Publications

2024

• Shape Optimization: Theoretical, Numerical and Practical Aspects
  
  • Bogosel Beniamin
  , 2024.
• Stochastic foraging paths primarily drives within-species variations of prey consumption rates
  
  • Bansaye Vincent
  • Berthelot Geoffroy
  • El Bachari Amina
  • Chazottes Jean-René
  • Billiard Sylvain
  , 2024. The speed at which individuals interact, in particular prey and predators, affects ecological processes at all scales, including how fast matter and energy flow through ecosystems, and how stable communities are. Environmental heterogeneity and individual variabilities are generally believed to be the main factors underlying the variation of consumption rates of prey by predators. We challenge this view by comparing predicted variability from a stochastic model to experimental data. We first analyze a stochastic model of a simple random walk with elementary ecological processes involved in prey consumption, including prey depletion, predator movements and prey handling. We provide sharp approximations of the distribution of the consumption rate and a quantitative prediction of the coefficient of variation when stochastic foraging is the only source of variability. Predictions are then compared to the coefficients of variation estimated from data from dozens of various species and experimental contexts. We show that the predictions only accounting for intrinsic stochasticity in foraging are compatible with the range of observed values, in particular in 1 or 2 dimensional space. After evaluating the robustness of our model’s predictions through stochastic computer simulations, we conclude that the main driver of the variation of the consumption rate is the foraging process itself rather than environmental or between-individual variabilities. Our approach lays the foundations for unifying foraging theory and population ecology, and as such has many empirical and theoretical implications for both fields.
• ICI: a data-driven individual-based epidemic propagation simulator
  
  • Colomb Maxime
  • Cormier Quentin
  • Garnier Josselin
  • Graham Carl
  • Perret Julien
  • Talay Denis
  , 2024. ICI is a platform for the data-driven simulation of epidemic propagation in large urban areas. It couples a generator of a digital twin of the urban area and its population of mobile individuals with an individual-based disease contagion model. The contagion chains thus simulated resemble those in real life. The aim of the ICI platform is to allow health authorities to test, evaluate, and compare a variety of custom-designed sanitary policies in view of their possible implementations.
• On the simulation of extreme events with neural networks
  
  • Girard Stéphane
  • Allouche Michaël
  • Gobet Emmanuel
  , 2024. This work aims at investigating the use of generative methods based on neural networks to simulate extreme events. Although very popular, these methods are mainly invoked in empirical works. Therefore, providing theoretical guidelines for using such models in an extreme value context is of primal importance. To this end, we propose an overview of some generative methods dedicated to extremes, giving theoretical tips on their tail behaviour thanks to both extreme-value and copula tools. More specifically, we shall focus on a new parametrization for the generator of a Generative adversarial network (GAN) adapted to the heavy tail framework. An analysis of the uniform error between an extreme quantile and its GAN approximation is provided: We establish that the rate of convergence of the error is mainly driven by the second-order parameter of the data distribution. The above results are illustrated on simulated data and real financial data
• Prior de référence sous contraintes selon différentes mesures de dissimilarité
  
  • van Biesbroeck Antoine
  • Gauchy Clément
  • Feau Cyril
  • Garnier Josselin
  , 2024, pp.189-196. La théorie des priors de référence propose une solution à la question du choix du prior en analyse bayésienne, en construisant ce dernier comme celui qui minimise son influence a posteriori. Bien que permettant la construction d'un cadre alors qualifié d'objectif, l'expression du prior de référence prend souvent une forme proche d'un prior non informatif de Jeffreys. Cette dernière est parfois perçue comme encombrante à l'implémentation et rigide quant à l'introduction d'un jugement a priori. Dans cette communication, nous nous appuyons sur des travaux récents de la littérature, qui étendent la théorie des priors de référence. Leur formalisme et leurs résultats nous permettent en effet de définir un prior à la fois dit de référence et qui satisfait un certains nombres de contraintes pour lesquels nous définissons un cadre adéquat. Le résultat théorique que nous démontrons propose l'expression de ce prior de référence sous des contraintes linéaires, dont le choix pratique reste large et inclusif.
• Adaptive probabilistic forecasting of French electricity spot prices
  
  • Dutot Grégoire
  • Zaffran Margaux
  • Féron Olivier
  • Goude Yannig
  , 2024. Electricity price forecasting (EPF) plays a major role for electricity companies as a fundamental entry for trading decisions or energy management operations. As electricity can not be stored, electricity prices are highly volatile which make EPF a particularly difficult task. This is all the more true when dramatic fortuitous events disrupt the markets. Trading and more generally energy management decisions require risk management tools which are based on probabilistic EPF (PEPF). In this challenging context, we argue in favor of the deployment of highly adaptive black-boxes strategies allowing to turn any forecasts into a robust adaptive predictive interval, such as conformal prediction and online aggregation, as a fundamental last layer of any operational pipeline. We propose to investigate a novel data set containing the French electricity spot prices during the turbulent 2020-2021 years, and build a new explanatory feature revealing high predictive power, namely the nuclear availability. Benchmarking state-of-the-art PEPF on this data set highlights the difficulty of choosing a given model, as they all behave very differently in practice, and none of them is reliable. However, we propose an adequate conformalisation, OSSCP-horizon, that improves the performances of PEPF methods, even in the most hazardous period of late 2021. Finally, we emphasize that combining it with online aggregation significantly outperforms any other approaches, and should be the preferred pipeline, as it provides trustworthy probabilistic forecasts.
• Predictive Uncertainty Quantification with Missing Covariates
  
  • Zaffran Margaux
  • Josse Julie
  • Romano Yaniv
  • Dieuleveut Aymeric
  , 2024. Predictive uncertainty quantification is crucial in decision-making problems. We investigate how to adequately quantify predictive uncertainty with missing covariates. A bottleneck is that missing values induce heteroskedasticity on the response's predictive distribution given the observed covariates. Thus, we focus on building predictive sets for the response that are valid conditionally to the missing values pattern. We show that this goal is impossible to achieve informatively in a distribution-free fashion, and we propose useful restrictions on the distribution class. Motivated by these hardness results, we characterize how missing values and predictive uncertainty intertwine. Particularly, we rigorously formalize the idea that the more missing values, the higher the predictive uncertainty. Then, we introduce a generalized framework, coined CP-MDA-Nested*, outputting predictive sets in both regression and classification. Under independence between the missing value pattern and both the features and the response (an assumption justified by our hardness results), these predictive sets are valid conditionally to any pattern of missing values. Moreover, it provides great flexibility in the trade-off between statistical variability and efficiency. Finally, we experimentally assess the performances of CP-MDA-Nested* beyond its scope of theoretical validity, demonstrating promising outcomes in more challenging configurations than independence.
• Reweighting the RCT for generalization: finite sample error and variable selection
  
  • Colnet Bénédicte
  • Josse Julie
  • Varoquaux Gaël
  • Scornet Erwan
  Journal of the Royal Statistical Society: Series A Statistics in Society, Royal Statistical Society, 2024. Randomized Controlled Trials (RCTs) may suffer from limited scope. In particular, samples may be unrepresentative: some RCTs over- or under- sample individuals with certain characteristics compared to the target population, for which one wants conclusions on treatment effectiveness. Re-weighting trial individuals to match the target population can improve the treatment effect estimation. In this work, we establish the exact expressions of the bias and variance of such reweighting procedures - also called Inverse Propensity of Sampling Weighting (IPSW) - in presence of categorical covariates for any sample size. Such results allow us to compare the theoretical performance of different versions of IPSW estimates. Besides, our results show how the performance (bias, variance, and quadratic risk) of IPSW estimates depends on the two sample sizes (RCT and target population). A by-product of our work is the proof of consistency of IPSW estimates. Results also reveal that IPSW performances are improved when the trial probability to be treated is estimated (rather than using its oracle counterpart). In addition, we study choice of variables: how including covariates that are not necessary for identifiability of the causal effect may impact the asymptotic variance. Including covariates that are shifted between the two samples but not treatment effect modifiers increases the variance while non-shifted but treatment effect modifiers do not. We illustrate all the takeaways in a didactic example, and on a semi-synthetic simulation inspired from critical care medicine. (10.1093/jrsssa/qnae043)
  DOI : 10.1093/jrsssa/qnae043
• Deterministic computation of quantiles in a Lipschitz framework
  
  • Gu Yurun
  • Rey Clément
  , 2024. In this article, our focus lies on computing the quantiles of a random variable $f(X)$, where $X$ is a $[0,1]^d$-valued random variable, $d \in \mathbb{N}^{\ast}$, and $f:[0,1]^d\to \mathbb{R}$ is a deterministic Lipschitz function. We are particularly interested in scenarios where the cost of a single function call is high, while assuming the law of $X$ is known. In this context, we propose a deterministic algorithm to obtain exact deterministic lower and upper bounds for the quantile of $f(X)$ at a given level $α\in (0,1)$. With a fixed budget of $N$ function calls, we demonstrate that our algorithm achieves exponential convergence rate for $d=1$ ($\sim ρ^N$ with $ρ\in (0,1)$) and polynomial convergence rate for $d>1$ ($\sim N^{-\frac{1}{d-1}}$) and show the optimality of those rates within the class of deterministic algorithms. Furthermore, we design two algorithms based on whether the Lipschitz constant of $f$ is known or unknown. (10.48550/arXiv.2405.10638)
  DOI : 10.48550/arXiv.2405.10638
• GenEO
  
  • Gouarin Loïc
  • Spillane Nicole
  , 2024. The open source software GenEO, written in Python, includes two new families of preconditioners for symmetric positive definite linear systems. 1) First, the AWG preconditioners (for Algebraic-Woodbury-GenEO) have the feature of being algebraic \cite{zbMATH07846109,10.1007/978-3-030-95025-5_81}. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes. The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable non-assembled form is known by construction. The extra cost is a second coarse solve in the preconditioner. 2) Second, the framework for Krylov subspace methods with adaptive multipreconditioning is implemented. Multipreconditiioning is a technique that allows to apply more than one preconditioner at each step. Domain decomposition is a natural application. Since a multipreconditioned iteration is more expensive than a classical iteration, it is advantageous to multiprecondition only when necessary. To this end, an adapativity scheme was proposed in \cite{zbMATH06601530} and is implemented in GenEO. GenEO uses Petsc4py and Dolfinx to solve 2D and 3D problems. Then, it is easy to compare this new family of preconditioners with those already defined in Petsc and see their impact on various problems with highly heterogeneous coefficients. @Article{zbMATH06601530, Author = {Spillane, Nicole}, Title = {An adaptive multipreconditioned conjugate gradient algorithm}, FJournal = {SIAM Journal on Scientific Computing}, Journal = {SIAM J. Sci. Comput.}, ISSN = {1064-8275}, Volume = {38}, Number = {3}, Pages = {a1896--a1918}, Year = {2016}, Language = {English}, DOI = {10.1137/15M1028534}, Keywords = {65F10,65N30,65N55}, zbMATH = {6601530}, Zbl = {1416.65087} } @Article{zbMATH07846109, Author = {Gouarin, Lo{\"{\i}}c and Spillane, Nicole}, Title = {Fully algebraic domain decomposition preconditioners with adaptive spectral bounds}, FJournal = {ETNA. Electronic Transactions on Numerical Analysis}, Journal = {ETNA, Electron. Trans. Numer. Anal.}, ISSN = {1068-9613}, Volume = {60}, Pages = {169--196}, Year = {2024}, Language = {English}, DOI = {10.1553/etna_vol60s169}, Keywords = {65F10,65N30,65N55}, URL = {etna.mcs.kent.edu/volumes/2021-2030/vol60/abstract.php?vol=60&pages=169-196}, zbMATH = {7846109} } @inproceedings{10.1007/978-3-030-95025-5_81, abstract = {The starting point for the algebraic preconditioner is to relax condition (1) by allowing symmetric, but possibly indefinite, matrices in the splitting of A.}, address = {Cham}, author = {Spillane, Nicole}, booktitle = {Domain Decomposition Methods in Science and Engineering XXVI}, editor = {Brenner, Susanne C. and Chung, Eric and Klawonn, Axel and Kwok, Felix and Xu, Jinchao and Zou, Jun}, isbn = {978-3-030-95025-5}, pages = {745--752}, publisher = {Springer International Publishing}, title = {Toward a New Fully Algebraic Preconditioner for Symmetric Positive Definite Problems}, year = {2022}}
• Structured dictionary learning of rating migration matrices for credit risk modeling
  
  • Allouche Michaël
  • Gobet Emmanuel
  • Lage Clara
  • Mangin Edwin
  , 2024.
• Directed Metric Structures arising in Large Language Models
  
  • Gaubert Stéphane
  • Vlassopoulos Yiannis
  , 2024. Large Language Models are transformer neural networks which are trained to produce a probability distribution on the possible next words to given texts in a corpus, in such a way that the most likely word predicted is the actual word in the training text. In this paper we find what is the mathematical structure defined by such conditional probability distributions of text extensions. Changing the view point from probabilities to -log probabilities we observe that the subtext order is completely encoded in a metric structure defined on the space of texts $L$, by -log probabilities. We then construct a metric polyhedron $P(L)$ and an isometric embedding (called Yoneda embedding) of $L$ into $P(L)$ such that texts map to generators of certain special extremal rays. We explain that $P(L)$ is a $(\min,+)$ (tropical) linear span of these extremal ray generators. The generators also satisfy a system of $(\min+)$ linear equations. We then show that $P(L)$ is compatible with adding more text and from this we derive an approximation of a text vector as a Boltzmann weighted linear combination of the vectors for words in that text. We then prove a duality theorem showing that texts extensions and text restrictions give isometric polyhedra (even though they look a priory very different). Moreover we prove that $P(L)$ is the lattice closure of (a version of) the so called, Isbell completion of $L$ which turns out to be the $(\max,+)$ span of the text extremal ray generators. All constructions have interpretations in category theory but we don't use category theory explicitly. The categorical interpretations are briefly explained in an appendix. In the final appendix we describe how the syntax to semantics problem could fit in a general well known mathematical duality.
• Learning out-of-sample Expected Shortfall and Conditional Tail Moments with neural networks. Application to cryptocurrency data
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2024. We propose new parameterizations for neural networks in order to estimate out-of-sample Expected Shortfall, and even more generally, out-of-sample conditional tail moments, in heavy-tailed settings as functions of confidence levels. The proposed neural network estimator is able to extrapolate in the distribution tails thanks to an extension of the usual extreme-value second-order condition to an arbitrary order. The convergence rate of the uniform error between the log-conditional tail moment and its neural network approximation is established. The finite sample performance of the neural network estimator is compared to bias-reduced extreme-value competitors on simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail. Finally, the neural network estimator is tested on real data to investigate the behavior of cryptocurrency extreme loss returns.
• Sensitivity analysis of a flow redistribution model for a multidimensional and multifidelity simulation of fuel assembly bow in a pressurized water reactor
  
  • Abboud Ali
  • de Lambert Stanislas
  • Garnier Josselin
  • Leturcq Bertrand
  , 2024, pp.272. <div><p>In the core of nuclear reactors, fluid-structure interaction and intense irradiation lead to progressive deformation of fuel assemblies. When this deformation is significant, it can lead to additional costs and longer fuel unloading and reloading operations. Therefore, it is preferable to adopt a fuel management that avoids excessive deformation and interactions between fuel assemblies. However, the prediction of deformation and interactions between fuel assemblies is uncertain. Uncertainties affect neutronics, thermohydraulics and thermomechanics parameters. Indeed, the initial uncertainties are propagated over several successive power cycles of twelve months each through the coupling of non-linear, nested and multidimensional thermal-hydraulic and thermomechanical simulations. In this article, we set out to study the hydraulic contribution and quantify the associated uncertainty. To achieve this objective, we develop a multi-stage approach to carry out an initial sensitivity analysis, highlighting the most influential parameters in the hydraulic model. By optimally adjusting these parameters, we aim to obtain a more accurate description of the flow redistribution phenomenon in the reactor core.The aim of the sensitivity analysis presented in this article is to construct an accurate and suitable surrogate model that represents the redistribution model in the core. This surrogate model will then be coupled with the thermomechanical model to quantify the final uncertainty in the simulation of fuel assembly deformation within a pressurised water reactor. This approach will provide a better understanding of the interactions between hydraulic and thermomechanical phenomena, thereby improving the reliability and accuracy of the simulation results.</p></div>
• A unified two-scale gas-liquid multi-fluid model with capillarity and interface regularization through a mass transfer between scales
  
  • Loison Arthur
  • Kokh Samuel
  • Pichard Teddy
  • Massot Marc
  International Journal of Multiphase Flow, Elsevier, 2024. In this contribution, we derive a gas-liquid two-scale multi-fluid model with capillarity effects to enable a novel interface regularization approach for multi-fluid models. As this unified modelling is capable of switching from the interface representation of a separated to a disperse regime it lays a new way of modelling regime transitions as it occurs in atomization processes. Above a preset length threshold at large scale, a multi-fluid diffuse interface model resolves the dynamics of the interface while, at small-scale, a set of geometric variables is used to characterize the interface geometry. These variables result from a reduced-order modelling of the small-scale kinetic equation that describes a collection of liquid inclusions. The flow model can be viewed as a two-phase two-scale mixture, and the equations of motion are obtained thanks to the Hamilton’s Stationary Action Principle, which requires to specify the kinetic and potential energies at play. We particularly focus on modelling the effects of capillarity on the mixture’s energy by including dependencies on additional variables accounting for the interface’s geometry at both scales. The regularization of the large-scale interface is then introduced as a local and dissipative process. The local curvature is limited via a relaxation toward a modified Laplace equilibrium such that an inter-scale mass transfer is triggered when the mean curvature is too high. We propose an original numerical method and assess the properties and potential of the modelling strategy on the relevant test-case of a two-dimensional liquid column in a compressible gas flow. (10.1016/j.ijmultiphaseflow.2024.104857)
  DOI : 10.1016/j.ijmultiphaseflow.2024.104857
• Decompounding with unknown noise through several independents channels
  
  • Garnier Guillaume
  , 2024. In this article, we consider two different statistical models. First, we focus on the estimation of the jump intensity of a compound Poisson process in the presence of unknown noise. This problem combines both the deconvolution problem and the decompounding problem. More specifically, we observe several independent compound Poisson processes but we assume that all these observations are noisy due to measurement noise. We construct an Fourier estimator of the jump density and we study its mean integrated squared error. Then, we propose an adaptive method to correctly select the cutoff of the estimator and we illustrate the efficiency of the method with numerical results. Secondly, we introduce in this paper the multiplicative decompounding problem. We study this problem with Mellin density estimators. We develop an adaptive procedure to select the optimal cutoff parameter.
• On the variational interpretation of local logarithmic Sobolev inequalities
  
  • Clerc Gauthier
  • Conforti Giovanni
  • Gentil Ivan
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2024, 32 (5), pp.823-837. The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $\Gamma$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schrödinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly unlock novel class of local inequalities. (10.5802/afst.1754)
  DOI : 10.5802/afst.1754
• ESTIMATION OF THE INVARIANT MEASURE OF A MULTIDIMENSIONAL DIFFUSION FROM NOISY OBSERVATIONS
  
  • Maillet Raphaël
  • Szymanski Grégoire
  , 2024. We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noises. We consider the intermediate regime, where observations occur at discrete time instances $k\Delta_n$ for $k=0,\dots,n$, under the conditions $\Delta_n\to 0$ and $n\Delta_n\to\infty$. Our methodology involves the construction of a kernel density estimator that uses a pre-averaging technique to proficiently remove noise from the data while preserving the analytical characteristics of the underlying signal and its asymptotic properties. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection.
• Contrôle de courbes de réponses isolées par optimisation structurelle
  
  • Mélot Adrien
  • Denimal Goy Enora
  • Renson Ludovic
  , 2024. Nous introduisons un cadre d’analyse numérique permettant de contrôler des courbes de réponse isolées (isolas), c’est-à-dire formant des courbes fermées et non connectées à la branche principale de solutions. La méthodologie repose sur des analyses de suivi de bifurcation afin de suivre l’évolution de bifurcations nœud-col dans un espace de codimension-2. La théorie des singularités est mise en œuvre afin de distinguer les points de formation et de fusion d’isolas des bifurcations de codimension-2. Un problème d’optimisation est défini pour avancer ou retarder la formation ou la fusion d’isolas.
• Modèle 1D enrichi pour le calcul mécanique rapide : approche multiparticulaire appliquée à la fabrication additive par dépôt de cordon
  
  • Preumont Laurane
  • Viano Rafaël
  • Weisz-Patrault Daniel
  • Margerit Pierre
  • Allaire Grégoire
  , 2024. Une nouvelle approche multiparticulaire (i.e., plusieurs champs de vitesse par point de matière) est présentée pour l’analyse mécanique des procédés de fabrication additive par dépôt de cordons. En effet, le modèle que nous proposons est un fil enrichi à quatre particules par point de matière permettant un maillage plus grossier dans la direction tangente du cordon, ce qui permet de réduire significativement le coût calculatoire. Le fondement théorique du modèle est rappelé brièvement et ses performances numériques (discrétisation aux éléments finis) sont comparées à un modèle 3D classique. Enfin, une étude paramétrique est proposée pour démontrer la pertinence et l’utilité de notre approche. Ce modèle mécanique rapide permet d’envisager des boucles d’optimisation sur des paramètres procédé tels que la vitesse de balayage et la puissance du laser.
• Iterative data-driven construction of surrogates for an efficient Bayesian identification of oil spill source parameters from image contours
  
  • El Mohtar Samah
  • Le Maître Olivier
  • Knio Omar
  • Hoteit Ibrahim
  Computational Geosciences, Springer Verlag, 2024, 28 (4), pp.681-696. Identifying the source of an oil spill is an essential step in environmental forensics. The Bayesian approach allows to estimate the source parameters of an oil spill from available observations. Sampling the posterior distribution, however, can be computationally prohibitive unless the forward model is replaced by an inexpensive surrogate. Yet the construction of globally accurate surrogates can be challenging when the forward model exhibits strong nonlinear variations. We present an iterative data-driven algorithm for the construction of polynomial chaos surrogates whose accuracy is localized in regions of high posterior probability. Two synthetic oil spill experiments, in which the construction of prior-based surrogates is not feasible, are conducted to assess the performance of the proposed algorithm in estimating five source parameters. The algorithm successfully provided a good approximation of the posterior distribution and accelerated the estimation of the oil spill source parameters and their uncertainties by an order of 100 folds. (10.1007/s10596-024-10288-9)
  DOI : 10.1007/s10596-024-10288-9
• Modified Lawson methods for Vlasov equations
  
  • Boutin Benjamin
  • Crestetto Anaïs
  • Crouseilles Nicolas
  • Massot Josselin
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2024, 46 (3), pp.A1574-A1598. In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov-Maxwell examples are presented to illustrate the good behavior of the approach. (10.1137/22M154301X)
  DOI : 10.1137/22M154301X
• Sharp convergence rates for the homogenization of the Stokes equations in a perforated domain
  
  • Balazi Loïc
  • Allaire Grégoire
  • Omnes Pascal
  , 2024. This paper is concerned with the homogenization of the Stokes equations in a periodic perforated domain. The homogenized model is known to be Darcy's law in the full domain. We establish a sharp convergence rate $O(\sqrt{\varepsilon})$ for the energy norm of the difference of the velocities, where $\varepsilon$ represents the size of the solid obstacles. This is achieved by using a two-scale asymptotic expansion of the Stokes equations and a new construction of a cut-off function which avoids the introduction of boundary layers. The main novelty is that our analysis applies for the physically relevant case of a porous medium where each of the fluid and solid parts is a connected subdomain.
• Demonstration-Regularized RL
  
  • Tiapkin Daniil
  • Belomestny Denis
  • Calandriello Daniele
  • Moulines Eric
  • Naumov Alexey
  • Perrault Pierre
  • Valko Michal
  • Menard Pierre
  , 2024. Incorporating expert demonstrations has empirically helped to improve the sample efficiency of reinforcement learning (RL). This paper quantifies theoretically to what extent this extra information reduces RL's sample complexity. In particular, we study the demonstration-regularized reinforcement learning that leverages the expert demonstrations by KL-regularization for a policy learned by behavior cloning. Our findings reveal that using $N^{\mathrm{E}}$ expert demonstrations enables the identification of an optimal policy at a sample complexity of order $\widetilde{O}(\mathrm{Poly}(S,A,H)/(\varepsilon^2 N^{\mathrm{E}}))$ in finite and $\widetilde{O}(\mathrm{Poly}(d,H)/(\varepsilon^2 N^{\mathrm{E}}))$ in linear Markov decision processes, where $\varepsilon$ is the target precision, $H$ the horizon, $A$ the number of action, $S$ the number of states in the finite case and $d$ the dimension of the feature space in the linear case. As a by-product, we provide tight convergence guarantees for the behaviour cloning procedure under general assumptions on the policy classes. Additionally, we establish that demonstration-regularized methods are provably efficient for reinforcement learning from human feedback (RLHF). In this respect, we provide theoretical evidence showing the benefits of KL-regularization for RLHF in tabular and linear MDPs. Interestingly, we avoid pessimism injection by employing computationally feasible regularization to handle reward estimation uncertainty, thus setting our approach apart from the prior works. (10.48550/arXiv.2310.17303)
  DOI : 10.48550/arXiv.2310.17303
• Optimal Portfolio Choice with Cross-Impact Propagators
  
  • Abi Jaber Eduardo
  • Neuman Eyal
  • Tuschmann Sturmius
  , 2024. We consider a class of optimal portfolio choice problems in continuous time where the agent's transactions create both transient cross-impact driven by a matrix-valued Volterra propagator, as well as temporary price impact. We formulate this problem as the maximization of a revenue-risk functional, where the agent also exploits available information on a progressively measurable price predicting signal. We solve the maximization problem explicitly in terms of operator resolvents, by reducing the corresponding first order condition to a coupled system of stochastic Fredholm equations of the second kind and deriving its solution. We then give sufficient conditions on the matrix-valued propagator so that the model does not permit price manipulation. We also provide an implementation of the solutions to the optimal portfolio choice problem and to the associated optimal execution problem. Our solutions yield financial insights on the influence of cross-impact on the optimal strategies and its interplay with alpha decays.