Publications

2024

• Topological interface modes in systems with damping
  
  • Alexopoulos Konstantinos
  • Davies Bryn
  • Hiltunen Erik Orvehed
  , 2024. We extend the theory of topological localised interface modes to systems with damping. The spectral problem is formulated as a root-finding problem for the interface impedance function and Rouché's theorem is used to track the zeros when damping is introduced. We show that the localised eigenfrequencies, corresponding to interface modes, remain for non-zero dampings. Using the transfer matrix method, we explicitly characterise the decay rate of the interface mode.
• Time-uniform log-Sobolev inequalities and applications to propagation of chaos
  
  • Monmarché Pierre
  • Ren Zhenjie
  • Wang Songbo
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2024, 29, pp.154. Time-uniform log-Sobolev inequalities (LSI) satisfied by solutions of semi-linear meanfield equations have recently appeared to be a key tool to obtain time-uniform propagation of chaos estimates. This work addresses the more general settings of timeinhomogeneous Fokker-Planck equations. Time-uniform LSI are obtained in two cases, either with the bounded-Lipschitz perturbation argument with respect to a reference measure, or with a coupling approach at high temperature. These arguments are then applied to mean-field equations, where, on the one hand, sharp marginal propagation of chaos estimates are obtained in smooth cases and, on the other hand, time-uniform global propagation of chaos is shown in the case of vortex interactions with quadratic confinement potential on the whole space. In this second case, an important point is to establish global gradient and Hessian estimates, which is of independent interest. We prove these bounds in the more general situation of non-attractive logarithmic and Riesz singular interactions. (10.1214/24-EJP1217)
  DOI : 10.1214/24-EJP1217
• State abstraction discovery from progressive disaggregation methods
  
  • Forghieri Orso
  • Castel Hind
  • Hyon Emmanuel
  • Le Pennec Erwan
  , 2024. The high dimensionality of model-based Reinforcement Learning (RL) and Markov Decision Processes (MDPs) can be reduced using abstractions of the state and action spaces. Although hierarchical learning and state abstraction methods have been explored over the past decades, explicit methods to build useful abstractions of models are rarely provided. In this work, we study the relationship between Approximate Dynamic Programming (ADP) and State Abstraction. We provide an estimation of the approximation made through abstraction, which can be explicitly calculated. We also introduce a way to solve large MDPs through an abstraction refinement process that can be applied to both discounted and total reward criteria. This method allows finding explicit state abstractions while solving any MDP with controlled error. We then integrate this state space disaggregation process into classical Dynamic Programming algorithms, namely Approximate Value Iteration, Q-Value Iteration, and Policy Iteration. We show that this method can decrease the solving time of a wide range of models and can also describe the underlying dynamics of the MDP without making any assumptions about the structure of the problem. We also conduct an extensive numerical comparison and compare our approach to existing aggregation methods to support our claims.
• Efficient and scalable atmospheric dynamics simulations using non-conforming meshes
  
  • Orlando Giuseppe
  • Benacchio Tommaso
  • Bonaventura Luca
  , 2025, 255, pp.33-42. We present the massively parallel performance of a $h$-adaptive solver for atmosphere dynamics that allows for non-conforming mesh refinement. The numerical method is based on a Discontinuous Galerkin (DG) spatial discretization, highly scalable thanks to its data locality properties, and on a second order Implicit-Explicit Runge-Kutta (IMEX-RK) method for time discretization, particularly well suited for low Mach number flows. Simulations with non-conforming meshes for flows over orography can increase the accuracy of the local flow description without affecting the larger scales, which can be solved on coarser meshes. We show that the local refining procedure has no significant impact on the parallel performance and, therefore, both efficiency and scalability can be achieved in this framework. (10.1016/j.procs.2025.02.258)
  DOI : 10.1016/j.procs.2025.02.258
• Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter
  
  • Goldman Michael
  • Merlet Benoît
  • Pegon Marc
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2024, 248 (6), pp.102. In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary which is uniform as the parameter $\varepsilon$ goes to $0$. As a consequence we obtain that volume-constrained minimizers of $\mathcal{F}_{\varepsilon,\gamma}$ are balls for any $\varepsilon$ small enough. For small $\varepsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable kernel $G$ with sufficiently fast decay at infinity. (10.1007/s00205-024-02048-x)
  DOI : 10.1007/s00205-024-02048-x
• Sensitivity analysis of colored-noise-driven interacting particle systems
  
  • Garnier Josselin
  • Ip Harry
  • Mertz Laurent
  Physical Review E, American Physical Society (APS), 2024, 110 (4), pp.044119. (10.1103/PhysRevE.110.044119)
  DOI : 10.1103/PhysRevE.110.044119
• Optimal Strategy Against Straightforward Bidding in Clock Auctions
  
  • Zeroual Jad
  • Akian Marianne
  • Bechler Aurélien
  • Chardy Matthieu
  • Gaubert Stéphane
  , 2025, Lecture Notes in Computer Science (LNCS-15185), pp.83-93. We study a model of auction representative of the 5G auction in France. We determine the optimal strategy of a bidder, assuming that the valuations of competitors are unknown to this bidder and that competitors adopt the straightforward bidding strategy. Our model is based on a Partially Observable Markov Decision Process (POMDP). We show in particular that this special POMDP admits a concise statistics, avoiding the solution of a dynamic programming equation in the space of beliefs. We illustrate our results by numerical experiments, comparing the value of the bidder with the value of a perfectly informed one. (10.1007/978-3-031-78600-6_8)
  DOI : 10.1007/978-3-031-78600-6_8
• Set-decomposition of normal rectifiable G-chains via an abstract decomposition principle
  
  • Goldman Michael
  • Merlet Benoît
  Revista Matemática Iberoamericana, European Mathematical Society, 2024, 40 (6), pp.2073-2094. We introduce the notion of set-decomposition of a normal G-flat chain. We show that any normal rectifiable $G$-flat chain admits a decomposition in set-indecomposable sub-chains. This generalizes the decomposition of sets of finite perimeter in their ``measure theoretic" connected components due to Ambrosio, Caselles, Masnou and Morel. It can also be seen as a variant of the decomposition of integral currents in indecomposable components by Federer. As opposed to previous results, we do not assume that G is boundedly compact. Therefore we cannot rely on the compactness of sequences of chains with uniformly bounded N-norms. We deduce instead the result from a new abstract decomposition principle. As in earlier proofs a central ingredient is the validity of an isoperimetric inequality. We obtain it here using the finiteness of some h-mass to replace integrality. (10.4171/RMI/1504)
  DOI : 10.4171/RMI/1504
• Study of neutron correlations measurements, model creation and supervised learning, bayesian inference of nuclear parameters.
  
  • Lartaud Paul
  , 2024. Nuclear security is the task of protecting the population and the environment against malicious acts involving radioactive substances. The International Atomic Energy Agency (IAEA) exhorts states to present a fast, robust, and reliable nuclear security strategy, in response to the increasing geopolitical tensions. In particular, the identification of nuclear fissile matter is a foundational element of any nuclear security strategy. To ensure the reliability of the response to any threat, the quantification and control of the uncertainties embedded in the underlying mathematical methods are mandatory. This thesis is located at the crossroads of fissile matter identification and uncertainty quantification. The general objective is to develop mathematical and numerical methods adapted to the neutron noise analysis in zero-power subcritical systems. This passive measurement technique is, along with gamma spectroscopy, a focal point of fissile matter identification.The methodology presented in this manuscript is based on a Bayesian resolution of an inverse problem, whose observations come from the study of temporal correlations between fission-induced neutrons. The standard resolution of this problem is intractable due to the cost of the Monte Carlo code for neutron transport. This thesis presents a framework in which the computer model is replaced by various surrogates, whose intrinsic uncertainties are fed into the inverse problem. The uncertainty quantification procedure encompasses both epistemic and aleatoric uncertainties into a common framework. This strategy can be improved with the help of sequential design strategies built specifically for the inverse problem, or with the introduction of gamma correlations which helps in reducing the residual uncertainties. Finally, we discuss a connex approach that circumvents the resolution of the inverse problem with a parametrization of the class of posterior distributions with the help of generalized lambda distributions.
• Variational Inference : theory and large scale applications.
  
  • Huix Tom
  , 2024. This thesis explores Variational Inference methods for high-dimensional Bayesian learning. In Machine Learning, the Bayesian approach allows one to deal with epistemic uncertainty and provides and a better uncertainty quantification, which is necessary in many machine learning applications. However, Bayesian inference is often not feasible because the posterior distribution of the model parameters is generally untractable. Variational Inference (VI) allows to overcome this problem by approximating the posterior distribution with a simpler distribution called the variational distribution.In the first part of this thesis, we worked on the theoretical guarantees of Variational Inference. First, we studied VI when the Variational distribution is a Gaussian and in the overparameterized regime, i.e., when the models are high dimensional. Finally, we explore the Gaussian mixtures Variational distributions, as it is a more expressive distribution. We studied both the optimization error and the approximation error of this method.In the second part of the thesis, we studied the theoretical guarantees for contextual bandit problems using a Bayesian approach called Thompson Sampling. First, we explored the use of Variational Inference for Thompson Sampling algorithm. We notably showed that in the linear framework, this approach allows us to obtain the same theoretical guarantees as if we had access to the true posterior distribution. Finally, we consider a variant of Thompson Sampling called Feel-Good Thompson Sampling (FG-TS). This method allows to provide better theoretical guarantees than the classical algorithm. We then studied the use of a Monte Carlo Markov Chain method to approximate the posterior distribution. Specifically, we incorporated into FG-TS a Langevin Monte Carlo algorithm and a Metropolized Langevin Monte Carlo algorithm. Moreover, we obtained the same theoretical guarantees as for FG-TS when the posterior distribution is known.
• Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation
  
  • Assogba Kenneth
  • Allaire Grégoire
  • Bourhrara Lahbib
  Computational Methods in Applied Mathematics, De Gruyter, 2024. (10.1515/cmam-2024-0021)
  DOI : 10.1515/cmam-2024-0021
• The Competive Spectral Radius of Families of Nonexpansive Mappings
  
  • Akian Marianne
  • Gaubert Stéphane
  • Marchesini Loïc
  , 2024. We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Ma\~ne's lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the second author (2012), concerning the asymptotic behavior of nonexpansive mappings. We discuss two special cases of this game: order preserving and positively homogeneous self-maps of a cone equipped with Funk's and Thompson's metrics, and groups of translations.
• Robust and accurate simulations of flows over orography using non-conforming meshes
  
  • Orlando Giuseppe
  • Benacchio Tommaso
  • Bonaventura Luca
  Quarterly Journal of the Royal Meteorological Society, Wiley, 2024, 150 (765), pp.4750-4770. We systematically validate the static local mesh refinement capabilities of a recently proposed IMEX-DG scheme implemented in the framework of the deal.II library. Non-conforming meshes are employed in atmospheric flow simulations to increase the resolution around complex orography. The proposed approach is fully mass and energy conservative and allows local mesh refinement in the vertical and horizontal direction without relaxation at the internal coarse/fine mesh boundaries. A number of numerical experiments based on classical benchmarks with idealized as well as more realistic orography profiles demonstrate that simulations with the locally refined mesh are stable for long lead times and that no spurious effects arise at the interfaces of mesh regions with different resolutions. Moreover, correct values of the momentum flux are retrieved and the correct large-scale orographic response is reproduced. Hence, large-scale orography-driven flow features can be simulated without loss of accuracy using a much lower total amount of degrees of freedom. (10.1002/qj.4839)
  DOI : 10.1002/qj.4839
• Super-resolution reconstruction from truncated Hankel transform
  
  • Goncharov Fedor
  • Isaev Mikhail
  • Novikov Roman
  • Zaytsev Rodion
  , 2024. We present the algorithm from our recent work Goncharov, Isaev, Novikov, Zaytsev (ArXiv preprint, 2024) for recovering a compactly supported function on $R_+$ from its Hankel transform given on a finite interval [0, r]. This work employs the PSWF-Radon approach that combines the theory of classical one-dimensional prolate spheroidal wave functions with the Radon transform theory, which was originally developed for reconstructing signals from their truncated Fourier transforms. Adapted to the Hankel transform, it achieves what is known as 'super-resolution' (the ability to recover details smaller than π/r), even in the presence of moderate noise in the data. In particular, our numerical examples show that the PSWF-Radon approach is consistently as good as, and often outperforms, the conventional approach that complements missing data with zeros. In this review, to illustrate the efficiency of our algorithm to simultaneously operate with the Hankel transform of several different orders, we also include a new application involving truncated multiple angle expansions for functions on $R^2$ .
• On the irreducibility and convergence of a class of nonsmooth nonlinear state-space models on manifolds
  
  • Gissler Armand
  • Durmus Alain
  • Auger Anne
  , 2024. In this paper, we analyze a large class of general nonlinear state-space models on a state-space X, defined by the recursion ϕ k+1 = F (ϕ k , α(ϕ k , U k+1 )), k ∈ N, where F, α are some functions and {U k+1 } k∈N is a sequence of i.i.d. random variables. More precisely, we extend conditions under which this class of Markov chains is irreducible, aperiodic and satisfies important continuity properties, relaxing two key assumptions from prior works. First, the state-space X is supposed to be a smooth manifold instead of an open subset of a Euclidean space. Second, we only suppose that F is locally Lipschitz continuous. We demonstrate the significance of our results through their application to Markov chains underlying optimization algorithms. These schemes belong to the class of evolution strategies with covariance matrix adaptation and step-size adaptation.
• The epidemiological footprint of contact structures in models with two levels of mixing
  
  • Bansaye Vincent
  • Deslandes François
  • Kubasch Madeleine
  • Vergu Elisabeta
  Journal of Mathematical Biology, Springer, 2024. Models with several levels of mixing (households, workplaces), as well as various corresponding formulations for R0, have been proposed in the literature. However, little attention has been paid to the impact of the distribution of the population size within social structures, effect that can help plan effective interventions. We focus on the influence on the model outcomes of teleworking strategies, consisting in reshaping the distribution of workplace sizes. We consider a stochastic SIR model with two levels of mixing, accounting for a uniformly mixing general population, each individual belonging also to a household and a workplace. The variance of the workplace size distribution appears to be a good proxy for the impact of this distribution on key outcomes of the epidemic, such as epidemic size and peak. In particular, our findings suggest that strategies where the proportion of individuals teleworking depends sublinearly on the size of the workplace outperform the strategy with linear dependence. Besides, one drawback of the model with multiple levels of mixing is its complexity, raising interest in a reduced model. We propose a homogeneously mixing SIR ODE-based model, whose infection rate is chosen as to observe the growth rate of the initial model. This reduced model yields a generally satisfying approximation of the epidemic. These results, robust to various changes in model structure, are very promising from the perspective of implementing effective strategies based on social distancing of specific contacts. Furthermore, they contribute to the effort of building relevant approximations of individual based models at intermediate scales. (10.1007/s00285-024-02147-z)
  DOI : 10.1007/s00285-024-02147-z
• Irreducibility of nonsmooth state-space models with an application to CMA-ES
  
  • Gissler Armand
  • Wolf Shan-Conrad
  • Auger Anne
  • Hansen Nikolaus
  , 2024. We analyze a stochastic process resulting from the normalization of states in the zeroth-order optimization method CMA-ES. On a specific class of minimization problems where the objective function is scaling-invariant, this process defines a time-homogeneous Markov chain whose convergence at a geometric rate can imply the linear convergence of CMA-ES. However, the analysis of the intricate updates for this process constitute a great mathematical challenge. We establish that this Markov chain is an irreducible and aperiodic T-chain. These contributions represent a first major step for the convergence analysis towards a stationary distribution. We rely for this analysis on conditions for the irreducibility of nonsmooth state-space models on manifolds. To obtain our results, we extend these conditions to address the irreducibility in different hyperparameter settings that define different Markov chains, and to include nonsmooth state spaces.
• Scaling limit of trees with vertices of fixed degrees and heights
  
  • Blanc-Renaudie Arthur
  • Kammerer Emmanuel
  , 2024. We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé-Galton-Watson trees in varying environment. (10.48550/arXiv.2409.12897)
  DOI : 10.48550/arXiv.2409.12897
• A fictitious domain method with enhanced interfacial mass conservation for immersed FSI with thin-walled solids
  
  • Corti Daniele
  • Diaz Jérôme
  • Vidrascu Marina
  • Chapelle Dominique
  • Moireau Philippe
  • Fernández Miguel Angel
  , 2024. In this paper, we extend the low-order fictitious domain method with enhanced mass conservation, introduced in [ESAIM: Math. Model. Numer. Anal., 58(1):303--333, 2024], to fluid-structure interaction with immersed thin-walled solids. The key idea is to improve mass conservation across the interface by imposing a single global velocity constraint on one side of the interface using a scalar Lagrange multiplier. Both 2D and 3D shell models are considered for the description of the solid, including contact between solids. For both models, the interface coupling is enforced on the mid-surface of the shell using a stabilized Lagrange multiplier formulation. Numerical examples in two and three dimensions illustrate the effectiveness of the proposed method, including its successful application to the simulation of aortic heart valve dynamics.
• Modèles de filaments pour la natation à l'échelle microscopique
  
  • Levillain Jessie
  , 2024. Les mathématiques associées à la natation à l'échelle microscopique constituent un domaine de recherche actif depuis une quinzaine d'années, avec de nombreuses applications en biologie et en physique. En effet, les micro-organismes se déplaçant dans l'eau jouent un rôle crucial dans l'origine et le maintien de la vie, et les principes physiques régissant leurs mouvements diffèrent grandement de ceux qui gouvernent la natation humaine. Les recherches dans ce domaine peuvent aussi être appliquées à la conception de micro sous-marins, ouvrant la voie à des applications innovantes en médecine, comme la chirurgie non-invasive.Cette thèse porte sur l'étude de tels micro-organismes, dans un contexte où les forces inertielles sont négligeables par rapport aux forces visqueuses dans le fluide environnant, phénomène caractérisé par un faible nombre de Reynolds.Les deux premières parties traitent de modèles mathématiques de nageurs, composés de bras actifs, sphères, et ressorts passifs. Ces modèles permettent de contourner le théorème de la coquille Saint-Jacques de Purcell, qui garantit qu'un nageur dont la brassée est un mouvement réciproque ne pourra jamais se déplacer, car il revient toujours à sa position initiale en l'absence d'inertie.Le premier modèle conçu au cours de cette thèse est un nageur à quatre sphères avec deux bras élastiques passifs. Faire varier la fréquence d'oscillation du bras actif permet de changer le signe du déplacement du nageur, rendant le système contrôlable, tout en n'ayant qu'un seul degré de liberté actif. Le modèle étudié dans le deuxième chapitre comporte, quant à lui, un grand nombre de ressorts. Un modèle limite de ce nageur où le nombre de ressorts tend vers l'infini a ensuite été considéré, transformant le nageur en une queue élastique se compressant et s'étendant unidimensionnellement.Au cours du troisième chapitre, on présente une preuve de la convergence et du caractère bien posé d'un modèle discret de micro-filament élastique à bas nombre de Reynolds. Ce modèle en trois dimensions est composé de N filaments rigides, dont le mouvement est en deux dimensions.Enfin, dans le dernier chapitre, des modèles des mécanismes d'activation le long d'un flagelle sont présentés. Du point de vue de la biologie, ces mécanismes d'activation sont présents dans le flagelle dans une structure interne, appelée axonème, sous la forme de moteurs moléculaires arrangés en plusieurs rangées.En se basant sur un modèle de ces moteurs issu de la biophysique, deux nouveaux systèmes sont ensuite étudiés. Le premier représente une projection de l'axonème dans laquelle deux rangées de moteurs sont présentes, et est étudié aussi bien théoriquement que numériquement. Le second prend en compte la totalité des rangées de moteurs dans l'axonème, son comportement est illustré par des simulations numériques.
• A mean-field model of Integrate-and-Fire neurons: non-linear stability of the stationary solutions
  
  • Cormier Quentin
  Mathematical Neuroscience and Applications, Inria, 2024. We investigate a stochastic network composed of Integrate-and-Fire spiking neurons, focusing on its mean-field asymptotics. We consider an invariant probability measure of the McKean-Vlasov equation and establish an explicit sufficient condition to ensure the local stability of this invariant distribution. Furthermore, we prove a conjecture proposed initially by J. Touboul and P. Robert regarding the bistable nature of a specific instance of this neuronal model. (10.46298/mna.12583)
  DOI : 10.46298/mna.12583
• ponio
  
  • Massot Josselin
  • Massot Marc
  • Series Laurent
  , 2024. The aim of ponio is to provide a set of schemes in time for solving a whole collection of ODEs and PDEs. It is initially written in C++, but various interfaces will later be available for use in other languages widely used in the scientific community (Python and Julia, for example). The idea here is to discuss the various strategies for the temporal integration of PDEs. The simplest is the combination of an operator separation strategy and a method of line involving various classical time integrators like Runge-Kutta methods, or optimized one (RADAU5, ROCK4) and also splitting operator methods (IMEX, Strang splitting) ; the long-term objective is also to be able to tackle innovative adaptive code coupling techniques through an interface as well as classes of time-space coupled schemes (Lax-Wendroff, OSMP, time-space coupled IMEX with good asymptotic preserving and stability properties...).
• Rare Event Simulation for Piecewise Deterministic Markov Processes, application in reliability assessment with PyCATSHOO tool.
  
  • Chennetier Guillaume
  , 2024. The purpose of this thesis is to provide new methods for estimating rare event probabilities for Piecewise Deterministic Markov Processes (PDMPs). This very general class of stochastic processes offers the flexibility needed to accurately represent complex dynamic industrial systems. In particular, it allow for the joint modeling of the deterministic and continuous dynamics of the physical variables of the system (temperature, pressure, liquild levels, etc.), and the random jump dynamics that govern the change in status of its components (failures, repairs, control mechanisms, etc.). The industrial challenge is to enable the tool PyCATSHOO, used by the company Électricité de France for its probabilistic safety assessment studies, to efficiently estimate the failure probability of such systems with guaranteed accuracy. A classical Monte Carlo approach requires, for a fixed level of accuracy, a number of simulations inversely proportional to the probability sought. It is therefore not suitable for highly reliable systems with high simulation costs. Importance sampling is a popular variance reduction method in the rare event context. It consists of generating simulations under a biased distribution that favors the occurrence of the event, and correcting the bias a posteriori. Recent work has proposed a theoretical framework for implementing importance sampling of PDMPs, and has highlighted the connection between the optimal biased distribution and the so-called "committor function" of the process. Using tools from reliability analysis and the theory of random walks on graphs, new families of approximations of the committor function are introduced in this thesis. The proposed methodology is adaptive: an approximation of the committor function is constructed a priori and then refined during the simulations of a cross-entropy procedure. The simulations are then recycled to produce an importance sampling estimator of the target probability. Convergence results have been obtained, making it possible to overcome the dependence between simulations and construct asymptotic confidence intervals. This method produces excellent results in practice on the tested industrial systems.
• Learning extreme Expected Shortfall and Conditional Tail Moments with neural networks. Application to cryptocurrency data
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2024. We propose a neural networks method to estimate extreme Expected Shortfall, and even more generally, extreme conditional tail moments as functions of confidence levels, in heavy-tailed settings. The convergence rate of the uniform error between the log-conditional tail moment and its neural network approximation is established leveraging extreme-value theory (in particular the high-order condition on the distribution tails) and using critically two activation functions (eLU and ReLU) for neural networks. The finite sample performance of the neural network estimator is compared to bias-reduced extreme-value competitors using synthetic heavy-tailed data. The experiments reveal that our method largely outperforms others. In addition, the selection of the anchor point appears to be much easier and stabler than for other methods. Finally, the neural network estimator is tested on real data related to extreme loss returns in cryptocurrencies: here again, the accuracy obtained by cross-validation is excellent, and is much better compared with competitors.
• A Mean Field Game Approach to Bitcoin Mining
  
  • Bertucci Charles
  • Bertucci Louis
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2024, 15 (3), pp.960-987. (10.1137/23M1617813)
  DOI : 10.1137/23M1617813