Publications

2019

• A Surrogate-Assisted Multi-fidelity Measure Approximation Framework for Efficient Constrained Multiobjective Optimization Under Uncertainty
  
  • Rivier Mickael
  • Congedo Pietro Marco
  , 2019. The SABBa framework has been shown to tackle multi-objective optimization under uncertainty problems efficiently. It deals with robust and reliability-based optimization problems with approximated robustness and reliability measures. The recursive aspect of the Bounding-Box (BB) approach has notably been exploited in [1], [2] and [3] with an increasing number of additional features, allowing for little computational costs. In these contributions, robustness and reliability measures are approximated by a Bounding-Box (or conservative box), which is roughly a uniform-density representation of the unknown objectives and constraints. It is supplemented with a surrogate-assisting strategy, which is very effective to reduce the overall computational cost, notably during the last iterations of the optimization. In [3], SABBa has been quantitatively compared to more classical approaches with much success both concerning convergence rate and convergence robustness. We propose in this work to further improve the parsimony of the approach with a more general framework, SAMMA (Surrogate-Assisted Multi-fidelity Measure Approximation), allowing for objects other than Bounding-Boxes to be compared in the recursive strategy. Such non-uniform approximations have been proposed in some previous works like [4] and [5]. Among others, the empirical sampling and Gaussian measure approximations are presented and quantitatively compared in the following. We propose suitable Pareto dominance rules and POP (Pareto Optimal Probability) computations for these new measure approximations. To extend the framework applicability to complex industrial cases, and alongside the multi-fidelity between different UQs (Uncertainty Quantification) inherent to the recursive strategy, we propose to plug multi-fidelity approaches within the measure computations. This approach should allow tackling very complex industrial problems in an acceptable timeframe.
• On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates
  
  • Bourgeois Laurent
  • Chesnel Lucas
  , 2019. We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.
• Diffusions from Infinity
  
  • Bansaye Vincent
  • Collet Pierre
  • Martinez Servet
  • Méléard Sylvie
  • San Martin Jaime
  Transactions of the American Mathematical Society, American Mathematical Society, 2019, 372 (8), pp.5781-5823. In this paper we consider diffusions on the half line (0, ∞) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity, which takes finite values at positive times. We study the behaviour of hitting times of large barriers and in a dual way, the behaviour of the process starting at infinity for small time. In particular we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion. (10.1090/tran/7841)
  DOI : 10.1090/tran/7841
• Autour de l'algorithme du Langevin : extensions et applications
  
  • Brosse Nicolas
  , 2019. Cette thèse porte sur le problème de l'échantillonnage en grande dimension et est basée sur l'algorithme de Langevin non ajusté (ULA).Dans une première partie, nous proposons deux extensions d'ULA et fournissons des garanties de convergence précises pour ces algorithmes. ULA n'est pas applicable lorsque la distribution cible est à support compact; grâce à une régularisation de Moreau Yosida, il est néanmoins possible d'échantillonner à partir d'une distribution suffisamment proche de la distribution cible. ULA diverge lorsque les queues de la distribution cible sont trop fines; en renormalisant correctement le gradient, cette difficulté peut être surmontée.Dans une deuxième partie, nous donnons deux applications d'ULA. Nous fournissons un algorithme pour estimer les constantes de normalisation de densités log concaves à partir d'une suite de distributions dont la variance augmente graduellement. En comparant ULA avec la diffusion de Langevin, nous développons une nouvelle méthode de variables de contrôle basée sur la variance asymptotique de la diffusion de Langevin.Dans une troisième partie, nous analysons Stochastic Gradient Langevin Dynamics (SGLD), qui diffère de ULA seulement dans l'estimation stochastique du gradient. Nous montrons que SGLD, appliqué avec des paramètres habituels, peut être très éloigné de la distribution cible. Cependant, avec une technique appropriée de réduction de variance, son coût calcul peut être bien inférieur à celui d'ULA pour une précision similaire.
• Low-rank methods for heterogeneous and multi-source data
  
  • Robin Geneviève
  , 2019. In modern applications of statistics and machine learning, one often encounters many data imperfections. In particular, data are often heterogeneous, i.e. combine quantitative and qualitative information, incomplete, with missing values caused by machine failure or nonresponse phenomenons, and multi-source, when the data result from the compounding of diverse sources. In this dissertation, we develop several methods for the analysis of multi-source, heterogeneous and incomplete data. We provide a complete framework, and study all the aspects of the different methods, with thorough theoretical studies, open source implementations, and empirical evaluations. We study in details two particular applications from ecology and medical sciences.
• Deciphering the progression of PET alterations using surface-based spatiotemporal modeling
  
  • Koval Igor
  • Marcoux Arnaud
  • Burgos Ninon
  • Allassonnière Stéphanie
  • Colliot Olivier
  • Durrleman Stanley
  , 2019.
• MONK -- Outlier-Robust Mean Embedding Estimation by Median-of-Means
  
  • Lerasle Matthieu
  • Szabó Zoltán
  • Mathieu Timothée
  • Lecué Guillaume
  , 2019. Mean embeddings provide an extremely flexible and powerful tool in machine learning and statistics to represent probability distributions and define a semi-metric (MMD, maximum mean discrepancy; also called N-distance or energy distance), with numerous successful applications. The representation is constructed as the expectation of the feature map defined by a kernel. As a mean, its classical empirical estimator, however, can be arbitrary severely affected even by a single outlier in case of unbounded features. To the best of our knowledge, unfortunately even the consistency of the existing few techniques trying to alleviate this serious sensitivity bottleneck is unknown. In this paper, we show how the recently emerged principle of median-of-means can be used to design estimators for kernel mean embedding and MMD with excessive resistance properties to outliers, and optimal sub-Gaussian deviation bounds under mild assumptions.
• Nonatomic Aggregative Games with Infinitely Many Types
  
  • Jacquot Paulin
  • Wan Cheng
  , 2019. We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional varia-tional inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints.
• Imputation multiple pour données mixtes par analyse factorielle
  
  • Audigier Vincent
  • Husson François
  • Josse Julie
  • Resche-Rigon Matthieu
  , 2019. La prise en compte de données toujours plus nombreuses complexifie sans cesse leur analyse. Cette complexité se traduit notamment par des variables de types différents, la présence de données manquantes, et un grand nombre de variables et/ou d'observations. L'application de méthodes statistiques dans ce contexte est généralement délicate. L'objet de cette présentation est de proposer une nouvelle méthode d'imputation multiple basée sur l'analyse factorielle des données mixtes (AFDM). L'AFDM est une méthode d'analyse factorielle adaptée pour des jeux de données comportant des variables quantita-tives et qualitatives, dont le nombre peut excéder, ou non, le nombre d'observations. En vertu de ses propriétés, le développement d'une méthode d'imputation multiple basée sur l'AFDM permet l'inférence sur des variables quantitatives et qualitatives incomplètes, en grande et petite dimension. La méthode d'imputation multiple proposée utilise une approche bootstrap pour refléter l'incertitude sur les composantes principales et vecteurs propres de l'AFDM, utilisés ici pour prédire (imputer) les données. Chaque réplication bootstrap fournit alors une prédiction pour l'ensemble des données incomplètes du jeu de données. Ces prédictions sont ensuite bruitées pour refléter la distribution des données. On obtient ainsi autant de tableaux imputés que de réplications bootstrap. Après avoir rappelé les principes de l'imputation multiple, nous présenterons notre méthodologie. La méthode proposée seraévaluée par simulation et comparée aux méthodes de références : imputation séquentielle par modèle linéaire généralisé, imputation par modèle de mélanges et par "general location model". La méthode proposée permet d'ob-tenir des estimations ponctuelles sans biais de différents paramètres d'intérêt ainsi que des intervalles de confiance au taux de recouvrement attendu. De plus, elle peut s'appliquer 1 sur des jeux de données de nature variée et de dimensions variées, permettant notamment de traiter les cas où le nombre d'observations est plus petit que le nombre de variables.
• Modèles de classification non supervisée avec données manquantes non au hasard
  
  • Laporte Fabien
  • Biernacki Christophe
  • Celeux Gilles
  • Josse Julie
  , 2019. La difficulté de prise en compte des données manquantes est souvent con-tournée en supposant que leur occurrence est due au hasard. Dans cette communication, nous envisageons que l'absence de certaines données n'est pas due au hasard dans le contexte de la classification non supervisée et nous proposons des modèles logistiques pour traduire le fait que cette occurrence peutêtre associéeà la classification cherchée. Nous privilégions différents modèles que nous estimons par le maximum de vraisemblance et nous analysons leurs caractéristiques au travers de leur application sur des données hospitalières.
• Condensation in critical Cauchy Bienaymé–Galton–Watson trees
  
  • Kortchemski Igor
  • Richier Loïc
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2019, 29 (3), pp.1837-1877. (10.1214/18-AAP1447)
  DOI : 10.1214/18-AAP1447
• Application of the interacting particle system method to piecewise deterministic Markov processes used in reliability
  
  • Chraïbi Hassane
  • Dutfoy Anne
  • Galtier Thomas Antoine
  • Garnier Josselin
  Chaos: An Interdisciplinary Journal of Nonlinear Science, American Institute of Physics, 2019, 29 (6), pp.063119. (10.1063/1.5081446)
  DOI : 10.1063/1.5081446
• A DENSITY RESULT IN $GSBD^p$ WITH APPLICATIONS TO THE APPROXIMATION OF BRITTLE FRACTURE ENERGIES
  
  • Chambolle Antonin
  • Crismale Vito
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2019, 232 (3), pp.1329--1378. We prove that any function in $GSBD^p(\Omega)$, with $\Omega$ a $n$-dimensional open bounded set with finite perimeter, is approximated by functions $u_k\in SBV(\Omega;\mathbb{R}^n)\cap L^\infty(\Omega;\mathbb{R}^n)$ whose jump is a finite union of $C^1$ hypersurfaces. The approximation takes place in the sense of Griffith-type energies $\int_\Omega W(e(u)) dx +\mathcal{H}^{n-1}(J_u)$, $e(u)$ and $J_u$ being the approximate symmetric gradient and the jump set of $u$, and $W$ a nonnegative function with $p$-growth, $p>1$. The difference between $u_k$ and $u$ is small in $L^p$ outside a sequence of sets $E_k\subset \Omega$ whose measure tends to 0 and if $|u|^r \in L^1(\Omega)$ with $r\in (0,p]$, then $|u_k-u|^r \to 0$ in $L^1(\Omega)$. Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce $\Gamma$-convergence approximation <i>à la</i> Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are <i>a priori</i> not even in $L^1(\Omega;\mathbb{R}^n)$.
• Model-Uncertain Value-at-Risk, Expected Shortfall and Sharpe Ratio, Using Stochastic Approximation
  
  • Crépey Stéphane
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  , 2019.
• Uniqueness of the nonlinear Schrödinger equation driven by jump processes
  
  • de Bouard Anne
  • Hausenblas Erika
  • Ondrejat Martin
  Nonlinear Differential Equations and Applications, Springer Verlag, 2019, 26 (3). (10.1007/s00030-019-0569-3)
  DOI : 10.1007/s00030-019-0569-3
• Chaos and order in the bitcoin market
  
  • Garnier Josselin
  • Solna Knut
  Physica A: Statistical Mechanics and its Applications, Elsevier, 2019, 524, pp.708-721. (10.1016/j.physa.2019.04.164)
  DOI : 10.1016/j.physa.2019.04.164
• Quantitative bounds for concentration-of-measure inequalities and empirical regression: the independent case
  
  • Barrera David
  • Gobet Emmanuel
  Journal of Complexity, Elsevier, 2019, 52, pp.45-81. This paper is devoted to the study of the deviation of the (random) average $L^{2}-$error associated to the least--squares regressor over a family of functions ${\cal F}_{n}$ (with controlled complexity) obtained from $n$ independent, but not necessarily identically distributed, samples of explanatory and response variables, from the minimal (deterministic) average $L^{2}-$error associated to this family of functions, and to some of the corresponding consequences for the problem of consistency. In the i.i.d. case, this specializes as classical questions on least--squares regression problems, but in more general cases, this setting permits a precise investigation in the direction of the study of nonasymptotic errors for least--squares regression schemes in nonstationary settings, which we motivate providing background and examples. More precisely, we prove first two nonasymptotic deviation inequalities that generalize and refine corresponding known results in the i.i.d. case. We then explore some consequences for nonasymptotic bounds of the error both in the weak and the strong senses. Finally, we exploit these estimates to shed new light into questions of consistency for least--squares regression schemes in the distribution--free, nonparametric setting. As an application to the classical theory, we provide in particular a result that generalizes the link between the problem of consistency and the Glivenko-Cantelli property, which applied to regression in the i.i.d. setting over non--decreasing families $({\cal F}_{n})_{n}$ of functions permits to create a scheme which is strongly consistent in $L^{2}$ under the sole (necessary) assumption of the existence of functions in $\cup_{n}{\cal F}_{n}$ which are arbitrarily close in $L^{2}$ to the corresponding regressor. (10.1016/j.jco.2019.01.003)
  DOI : 10.1016/j.jco.2019.01.003
• Examples of solving the inverse scattering problem and the equations of the Veselov-Novikov hierarchy from the scattering data of point potentials
  
  • Agaltsov Alexey
  • Novikov Roman
  Russian Mathematical Surveys, Turpion, 2019, 74 (3), pp.373-386. We consider the inverse scattering problem for the two-dimensional Schrödinger equation at fixed positive energy. Our results include inverse scattering reconstructions from the simplest scattering amplitudes. In particular, we give a complete analytic solution of the phased and phaseless inverse scattering problems for the single-point potentials (of the Bethe-Peierls-Fermi-Zeldovich-Berezin-Faddeev type). Then we study numerical inverse scattering reconstructions from the simplest scattering amplitudes using the Riemann-Hilbert-Manakov problem of the soliton theory. Finally, we apply the later numerical inverse scattering results for constructing related numerical solutions for equations of the Novikov-Veselov hierarchy at fixed positive energy . (10.1070/RM9867)
  DOI : 10.1070/RM9867
• Tutorial on "Monte-Carlo methods for tail risks
  
  • Gobet Emmanuel
  , 2019.
• Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
  
  • Bonazzoli Marcella
  • Dolean Victorita
  • Graham Ivan G.
  • Spence Euan A.
  • Tournier Pierre-Henri
  Mathematics of Computation, American Mathematical Society, 2019, 88, pp.2559-2604. This paper rigorously analyses preconditioners for the time-harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using Additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping Additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally -- in the sense that GMRES converges in a wavenumber-independent number of iterations -- for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors. (10.1090/mcom/3447)
  DOI : 10.1090/mcom/3447
• A five equation model for the simulation of the two-phase flow in cryogenic coaxial injector
  
  • Murrone Angelo
  • Boucher Aymeric
  • Cordesse Pierre
  , 2019. In this paper, we propose models and methods for the simulation of two-phase flows in Liquid Rocket Engines (LRE) under subcritical conditions. The numerical strategy consists into coupling models dedicated to different topologies. Actually, we propose a five equation diffuse interface model for the treatment of the dense "separated two-phase flow" near the injector and an Eulerian kinetic based model for the "dispersed two-phase flow" in the chamber. We derive a novel formulation of the 5 equation system to build a robust HLLC type scheme. Then we use a fully Eulerian coupling strategy to take into account for primary atomization. We first run classical test cases in order to validate the numerical methods. Then a simulation on a test case representative to one coaxial injector is performed under subcritical conditions.
• Laser Path Optimization For Additive Manufacturing
  
  • Boissier M
  • Allaire G.
  • Tournier Christophe
  , 2019. Additive Manufacturing (AM) through a Laser Powder Bed Fusion (LPBF) process consists in building objects layer by layer, by deposing energy in metallic powder with a laser along a chosen path. Despite huge advantages, such as the freedom in the design, this manufacturing process causes defects in the resulting object. Choosing the laser path is of high significance, since it is tightly related to both the manufacturing speed and the temperature distribution in each layer. The design of the laser path is often based on optimizing parameters of existing patterns and combining them. A different approach is proposed here. Without presupposing any specific pattern, our method is based on shape optimization theory and a descent algorithm is drafted to adapt the path to the manufacturing requirements.
• Post-processing of two-phase DNS simulations exploiting geometrical features and topological invariants to extract flow statistics: application to canonical objects and the collision of two droplets
  
  • Di Battista Ruben
  • Bermejo-Moreno Iván
  • Ménard Thibaut
  • de Chaisemartin Stéphane
  • Massot Marc
  , 2019. This work presents a methodology to collect useful flow statistics over DNS simulations exploiting geometrical properties maps and topological invariants. The procedure is based on estimating curvatures on triangulated surfaces as as averaged values around a given point and its first neighbours (the 1-ring of such a point). In the case of two-phase flow high-fidelity simulations, the surfaces are obtained after an iso-contouring procedure of the volumetric level-set field. The estimation of the curvatures on the surface allows the possibility of characterizing the 3D objects that are created in a high-fidelity simulation in terms of their area-weighted geometrical maps. In this work we provide an assessment of the robustness of the curvature estimation algorithm applied to some canonical 3D objects and to the Direct Numerical Simulation of the collision of two droplets. We provide the tracking of the topological evolution of such objects in terms of geometrical maps and we highlight the effect of mesh resolution on those topological changes.
• Validation strategy of reduced-order two-fluid flow models based on a hierarchy of direct numerical simulations
  
  • Cordesse Pierre
  • Murrone Angelo
  • Ménard Thibaut
  • Massot Marc
  , 2019. In industrial applications, the use of reduced-order models to conduct numerical simulations on realistic configurations as a predictive tool strengthen the need of assessing them. In the context of cryogenic atomization, we propose to build a validation strategy of large scale two fluid models with subscale modelling based on a hierarchy of direct numerical simulation test cases to qualitatively and quantitatively assess these models. In the present work, we propose a validation of these reduced-order model relying on DNS on an hierarchy of specific test cases. We propose in this work to investigate an air-assisted water atomization using a planar injector. This test case offers an atomization regime, which makes it worthy to eventually validate our reduced-order models on a cryogenic coaxial injection.
• Impact of particle field heterogeneity on the dynamics of turbulent two-way coupled particulate flows
  
  • Letournel Roxane
  • Laurent Frédérique
  • Massot Marc
  • Vié A.
  , 2019. A series of coupled direct numerical simulations of decaying Homogeneous Isotropic Turbulence are performed on a 128 3 perodic box, with a size monodisperse population of particles at different Stokes numbers St, mass loading φ and particle number densities n 0 (mean number of particles per unit volume). Indeed, mechanisms for turbulence enhancement or suppression depend on these three combined parameters. Like previous investigators, crossover and modulations of the fluid energy spectra were observed consistently with the change in Stokes number and mass loading. Additionally, we also investigate the impact of the particle number density, which was not properly and independently characterized. This parameter plays a key role in the statistical convergence of the disperse phase and is therefore of primary importance for modeling purposes. DNS results show a clear impact of this parameter, promoting the energy at small scales while reducing the energy at large scales. Our investigation focuses on the identification of the energy transfer mechanisms, to highlight the differences between the influence of a homogeneous disperse phase (very high particle number density where statistical convergence is obtained) and heterogeneous cases (low particle number density). In particular, different regimes have been identified and described in terms of particle number density.