we consider supervised-learning settings: predicting a target when missing values appear in both training and testing data. We show the consistency of two approaches in prediction. A striking result is that the widely-used method of imputing with a constant, such as the mean prior to learning is consistent when missing values are not informative. This contrasts with inferential settings where mean imputation is pointed at for distorting the distribution of the data. That such a simple approach can be consistent is important in practice. We also show that a predictor suited for complete observations can predict optimally on incomplete data, through multiple imputation. Finally, to compare imputation with learning directly with a model that accounts for missing values, we analyze further decision trees. These can naturally tackle empirical risk minimization with missing values, due to their ability to handle the half-discrete nature of incomplete variables. After comparing theoretically and empirically different missing values strategies in trees, we recommend using the "missing incorporated in attribute" method as it can handle both non-informative and informative missing values. (10.1007/s00362-024-01550-4)
  DOI : 10.1007/s00362-024-01550-4
• Progressive state space disaggregation for dynamic programming
  
  • Forghieri Orso
  • Castel Hind
  • Le Pennec Erwan
  • Hyon Emmanuel
  , 2024.
• A Mean Field Game Model for Renewable Investment under Long-Term Uncertainty and Risk Aversion
  
  • Escribe Célia
  • Garnier Josselin
  • Gobet Emmanuel
  Dynamic Games and Applications, Springer Verlag, 2024. We consider a stylized model for investment into renewable power plants under long-term uncertainty. We model risk-averse agents facing heterogeneous weather conditions and a common noise including uncertainty on demand trends, future fuel prices and the average national weather conditions. The objective of each agent is to maximize multistage profit by controlling investment in discrete time steps. We analyze this model in a noncooperative game setting with N players, where the interaction among agents occurs through the spot price mechanism. Our model extends to a mean field game with common noise when the number of agents is infinite. We prove that the N-player game admits a Nash equilibrium. Moreover, we prove that under proper assumptions, any sequence of Nash equilibria to the N-player game converges to the unique solution of the MFG game. Finally, our numerical experiments highlight the impact of the risk aversion parameter and emphasize the difference between our model which captures heterogeneity and representative agent models. (10.1007/s13235-024-00554-x)
  DOI : 10.1007/s13235-024-00554-x
• Mathematical modelling and analysis of Impermanent Loss and Fees in Uniswap v3
  
  • Echenim Mnacho
  • Gobet Emmanuel
  • Maurice Anne-Claire
  , 2024.
• Quadratic regularization of bilevel pricing problems and application to electricity retail markets
  
  • Jacquet Quentin
  • van Ackooij Wim
  • Alasseur Clémence
  • Gaubert Stéphane
  European Journal of Operational Research, Elsevier, 2024, 313 (3), pp.841-857. We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers’ choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer’s response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer’s response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case. (10.1016/j.ejor.2023.05.006)
  DOI : 10.1016/j.ejor.2023.05.006
• Social Learning and Monetary Policy at the Effective Lower Bound
  
  • Arifovic Jasmina
  • Grimaud Alex
  • Salle Isabelle
  • Vermandel Gauthier
  Journal of Money, Credit and Banking, Wiley, 2024. This paper develops a model that jointly accounts for the missing disinflation in the wake of the Great Recession and the subsequently observed inflation‐less recovery. The key mechanism works through heterogeneous expectations that may durably lose their anchoring to the central bank (CB)'s target and coordinate on particularly persistent below‐target paths. The welfare cost associated with persistent low inflation may be reduced if the CB announces to the agents its target or its own inflation forecasts, as communication helps coordinate expectations. However, the CB may lose its credibility whenever its announcements become decoupled from actual inflation. (10.1111/jmcb.13133)
  DOI : 10.1111/jmcb.13133
• Augmented Quantization: a General Approach to Mixture Models
  
  • Sire Charlie
  • Le Riche Rodolphe
  • Rullière Didier
  • Rohmer Jérémy
  • Pheulpin Lucie
  • Richet Yann
  , 2024. The investigation of mixture models is a key to understand and visualize the distribution of multivariate data. Most mixture models approaches are based on likelihoods, and are not adapted to distribution with finite support or without a well-defined density function. This study proposes the Augmented Quantization method, which is a reformulation of the classical quantization problem but which uses the p-Wasserstein distance. This metric can be computed in very general distribution spaces, in particular with varying supports. The clustering interpretation of quantization is revisited in a more general framework. The performance of Augmented Quantization is first demonstrated through analytical toy problems. Subsequently, it is applied to a practical case study involving river flooding, wherein mixtures of Dirac and Uniform distributions are built in the input space, enabling the identification of the most influential variables.
• Hörmander properties of discrete time Markov processes
  
  • Rey Clément
  , 2024. We present an abstract framework for establishing smoothing properties within a specific class of inhomogeneous discrete-time Markov processes. These properties, in turn, serve as a basis for demonstrating the existence of density functions for our processes or more precisely for regularized versions of them. They can also be exploited to show the total variation convergence towards the solution of a Stochastic Differential Equation as the time step between two observations of the discrete time Markov processes tends to zero. The distinctive feature of our methodology lies in the exploration of smoothing properties under some local weak Hörmander type conditions satisfied by the discrete-time Markov processes. Our Hörmander properties are demonstrated to align with the standard local weak Hörmander properties satisfied by the coefficients of the Stochastic Differential Equations which are the total variation limits of our discrete time Markov processes. (10.48550/arXiv.2401.01167)
  DOI : 10.48550/arXiv.2401.01167