Publications

2021

• Preconditioning of domain decomposition methods for stochastic elliptic equations
  
  • Felicio dos Reis Joao
  , 2021. This thesis presents a new numerical method to efficiently generate samples of the solution of stochastic elliptical equations with random coefficients. Particular emphasis is placed on coefficients with high variance and short correlation length.This work concerns the adaptation of some classical Domain Decomposition (DD) Methods to the sampling of stochastic problems.Classical deterministic DD methods are based on iterative approaches which require preconditioning strategies capable of maintaining a high rate of convergence when the number of subdomains increases. In our stochastic context, determining a classical preconditioner suitable for each sample can be expensive, and alternative strategies can be more efficient. Each sample amounts to solving a reduced linear system for the values of the solution at the interfaces of the subdomains, according to a finite element discretization. This reduced system is then solved by an iterative method. This thesis proposed three main contributions to efficient preconditioning, by introducing surrogates of 1) the reduced global operator, 2) the contribution of each subdomain to the reduced global operator, and 3) local preconditioners (multi-preconditioning).The first contribution focuses on the iterative Schwarz method and introduces a stochastic preconditioner consisting of a surrogate of the Schwarz system for the unknown values on the interface of the subdomains. In a preprocessing stage, a truncated Karhunen-Loève (KL) expansion of the coefficient field and a Polynomial Chaos (PC) expansion of the Schwarz system are constructed to form the stochastic preconditioner. At the sampling stage, the preconditioner of each sample is recovered thanks to the very efficient evaluation of the PC expansions. Numerical simulations on a one-dimensional problem illustrate the rapid convergence of the resulting approach, provided that the number of KL modes and the PC degree are both sufficiently large.The second contribution extends the previous idea to non-overlapping DD methods by building surrogates of the local components of the Schur complement. The structure of the Schur problem is used to exploit the local character of the DD setting. This leads to local PC expansions of Neumann-Neumann (NN) maps with a small number of local random variables to discretize the stochastic field on each subdomain. This set of local PC expansions are computed independently at a preprocessing step. Then, the PC extensions are evaluated and assembled during the sampling stage to form the preconditioner. A decomposition of the local operators is proposed to ensure that the preconditioner is symmetric and positive-definite, and therefore convergence is guaranteed almost surely.The preceding preconditioning strategy requires solving at each iteration a problem of size equal to the number of interface nodes. This can be a limitation for larger dimension problems. Thus, the third contribution concerns a totally local preconditioner: the two-level NN preconditioner. Again, we propose to use surrogates based on local PC expansions that substitute for NN maps instead of solving local preconditioning problems for each sample. Numerical experiments show that local preconditioning based on local surrogates is almost as efficient as the actual computation of NN maps for each sample.
• YUKI Algorithm and POD-RBF for Elastostatic and dynamic crack identification
  
  • Benaissa Brahim
  • Aït Hocine Nourredine
  • Khatir Samir
  • Riahi Mohamed Kamel
  • Mirjalili Seyedali
  Journal of computational science, Elsevier, 2021, 55, pp.101451. (10.1016/j.jocs.2021.101451)
  DOI : 10.1016/j.jocs.2021.101451
• Chance constraint optimization of a complex system : Application to the design of a floating offshore wind turbine
  
  • Cousin Alexis
  , 2021. In this thesis, we propose a methodology to optimize the configuration of the mooring lines by minimizing the material cost while satisfying Fatigue Limit State (FLS) constraints. These constraints inherit the randomness of the marine environment as well as uncertainties on material properties and model parameters. Therefore, we face an optimization problem with a deterministic cost function and constraints involving probabilities of threshold exceedance of the maximum and the integral over a period [0,T] of time-dependent random processes.Having to evaluate these failure probabilities at each loop of the optimization algorithm is the main difficulty. Indeed, reliability methods require many time-consuming simulations. The estimation of these probabilities is all the more challenging as we are dealing with rare events. To solve this problem efficiently, we propose a two-step methodology. First, considering T sufficiently large, we use the properties of the constraints and limit theorems of the extreme value theory and the ergodic theory to reformulate the original constraints into time-independent ones. We thus obtain an equivalent problem for which classical algorithms perform poorly. The second step of the procedure consists in solving the reformulated problem with a new method based on an adaptive kriging strategy well suited to the reformulated constraints. This method is called AK-ECO for Adaptive Kriging for Expectation Constraints Optimization.An academic case of a harmonic oscillator presenting all the characteristics of the industrial problem is introduced to illustrate the methodology. The procedure is then applied with success to the FOWT problem. The two steps composing this methodology are described in a general framework so that they can be applied to other optimization problems involving probabilistic constraints depending on the maximum or the integral of random processes.
• Shape optimization of an imperfect interface: steady-state heat diffusion
  
  • Allaire Grégoire
  • Bogosel Beniamin
  • Godoy Matías
  Journal of Optimization Theory and Applications, Springer Verlag, 2021, 191 (1), pp.169-201. In the context of a diffusion equation, this work is devoted to a two-phase optimal design problem where the interface, separating the phases, is imperfect, meaning that the solution is discontinuous while the normal flux is continuous and proportional to the jump of the solution. The shape derivative of an objective function with respect to the interface position is computed by the adjoint method. Numerical experiments are performed with the level set method and an exact remeshing algorithm so that the interface is captured by the mesh at each optimization iteration. Comparisons with a perfect interface are discussed in the setting of optimal design or inverse problems. (10.1007/s10957-021-01928-6)
  DOI : 10.1007/s10957-021-01928-6
• Le profil de méthylome du sang total comme biomarqueur de l’excès des glucocorticoïdes
  
  • Armignacco Roberta
  • Jouinot Anne
  • Bouys Lucas
  • Septier Amandine
  • Lartigue Thomas
  • Neou Mario
  • Gaspar Cassandra
  • Perlemoine Karine
  • Braun Leah
  • Riester Anne
  • Bonnet-Serrano Fidéline
  • Blanchard Anne
  • Amar Laurence
  • Scaroni Carla
  • Ceccato Filippo
  • Rossi Gian Paolo
  • Williams Tracy Ann
  • Larsen Casper K.
  • Allassonnière Stéphanie
  • Zennaro Maria Christina
  • Beuschlein Felix
  • Reincke Martin
  • Bertherat Jérôme
  • Assié Guillaume
  Annales d'Endocrinologie = Annals of Endocrinology, Société française d'endocrinologie [1939-....], 2021, 82 (5), pp.238. (10.1016/j.ando.2021.07.058)
  DOI : 10.1016/j.ando.2021.07.058
• Towards a unified eulerian modeling framework for two-phase flows : geometrical small scale phenomena and associated flexible computing strategies
  
  • Di Battista Ruben
  , 2021. In current times we are witnessing a “second space race”: private companies like SpaceX are paving the way to a new generation of space launcher systems optimized for cost effectiveness and extreme performances that will bring humankind to Mars for the first time in its existence. A key aspect of those systems is to provide a high level of reusability leading to a drastic drop of launch costs. This translates into propulsion systems that need to operate on wider flight envelopes, with more advantageous propellant pairs like cryogenic methane and liquid oxygen, therefore requiring tighter designs for the injection systems. The injectors are responsible for the correct nebulization of fuel and oxidizer and they have a direct impact on the performance of the engines. These kind of problems are shared across different applications and are somehow generic The current state of the art modeling strategies fail at predicting the correct distributions of droplets in the combustion chamber. Therefore, the target of this thesis is to contribute to the design of a unified modeling framework addressing the derivation of system of equations governing two-phase flow systems characterized by a sound mathematical structure via a variational approach named Stationary Action Principle (SAP) coupled to the second principle of thermodynamics. This effort is backed by a tailored computational toolset that allows the rational choice of modeling assumptions and the effective simulations of the developed models, possibly on modern computing architectures. This work identifies three main points of improvement: the development of reduced-order models via a variational procedure named the Stationary Action Principle (SAP) featuring a set of equations that include geometrical properties such as the interfacial surface density and the mean and Gauss curvatures; the implementation of a geometric DNS post-processing tool that is used to collect useful insight from high-fidelity simulations in order to craft an accurate reducedorder model, and the development of a Python library that acts as a prototyping playbook aimed at quickly testing ideas in the context of numerical schemes, boundary conditions, domain configurations, with the potential ability of leveraging modern computational architectures such as GPUs.
• Approches déterministes et stochastiques de modélisation de l'hétérogénéité métabolique chez les bactéries
  
  • Tchouanti Fotso Josué
  , 2021. Cette thèse porte sur la compréhension de l’hétérogénéité métabolique au sein de populations bactériennes grâce à des approches de modélisation déterministes et stochastiques. Dans un premier temps, nous nous intéressons à l’étude de la croissance diauxique pour une souche d’Escherichia coli cultivée sur un mélange homogène de glucose et xylose dans un batch. Nous commençons par un modèle compartimental simple dans lequel nous séparons la population en deux classes en fonction du sucre métabolisé. Nous donnons dans ce cas une approximation du lag-diauxique dont nous étudions numériquement la sensibilité par rapport aux paramètres métaboliques. Nous proposons ensuite un second modèle compartimental basé sur les observations et données expérimentales recueillies par nos collaborateurs de l’Institut de Biotechnologie de Toulouse qui mettent en lumière l’impact d’une protéine nommée xylR sur la transition glucose/xylose. Grâce à l’algorithme d’optimisation stochastique CMA-ES, nous calibrons ce modèle pour deux souches bactériennes différentes dont l’une sauvage et l’autre modifiée, ce qui nous permet de quantifier l’impact de la disponibilité de cette protéine.Motivée par les questions biologiques sur l’émergence et l’impact de l’hétérogénéité métabolique, la deuxième partie de cette thèse s’intéresse au rôle joué par les caractéristiques génétiques individuelles. Nous proposons deux approches de modélisation individu-centrées et établissons des résultats de convergence en grande population pour des cultures continues. Une première approche s’intéresse à la dynamique de la population bactérienne structurée suivant la masse d’une protéine jouant un rôle central dans le métabolisme du substrat (cas du xylR pour le xylose). Nous montrons qu’à l’échelle macroscopique, la distribution de cette masse de protéine au sein de la population peut être décrite par une solution fonction d’une équation de diffusion-croissance-fragmentation couplée à une ressource, et dont nous établissons des propriétés de régularité Besov. Une dernière approche s’intéresse à une modélisation multi-échelle de la dynamique de la population structurée en fonction des densités volumiques des protéines codées par un système de gènes intervenant dans le métabolisme individuel. Nous supposons que des erreurs de petites variances sont commises lors de la fragmentation des protéines au moment de la division cellulaire, puis distinguons un régime lent et un régime rapide en fonction de la vitesse des mécanismes démographiques. Dans le régime lent, nous montrons qu’une population N-morphique le demeure à l’échelle macroscopique avec des traits dynamiques. En particulier dans le cas monomorphique (N = 1), nous décrivons une sous-population hétérogène cachée dans la population macroscopique. Dans le régime rapide, on distingue trois cas : un cas sous-critique dans lequel une population N-morphique le demeure comme dans le régime lent, à la différence qu’il peut y avoir conservation du bruit à l’échelle macroscopique ; un cas critique dans lequel les effets des erreurs à la division sont observables dans l’échelle typique et expliquent l’émergence de l’hétérogénéité par un double effet de sauts et de diffusion du trait ; un cas surcritique dans lequel la dynamique couplée de la population et de la ressource devient lente/rapide. Dans ce dernier cas, nous utilisons les méthodes dites de moyennisation pour décrire la distribution du trait au sein de la population à chaque instant comme la limite stationnaire d’une martingale à coordonnées positives.
• Topology optimization in contact, plasticity, and fracture mechanics using a level-set method
  
  • Desai Jeet Samir
  , 2021. The main contribution of this thesis is the theoretical and the numerical study of linear elasticity with contact boundary conditions, plasticity with hardening, damage and fracture model in the context of shape and topology optimization. One application of the contact boundary condition for an idealized-bolt model is also proposed. The governing equations of the three physics dealt with in this thesis: contact, plasticity and damage, are theoretically not shape-differentiable. In each case, we construct an approximation by penalization, regularization or a combination of the two. The approximations for contact and plasticity are shown to be well-posed and to admit solutions that converge to the exact solution. For each physics, the shape sensitivity analysis is performed on the approximate model and the resultant adjoint problem is shown to be well-posed under technical assumptions. The shape optimization is implemented numerically using a level-set method with body-fitted remeshing, which captures the boundary of the shapes while allowing for topology changes. Numerical results are presented in 2D and 3D. We also discuss high-performance computing for linear elasticity and for fracture model and present a few 3D results.
• Uncertainty Characterization of ablative materials for atmospheric reentry of spacecrafts using PuMA
  
  • Girault Florian
  • Congedo Pietro Marco
  , 2021.
• Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals
  
  • Bungert Leon
  • Burger Martin
  • Chambolle Antonin
  • Novaga Matteo
  Analysis & PDE, Mathematical Sciences Publishers, 2021, 14 (3), pp.823-860. This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinitedimensional Hilbert space. This approach is both motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent results on finite-dimensional polyhedral semi-norms, where gradient flows can yield spectral decompositions into eigenvectors. We provide a geometric characterization of eigenvectors via a dual unit ball and prove them to be subgradients of minimal norm. This establishes the connection to gradient flows, whose time evolution is a decomposition of the initial condition into subgradients of minimal norm. If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow. Indeed we verify that all scenarios where these equivalences were known before by other arguments-such as one-dimensional total variation, multidimensional generalizations to vector fields, or certain polyhedral semi-norms-yield spectral decompositions, and we provide further examples. We also investigate extinction times and extinction profiles, which we characterize as eigenvectors in a very general setting, generalizing several results from literature. (10.2140/apde.2021.14.823)
  DOI : 10.2140/apde.2021.14.823
• Large deviations in interacting particle systems : out of equilibrium correlations and interface dynamics
  
  • Dagallier Benoît
  , 2021. The objective of this thesis is the study of rare dynamical events in some interacting particle systems. Two models are considered : the one dimensional symmetric simple exclusion process interacting with reservoirs, and an interface dynamics related to the zero temperature Glauber dynamics for the two dimensional Ising model.In the case of the simple exclusion process, the work presented in the manuscript concerns the study of the out of equilibrium two-point correlation field. More precisely, the objective of the work is to estimate the probability of observing anomalous time-averaged two-point correlations, in the hydrodynamics scaling and the long time limit simultaneously. Studying two-point correlations at a suitable level of precision requires improving existing techniques. A refinement of the relative entropy method initially due to Yau provides a sufficient toolbox, thanks to which a large deviation principle for time-averaged two-point correlations is obtained.The interface dynamics aims at modelling the evolution of the interface separating a droplet of - Ising spins in a sea of + spins in the zero temperature Ising model. In the zero temperature Ising case, the boundary of this droplet has been shown to follow an anisotropic motion by curvature by Lacoin, Simenhaus and Toninelli a few years ago, rigorously establishing a long standing conjecture. In the manuscript, we aim to investigate the structure of atypical interface trajectories. To do so, another interface dynamics, called the contour dynamics, is introduced. Very similar to the zero temperature Ising dynamics, it differs by the presence of an additional parameter, which plays the role of a (small) temperature acting locally on the interface. In particular, Ising and contour dynamics coincide when this parameter vanishes. We show that the typical interface trajectory in the contour dynamics is still given by an anisotropic motion by curvature, with an influence of the temperature-like parameter. A large deviation principle is also established, characterising atypical trajectories as perturbations of the anisotropic motion by curvature, again with an influence of the temperature-like parameter.
• Generalized conditional gradient and learning in potential mean field games
  
  • Frédéric Bonnans J
  • Lavigne Pierre
  • Pfeiffer Laurent
  , 2021. We apply the generalized conditional gradient algorithm to potential mean field games and we show its well-posedeness. It turns out that this method can be interpreted as a learning method called fictitious play. More precisely, each step of the generalized conditional gradient method amounts to compute the best-response of the representative agent, for a predicted value of the coupling terms of the game. We show that for the learning sequence δk = 2/(k + 2), the potential cost converges in O(1/k), the exploitability and the variables of the problem (distribution, congestion, price, value function and control terms) converge in O(1/ √ k), for specific norms.
• Introduction to Spectral Methods for Uncertainty Quantification
  
  • Gori Giulio
  • Reis João
  • Congedo Pietro
  • Maître Olivier Le
  , 2021, pp.1-34. (10.1007/978-3-030-60166-9_1)
  DOI : 10.1007/978-3-030-60166-9_1
• An introduction to the topological derivative
  
  • Amstutz Samuel
  Engineering Computations, Emerald, 2021, 39 (1), pp.3-33. This paper provides a self-contained introduction to the mathematical aspects of the topological derivative. Full justifications are given on simple model problems following a modern approach. Methodological aspects and extensions are discussed, in relation with the literature on the field. (10.1108/ec-07-2021-0433)
  DOI : 10.1108/ec-07-2021-0433
• AVIS en réponse à la saisine HCB - dossier 2020-172. Paris, le 06 septembre 2021
  
  • Comité Scientifique Du Haut Conseil Des Biotechnologies .
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • de Verneuil Hubert
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • Vilotte Jean-Luc
  , 2021, pp.26 p.. Le Haut Conseil des biotechnologies (HCB) a été saisi le 22 juin 2021 par les autorités compétentes françaises (Ministère de l'Agriculture et de l'Alimentation) d'une demande d'avis relative au dossier EFSA-GMO-NL-2020-172 de demande d'autorisation de mise sur le marché du maïs génétiquement modifié DP915635 à des fins d'importation, transformation et alimentation humaine et animale. Ce dossier a été déposé par la société Pioneer Hi-Bred International, Inc. auprès des autorités compétentes néerlandaises sur le fondement du règlement (CE) n° 1829/2003. Dans le cadre de ce règlement, l'évaluation des dossiers de demande de mise sur le marché est confiée à l'Autorité européenne de sécurité des aliments (EFSA). Les Etats membres disposent de trois mois pour envoyer leurs commentaires à l'EFSA en contribution à l'évaluation du dossier. Dans ce contexte, le HCB est invité à proposer des commentaires à destination de l'EFSA au plus tard le 07 septembre 2021.
• An error estimate for a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions
  
  • Nabet Flore
  Numerische Mathematik, Springer Verlag, 2021, 149 (1), pp.185-226. In the paper we consider a finite-volume approximation for the Cahn-Hilliard equation with dynamic boundary conditions. We prove an error estimate for the fully-discrete scheme. The convergence of the scheme is proved in Nabet (IMA J Numer Anal 36(4): 1898–1942, 2016), we prove here an error estimate for the fully-discrete scheme. We also give numerical simulations which validate the theoretical result. (10.1007/s00211-021-01230-7)
  DOI : 10.1007/s00211-021-01230-7
• The saturn ring effect in nematic liquid crystals with external field: effective energy and hysteresis
  
  • Alouges François
  • Chambolle Antonin
  • Stantejsky Dominik
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2021, 241, pp.1403--1457. In this work we consider the Landau-de Gennes model for liquid crystals with an external electromagnetic field to model the occurrence of the saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous director field exists. Studying the limit problem, we explain the transition between the dipole and saturn ring configuration and the occurence of a hysteresis phenomenon, giving a rigorous explanation of what was conjectured previously by [H. Stark, Eur. Phys. J. B 10, 311–321 (1999)]. (10.1007/s00205-021-01674-z)
  DOI : 10.1007/s00205-021-01674-z
• Gaussian Graphical Model exploration and selection in high dimension low sample size setting
  
  • Lartigue Thomas
  • Bottani Simona
  • Baron Stephanie
  • Colliot Olivier
  • Durrleman Stanley
  • Allassonnière Stéphanie
  IEEE Transactions on Pattern Analysis and Machine Intelligence, Institute of Electrical and Electronics Engineers, 2021, 43 (9), pp.3196-3213. Gaussian graphical models (GGM) are often used to describe the conditional correlations between the components of a random vector. In this article, we compare two families of GGM inference methods: the nodewise approach and the penalised likelihood maximisation. We demonstrate on synthetic data that, when the sample size is small, the two methods produce graphs with either too few or too many edges when compared to the real one. As a result, we propose a composite procedure that explores a family of graphs with a nodewise numerical scheme and selects a candidate among them with an overall likelihood criterion. We demonstrate that, when the number of observations is small, this selection method yields graphs closer to the truth and corresponding to distributions with better KL divergence with regards to the real distribution than the other two. Finally, we show the interest of our algorithm on two concrete cases: first on brain imaging data, then on biological nephrology data. In both cases our results are more in line with current knowledge in each field. (10.1109/TPAMI.2020.2980542)
  DOI : 10.1109/TPAMI.2020.2980542
• A tangent linear approximation of the ignition delay time. II: Sensitivity to thermochemical parameters
  
  • Hantouche Mireille
  • Almohammadi Saja
  • Le Maitre Olivier
  • Knio Omar M
  Combustion and Flame, Elsevier, 2021, pp.111677. The tangent linear approximation (TLA) developed in Almohammadi et al. (Combust. Flame 230, 111426) is extended to estimate the sensitivity of the ignition delay time with respect to species enthalpies and entropies. The proposed method relies on integrating the linearized system of equations governing the evolution of the state vector's partial derivatives with respect to variations in thermodynamic parameters. The sensitivity of the ignition delay time is estimated through a linearized approximation of a temperature functional. The TLA approach is applied to three gas mixtures, H 2 , n-butanol, and iso-octane, reacting in air under adiabatic, constant-volume conditions. The numerical experiments indicate that the linearized approximation of the ignition delay time's sensitivity is in excellent agreement with the finite-difference estimates. This is also the case for sensitivity estimates obtained using the TLA approach. Further, significant computational speed-ups are achieved with the TLA approach, and the method scales well with the number of perturbed parameters. In the case of the H 2 mechanism, TLA is about ten times faster than finite differences, and this enhancement becomes even more substantial when more complex mechanisms are considered. (10.1016/j.combustflame.2021.111677)
  DOI : 10.1016/j.combustflame.2021.111677
• Scaling-invariant functions versus positively homogeneous functions
  
  • Touré Cheikh
  • Gissler Armand
  • Auger Anne
  • Hansen Nikolaus
  Journal of Optimization Theory and Applications, Springer Verlag, 2021, 191 (1), pp.363-383. Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are scaling-invariant with respect to zero. We prove in this paper that the reverse is true for large classes of scaling-invariant functions. Specifically, we give necessary and sufficient conditions for scaling-invariant functions to be composites of a strictly monotonic function with a positively homogeneous function. We also study sublevel sets of scaling-invariant functions generalizing well-known properties of positively homogeneous functions. (10.1007/s10957-021-01943-7)
  DOI : 10.1007/s10957-021-01943-7
• A geometric representation of fragmentation processes on stable trees
  
  • Thévenin Paul
  The Annals of Probability, Institute of Mathematical Statistics, 2021, 49 (5). (10.1214/21-AOP1512)
  DOI : 10.1214/21-AOP1512
• Quantification des incertitudes en gestion d'actifs : méthodes à noyaux et fluctuations statistiques
  
  • Chamakh Linda
  , 2021. The treatment of uncertainties is a fundamental problem in the financial context, and more precisely in portfolio optimisation. The variables studied are often time dependent, with heavy tails. In this thesis, we are interested in tools allowing to take into account uncertainties in its main forms: statistical uncertainties, parametric uncertainties and model error, keeping in mind that we wish to apply them to the financial context.The first part is devoted to the establishment of concentration inequalities for variables with heavy tailed distributions. The objective of these inequalities is to quantify the confidence that can be given to an estimator based on observations of finite size. In this thesis, we establish new concentration inequalities which include the case of estimators with log-normal distribution.In the second part, we discuss the impact of the model error for the estimation of the covariance matrix on stock returns, under the assumption that there is an instantaneous covariance process between the returns whose present value depends on its past values. One can then explicitly construct the best estimate of the covariance matrix for a given time and investment horizon, and we show that this estimate gives the best performance with high probability in the minimum variance portfolio framework.In the third part, we propose an approach to estimate the Sharpe ratio and the portfolio allocation when they depend on parameters considered uncertain. Our approach involves the adaptation of a stochastic approximation technique for the computation of the polynomial decomposition of the quantity of interest.Finally, in the last part of this thesis, we focus on portfolio optimization with target distribution. This technique can be formalised without the need for any model assumptions on returns. We propose to find these portfolios by minimizing divergence measures based on kernels or optimal transport. Since these divergence measures can be unbounded and have not been studied much yet in the unbounded kernel case, we establish new convergence guarantees based on concentration inequalities.
• Reproducing segregation and particle dynamics in Large Eddy Simulation of particle-laden flows
  
  • Letournel Roxane
  • Laurent Frédérique
  • Massot Marc
  • Vié Aymeric
  International Conference on Liquid Atomization and Spray Systems (ICLASS), 2021, 1 (1). Lagrangian simulations are today widely used for simulating aeronautical chambers. The way droplets are spatially distributed strongly affects the combustion, and accurate modeling and simulation strategies are required. The objective of the present contribution is to investigate how to correctly reproduce preferential concentration in Large Eddy Simulation (LES) of particleladen flows. Looking for a way to recover the DNS statistics, we highlight that stochastic models can fail in retrieving the tracer limit for non-inertial particles. We suggest a new strategy in the spirit of kinematic modeling of turbulence, which makes use of a random field with enforced divergence-free condition and spatial and temporal correlations. We show that the model can retrieve some Lagrangian statistics and in particular, particle segregation. We also suggest another approach for the LES particle model, based on retrieving not DNS statistics but filtered DNS statistics. We show that in this case, stochastic models can be relevant and appropriate. (10.2218/iclass.2021.5949)
  DOI : 10.2218/iclass.2021.5949
• Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation
  
  • Alzahrani Hasnaa H
  • Lucchesi Marco
  • Mustapha Kassem
  • Le Maitre Olivier
  • Knio Omar M
  Journal of Physics Communications, IOP Publishing, 2021, 5 (8), pp.085014. This work focuses on parameter calibration of a variable-diffusivity fractional diffusion model. A random, spatially-varying diffusivity field with log-normal distribution is considered. The variance and correlation length of the diffusivity field are considered uncertain parameters, and the order of the fractional subdiffusion operator is also taken uncertain and uniformly distributed in the range (0, 1). A Karhunen-Loève (KL) decomposition of the random diffusivity field is used, leading to a stochastic problem defined in terms of a finite number of canonical random variables. Polynomial chaos (PC) techniques are used to express the dependence of the stochastic solution on these random variables. A non-intrusive methodology is used, and a deterministic finite-difference solver of the fractional diffusion model is utilized for this purpose. The PC surrogates are first used to assess the sensitivity of quantities of interest (QoIs) to uncertain inputs and to examine their statistics. In particular, the analysis indicates that the fractional order has a dominant effect on the variance of the QoIs considered, followed by the leading KL modes. The PC surrogates are further exploited to calibrate the uncertain parameters using a Bayesian methodology. Different setups are considered, including distributed and localized forcing functions and data consisting of either noisy observations of the solution or its first moments. In the broad range of parameters addressed, the analysis shows that the uncertain parameters having a significant impact on the variance of the solution can be reliably inferred, even from limited observations. (10.1088/2399-6528/ac1507)
  DOI : 10.1088/2399-6528/ac1507
• Monte Carlo Variational Auto-Encoders
  
  • Thin Achille
  • Kotelevskii Nikita
  • Durmus Alain
  • Panov Maxim
  • Moulines Eric
  • Doucet Arnaud
  , 2021, 139. Variational auto-encoders (VAE) are popular deep latent variable models which are trained by maximizing an Evidence Lower Bound (ELBO). To obtain tighter ELBO and hence better variational approximations, it has been proposed to use importance sampling to get a lower variance estimate of the evidence. However, importance sampling is known to perform poorly in high dimensions. While it has been suggested many times in the literature to use more sophisticated algorithms such as Annealed Importance Sampling (AIS) and its Sequential Importance Sampling (SIS) extensions, the potential benefits brought by these advanced techniques have never been realized for VAE: the AIS estimate cannot be easily differentiated, while SIS requires the specification of carefully chosen backward Markov kernels. In this paper, we address both issues and demonstrate the performance of the resulting Monte Carlo VAEs on a variety of applications. (10.48550/arXiv.2106.15921)
  DOI : 10.48550/arXiv.2106.15921