Publications

2020

• Machine Learning and Big Data for outlier detection, and applications
  
  • Virouleau Alain
  , 2020. The problems of outliers detection and robust regression in a high-dimensional setting are fundamental in statistics, and have numerous applications.Following a recent set of works providing methods for simultaneous robust regression and outliers detection,we consider in a first part a model of linear regression with individual intercepts, in a high-dimensional setting.We introduce a new procedure for simultaneous estimation of the linear regression coefficients and intercepts, using two dedicated sorted-l1 convex penalizations, also called SLOPE.We develop a complete theory for this problem: first, we provide sharp upper bounds on the statistical estimation error of both the vector of individual intercepts and regression coefficients.Second, we give an asymptotic control on the False Discovery Rate (FDR) and statistical power for support selection of the individual intercepts.Numerical illustrations, with a comparison to recent alternative approaches, are provided on both simulated and several real-world datasets.Our second part is motivated by a genetic problem. Among some particular DNA sequences called multi-satellites, which are indicators of the development or colorectal cancer tumors, we want to find the sequences that have a much higher (resp. much lower) rate of mutation than expected by biologist experts. This problem leads to a non-linear probabilistic model and thus goes beyond the scope of the first part. In this second part we thus consider some generalized linear models with individual intercepts added to the linear predictor, and explore the statistical properties of a new procedure for simultaneous estimation of the regression coefficients and intercepts, using again the sorted-l1 penalization. We focus in this part only on the low-dimensional case and are again interested in the performance of our procedure in terms of statistical estimation error and FDR.
• Quasi-Stationary Distributions and Resilience: What to get from a sample?
  
  • Chazottes Jean-René
  • Collet Pierre
  • Méléard Sylvie
  • Martínez Servet
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2020. We study a class of multi-species birth-and-death processes going almost surely to extinction and admitting a unique quasi-stationary distribution (qsd for short). When rescaled by K and in the limit K ! +1, the realizations of such processes get close, in any fixed finite-time window, to the trajectories of a dynamical system whose vector field is defined by the birth and death rates. Assuming that this dynamical has a unique attracting fixed point, we analyzed in a previous work what happens for large but finite K, especially the different time scales showing up. In the present work, we are mainly interested in the following question: Observing a realization of the process, can we determine the so-called engineering resilience? To answer this question, we establish two relations which intermingle the resilience, which is a macroscopic quantity defined for the dynamical system, and the fluctuations of the process, which are microscopic quantities. Analogous relations are well known in nonequilibrium statistical mechanics. To exploit these relations, we need to introduce several estimators which we control for times between log K (time scale to converge to the qsd) and $\exp(K)$ (time scale of mean time to extinction).
• Jeux différentiels stochastiques non-Markoviens etdynamiques de Langevin à champ-moyen
  
  • Hu Kaitong
  , 2020. Cette thèse se compose de deux parties indépendantes et la première regroupant deux problématiques distinctes. Dans la première partie, nous étudions d’abord le problème de Principal-Agent dans des systèmes dégénérés, qui apparaissent naturellement dans des environnements à l’observation partielle où l’Agent et le Principal n’observent qu’une partie du système. Nous présentons une approche se basant sur le principe du maximum stochastique, dont le but est d’étendre les travaux existants qui utilisent le principe de la programmation dynamique dans des systèmes non-dégénérés. D’abord nous résolvons le problème du Principal dans un ensembledes contrats élargi donné par la condition du premier ordre du problème de l’Agent sous forme d’une équation différentielle stochastique progressive-rétrograde (abrégée EDSPR) dépendante de la trajectoire. Ensuite nous utilisons la condition suffisante du problème de l’Agent pour vérifier que le contrat optimal obtenu est bien implémentable. Une étude parallèle est consacrée à l’existence et l’unicité de la solution d'EDSPRs dépendantes de la trajectoire dans le chapitre IV. Nous étendons la méthode de champ de découplage aux cas où les coefficients des équations peuvent dépendre de la trajectoire du processus forward. Nous démontrons également une propriété de stabilité pour ce genre d'EDSPRs. Enfin, nous étudions le problème de hasard moral avec plusieurs Principals. L’Agent ne peut travailler que pour un seul Principal à la fois et fait donc face à un problème de switching optimal. En utilisant la méthode de randomisation nous montrons que la fonction valeur de l’Agent et son effort optimal sont donnés par un processus d’Itô. Cette représentation nous aide à résoudre ensuite le problème du Principal lorsqu’il y a une infinité de Principals en équilibre selon un jeu à champ-moyen. Nous justifions la formulation à champ-moyen par un argument de propagation de chaos.La deuxième partie de cette thèse est constituée des chapitres V et VI. La motivation de ces travaux est de donner un fondement théorique rigoureux pour la convergence des algorithmes du type descente de gradient très souvent utilisés dans la résolution des problème non-convexes comme la calibration d’un réseau de neurones. Pour les problèmes non-convexes du type réseaux de neurones à une couche cachée, l’idée clé est de transformer le problème en un problème convexe en le relevant dans l’espace des mesures. Nous montrons que la fonction d’énergie correspondante admet un unique minimiseur qui peut être caractérisé par une condition du premier ordre utilisant la dérivation dans l’espace des mesures au sens de Lions. Nous présentons ensuite une analyse du comportement à long terme de la dynamique de Langevin à champ-moyen, qui possède une structure de flot de gradient dans la métrique de 2-Wasserstein. Nous montrons que le flot de la loi marginale induite par la dynamique de Langevin à champ-moyen converge vers une loi stationnaire en utilisant le principe d’invariance de La Salle, qui est le minimiseur de la fonction d’énergie.Dans le cas des réseaux de neurones profonds, nous les modélisons à l’aide d’un problème de contrôle optimal en temps continu. Nous donnons d’abord la conditiondu premier ordre à l’aide du principe de Pontryagin, qui nous aidera ensuiteà introduire le système d’équation de Langevin à champ-moyen, dont la mesure invariante correspond au minimiseur du problème de contrôle optimal. Enfin, avec la méthode de couplage par réflexion nous montrons que la loi marginale du système de Langevin à champ-moyen converge vers la mesure invariante avec une vitesse exponentielle.
• Assessment of robust optimization for design of rotorcraft airfoils in forward flight
  
  • Fusi Francesca
  • Congedo Pietro Marco
  • Guardone Alberto
  • Quaranta Giuseppe
  Aerospace Science and Technology, Elsevier, 2020, 107, pp.106355. The paper compares the deterministic and robust approaches to improve the aerodynamic design of helicopter airfoils. The two formulations are different due to the characteristics of each approach. In the deterministic case, the objective of optimization is the minimization of drag while maintaining a level of lift that guarantees satisfaction of trimming condition. In the case of robust design, a range of angles of attack and not a single trim condition is considered. Thus, the robust optimization takes the lift-to-drag ratio as a measure of the performance of the airfoil, imposing at the same time inequality constraint on the lift coefficient to guarantee a sufficient level of lift, and then checking after optimization that the trimming condition can be satisfied. The two approaches are compared showing pros and cons of the robust framework. In general, the robust approach shows the capability to reach the same mean performance of the deterministic one, but with a lower degradation of performances in off-design situations considered through the uncertainty. On the other side, the difficulties in imposing the lift trim condition for the robust formulation may lead to results of limited use. (10.1016/j.ast.2020.106355)
  DOI : 10.1016/j.ast.2020.106355
• Machine learning surrogate models for prediction of point defect vibrational entropy
  
  • Lapointe Clovis
  • Swinburne T D
  • Thiry Louis
  • Mallat Stéphane
  • Proville Laurent
  • Becquart Charlotte
  • Marinica Mihai-Cosmin
  Physical Review Materials, American Physical Society, 2020, 4 (6). The temperature variation of the defect densities in a crystal depends on vibrational entropy. This contribution to the system thermodynamics remains computationally challenging as it requires a diagonalisation of the system's Hessian which scales as O(N 3) for a crystal made of N atoms. Here, to circumvent such an heavy computational task and make it feasible even for systems containing millions of atoms the harmonic vibrational entropy of point defects is estimated directly from the relaxed atomic positions through a linear-in-descriptor machine learning approach of order O(N). With a size-independent descriptor dimension and fixed model parameters, an excellent predictive power is demonstrated on a wide range of defect configurations, supercell sizes and external deformations well outside of the training database. In particular, formation entropies in a range of 250 kB are predicted with less than 1.6 kB error from a training database whose formation entropies span only 25 kB (train error less than 1.0 kB). This exceptional transferability is found to hold even when the training is limited to a low energy superbasin in the phase space while the tests are performed for a different liquid-like superbasin at higher energies. (10.1103/PhysRevMaterials.4.063802)
  DOI : 10.1103/PhysRevMaterials.4.063802
• Energy stable discretization for two-phase porous media flows
  
  • Cancès Clément
  • Nabet Flore
  , 2020. We propose a P1 finite-element scheme with mass-lumping for a model of two incompressible and immiscible phases in a porous media flow. We prove the dissipation of the free energy and the existence of a solution to the nonlinear scheme. We also present numerical simulations to illustrate the behavior of the scheme.
• Option valuation and hedging using asymmetric risk function: asymptotic optimality through fully nonlinear Partial Differential Equations
  
  • Gobet Emmanuel
  • Pimentel Isaque
  • Warin Xavier
  Finance and Stochastics, Springer Verlag (Germany), 2020, 24 (3), pp.633-675. Discrete time hedging produces a residual risk, namely, the tracking error. The major problem is to get valuation/hedging policies minimizing this error. We evaluate the risk between trading dates through a function penalizing asymmetrically profits and losses. After deriving the asymptotics within a discrete time risk measurement for a large number of trading dates, we derive the optimal strategies minimizing the asymptotic risk in the continuous time setting. We characterize the optimality through a class of fully nonlinear Partial Differential Equations (PDE). Numerical experiments show that the optimal strategies associated with discrete and asymptotic approach coincides asymptotically. (10.1007/s00780-020-00428-1)
  DOI : 10.1007/s00780-020-00428-1
• Procédé de gestion décentralisée de consommation électrique non-intrusif
  
  • Jacquot Paulin
  • Oudjane Nadia
  • Beaude Olivier
  • Benchimol Pascal
  • Gaubert Stéphane
  , 2020. La présente invention concerne un procédé de gestion décentralisée d’au moins une partie de consommation électrique par des consommateurs raccordés à un réseau de distribution électrique, caractérisé par le fait qu’il comprend les étapes consistant à : i) établir au niveau d’un coordinateur central un profil optimal agrégé pour l’ensemble des consommateurs considérés avec une estimation de contraintes agrégées, ii) transmettre le profil optimal agrégé aux consommateurs par le coordinateur central, iii) analyser localement par projection le profil optimal agrégé, iv) transmettre par chaque consommateur au coordinateur central le résultat de la projection établie à l’étape iii), v) réitérer l’étape i), pour déterminer un nouveau profil optimal agrégé sur la base d’une nouvelle contrainte si le coordinateur central détermine que la désagrégation est impossible sur la base du profil optimal agrégé antérieur, puis réitérer les étapes ii) à v) jusqu’à convergence pour l’ensemble des consommateurs
• Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches
  
  • Calvez Vincent
  • Figueroa Iglesias Susely
  • Hivert Hélène
  • Méléard Sylvie
  • Melnykova Anna
  • Nordmann Samuel
  ESAIM: Proceedings and Surveys, EDP Sciences, 2020, 67, pp.135-160. Horizontal gene Transfer (HT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integrod-ifferential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal of this paper is to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular. (10.1051/proc/202067009)
  DOI : 10.1051/proc/202067009
• On the Convergence Rate of some Nonlocal Energies
  
  • Chambolle Antonin
  • Novaga Matteo
  • Pagliari Valerio
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2020, 200 (112016), pp.23pp. We study the rate of convergence of some nonlocal functionals recently considered by Bourgain, Brezis and Mironescu. In particular, we establish the Γ-convergence of the corresponding rate functionals, suitably rescaled, to a limit functional of second order. (10.1016/j.na.2020.112016)
  DOI : 10.1016/j.na.2020.112016
• Contributions à l'apprentissage statistique : estimation de densité, agrégation d'experts et forêts aléatoires
  
  • Mourtada Jaouad
  , 2020. L’apprentissage statistique fournit un cadre aux problèmes de prédiction, où l’on cherche à prédire des quantités inconnues à partir d’exemples.La première partie de cette thèse porte sur les méthodes de Forêts aléatoires, une famille d'algorithmes couramment utilisés en pratique, mais dont l'étude théorique s'avère délicate. Notre principale contribution est l'analyse précise d'une variante stylisée, les forêts de Mondrian, pour lesquelles nous établissons des vitesses de convergence non paramétriques minimax ainsi qu'un avantage des forêts sur les arbres. Nous étudions également une variante "en ligne" des forêts de Mondrian.La seconde partie est dédiée à l'agrégation d'experts, où il s'agit de combiner plusieurs sources de prédictions (experts) afin de prédire aussi bien que la meilleure d'entre elles. Nous analysons l'algorithme classique d'agrégation à poids exponentiels dans le cas stochastique, où il exhibe une certaine adaptativité à la difficulté du problème. Nous étudions également une variante du problème avec une classe croissante d'experts.La troisième partie porte sur des problèmes de régression et d'estimation de densité. Notre première contribution principale est une analyse minimax détaillée de la prédiction linéaire avec design aléatoire, en fonction de la loi des variables prédictives; nos bornes supérieures reposent sur un contrôle de la queue inférieure de matrices de covariance empiriques. Notre seconde contribution principale est l'introduction d'une procédure générale pour l'estimation de densité avec perte logarithmique, qui admet des bornes optimales d'excès de risque ne se dégradant pas dans le cas mal spécifié. Dans le cas de la régression logistique, cette procédure admet une forme simple et atteint des vitesses de convergence rapides inaccessibles aux estimateurs de type plug-in.
• MUMFORD -SHAH FUNCTIONALS ON GRAPHS AND THEIR ASYMPTOTICS
  
  • Caroccia Marco
  • Chambolle Antonin
  • Slepčev Dejan
  Nonlinearity, IOP Publishing, 2020, 33 (8), pp.3846--3888. We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the graph constructions under which the minimizers of graph Mumford-Shah functionals converge to a minimizer of a continuum Mumford-Shah functional. Furthermore we explicitly identify the limiting functional. Moreover we describe an efficient algorithm for computing the approximate minimizers of the graph Mumford-Shah functional. (10.1088/1361-6544/ab81ee)
  DOI : 10.1088/1361-6544/ab81ee
• Error estimates for phaseless inverse scattering in the Born approximation at high energies
  
  • Agaltsov Alexey
  • Novikov Roman
  The Journal of Geometric Analysis, Springer, 2020, 30 (3), pp.2340-2360. We study explicit formulas for phaseless inverse scattering in the Born approximation at high energies for the Schrödinger equation with compactly supported potential in dimension d ≥ 2. We obtain error estimates for these formulas in the configuration space.
• Microstructural organization of human insula is linked to its macrofunctional circuitry and predicts cognitive control
  
  • Menon Vinod
  • Gallardo Guillermo
  • Pinsk Mark
  • Nguyen Van-Dang
  • Li Jing-Rebecca
  • Cai Weidong
  • Wassermann Demian
  eLife, eLife Sciences Publication, 2020, 9. The human insular cortex is a heterogeneous brain structure which plays an integrative role in guiding behavior. The cytoarchitectonic organization of the human insula has been investigated over the last century using postmortem brains but there has been little progress in noninvasive in vivo mapping of its microstructure and large-scale functional circuitry. Quantitative modeling of multi-shell diffusion MRI data from 413 participants revealed that human insula microstructure differs significantly across subdivisions that serve distinct cognitive and affective functions. Insular microstructural organization was mirrored in its functionally interconnected circuits with the anterior cingulate cortex that anchors the salience network, a system important for adaptive switching of cognitive control systems. Furthermore, insular microstructural features, confirmed in Macaca mulatta, were linked to behavior and predicted individual differences in cognitive control ability. Our findings open new possibilities for probing psychiatric and neurological disorders impacted by insular cortex dysfunction, including autism, schizophrenia, and fronto-temporal dementia. (10.7554/eLife.53470)
  DOI : 10.7554/eLife.53470
• KORN AND POINCARÉ-KORN INEQUALITIES FOR FUNCTIONS WITH SMALL JUMP SET
  
  • Cagnetti Filippo
  • Chambolle Antonin
  • Scardia Lucia
  , 2020. In this paper we prove a regularity and rigidity result for displacements in GSBD^p , for every p > 1 and any dimension n ≥ 2. We show that a displacement in GSBD^p with a small jump set coincides with a W^{1,p} function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in GSBD^p.
• Learning the clustering of longitudinal shape data sets into a mixture of independent or branching trajectories
  
  • Debavelaere Vianney
  • Durrleman Stanley
  • Allassonnière Stéphanie
  International Journal of Computer Vision, Springer Verlag, 2020. Given repeated observations of several subjects over time, i.e. a longitudinal data set, this paper introduces a new model to learn a classification of the shapes progression in an unsupervised setting: we automatically cluster a longitudinal data set in different classes without labels. Our method learns for each cluster an average shape trajectory (or representative curve) and its variance in space and time. Representative trajectories are built as the combination of pieces of curves. This mixture model is flexible enough to handle independent trajectories for each cluster as well as fork and merge scenarios. The estimation of such non linear mixture models in high dimension is known to be difficult because of the trapping states effect that hampers the optimisation of cluster assignments during training. We address this issue by using a tempered version of the stochastic EM algorithm. Finally, we apply our algorithm on different data sets. First, synthetic data are used to show that a tempered scheme achieves better convergence. We then apply our method to different real data sets: 1D RECIST score used to monitor tumors growth, 3D facial expressions and meshes of the hippocampus. In particular, we show how the method can be used to test different scenarios of hip-pocampus atrophy in ageing by using an heteregenous population of normal ageing individuals and mild cog-nitive impaired subjects. (10.1007/s11263-020-01337-8)
  DOI : 10.1007/s11263-020-01337-8
• Quantile-based robust optimization of a supersonic nozzle for organic rankine cycle turbines
  
  • Razaaly Nassim
  • Persico Giacomo
  • Gori Giulio
  • Congedo Pietro Marco
  Applied Mathematical Modelling, Elsevier, 2020, 82, pp.802-824. Organic Rankine Cycle (ORC) turbines usually operate in thermodynamic regions characterized by high-pressure ratios and strong non-ideal gas effects, complicating the aerodynamic design significantly. Systematic optimization methods accounting for multiple uncertainties due to variable operating conditions, referred to as Robust Optimization may benefit to ORC turbines aerodynamic design. This study presents an original and fast robust shape optimization approach to overcome the limitation of a deterministic optimization that neglects operating conditions variability, applied to a well-known supersonic turbine nozzle for ORC applications. The flow around the blade is assumed inviscid and adiabatic and it is reconstructed using the open-source SU2 code. The non-ideal gasdynamics is modeled through the Peng-Robinson-Stryjek-Vera equation of state. We propose here a mono-objective formulation which consists in minimizing the α-quantile of the targeted Quantity of Interest (QoI) under a probabilistic constraint, at a low computational cost. This problem is solved by using an efficient robust optimization approach, coupling a stateof-the-art quantile estimation and a classical Bayesian optimization method. First, the advantages of a quantile-based formulation are illustrated with respect to a conventional mean-based robust optimization. Secondly, we demonstrate the effectiveness of applying this robust optimization framework with a low-fidelity inviscid solver by comparing the resulting optimal design with the ones obtained with a deterministic optimization using a fully turbulent solver. (10.1016/j.apm.2020.01.048)
  DOI : 10.1016/j.apm.2020.01.048
• Convergence of a finite-volume scheme for a heat equation with a multiplicative stochastic force
  
  • Bauzet Caroline
  • Nabet Flore
  , 2020. We present here the discretization by a finite-volume scheme of a heat equation perturbed by a multiplicative noise of Itô type and under homogeneous Neumann boundary conditions. The idea is to adapt well-known methods in the de-terministic case for the approximation of parabolic problems to our stochastic PDE. In this paper, we try to highlight difficulties brought by the stochastic perturbation in the adaptation of these deterministic tools. (10.1007/978-3-030-43651-3_24)
  DOI : 10.1007/978-3-030-43651-3_24
• Bridging the Gap between Constant Step Size Stochastic Gradient Descent and Markov Chains
  
  • Dieuleveut Aymeric
  • Durmus Alain
  • Bach Francis
  Annals of Statistics, Institute of Mathematical Statistics, 2020, 48 (3). We consider the minimization of an objective function given access to unbiased estimates of its gradient through stochastic gradient descent (SGD) with constant step-size. While the detailed analysis was only performed for quadratic functions, we provide an explicit asymptotic expansion of the moments of the averaged SGD iterates that outlines the dependence on initial conditions, the effect of noise and the step-size, as well as the lack of convergence in the general (non-quadratic) case. For this analysis, we bring tools from Markov chain theory into the analysis of stochastic gradient. We then show that Richardson-Romberg extrapolation may be used to get closer to the global optimum and we show empirical improvements of the new extrapolation scheme. (10.1214/19-AOS1850)
  DOI : 10.1214/19-AOS1850
• An electronic travel aid device to help blind people playing sport
  
  • Ferrand Sylvain
  • Alouges François
  • Aussal Matthieu
  IEEE Instrumentation and Measurement Magazine, Institute of Electrical and Electronics Engineers, 2020, 23 (4), pp.14-21. (10.1109/MIM.2020.9126047)
  DOI : 10.1109/MIM.2020.9126047
• Cheating in arbuscular mycorrhizal mutualism: a network and phylogenetic analysis of mycoheterotrophy
  
  • Perez‐lamarque Benoît
  • Selosse Marc‐andré
  • Öpik Maarja
  • Morlon Hélène
  • Martos Florent
  New Phytologist, Wiley, 2020, 226 (6), pp.1822-1835. Although mutualistic interactions are widespread and essential in ecosystem functioning, the emergence of uncooperative cheaters threatens their stability, unless there are some physiological or ecological mechanisms limiting interactions with cheaters. In this framework, we investigated the patterns of specialization and phylogenetic distribution of mycoheterotrophic cheaters vs noncheating autotrophic plants and their respective fungi, in a global arbuscular mycorrhizal network with> 25 000 interactions. We show that mycoheterotrophy evolved repeatedly among vascular plants, suggesting low phylogenetic constraints for plants. However, mycoheterotrophic plants are significantly more specialized than autotrophic plants, and they tend to be associated with specialized and closely related fungi. These results raise new hypotheses about the mechanisms (e.g. sanctions, or habitat filtering) that actually limit the interaction of mycoheterotrophic plants and their associated fungi with the rest of the autotrophic plants. Beyond mycorrhizal symbiosis, this unprecedented comparison of mycoheterotrophic vs autotrophic plants provides a network and phylogenetic framework to assess the presence of constraints upon cheating emergences in mutualisms. (10.1111/nph.16474)
  DOI : 10.1111/nph.16474
• Optimal assignments with supervisions
  
  • Niv Adi
  • Maccaig Marie
  • Sergeev Sergeĭ
  Linear Algebra and its Applications, Elsevier, 2020, 595, pp.72-100. (10.1016/j.laa.2020.02.032)
  DOI : 10.1016/j.laa.2020.02.032
• Bayesian inference of thermodynamic models from vapor flow experiments
  
  • Gori Giulio
  • Zocca Marta
  • Guardone Alberto
  • Le Maitre Olivier
  • Congedo Pietro Marco
  Computers and Fluids, Elsevier, 2020, 205, pp.104550. The present work concerns the inference of the coefficients of fluid-dependent thermodynamic models, applicable to complex molecular compounds with non-ideal effects. The main objective is to numerically assess the potential of using experimental measurements of some expansion flows to infer the model parameters. The Bayesian formulation incorporates uncertainties in the flow conditions and measurement errors and compares the measurements with the predictions of Computational Fluid Dynamics (CFD) simulations which depend on the parameter values. The resulting parameters posterior distribution is sampled using a Markov-Chain Monte-Carlo method. Polynomial-Chaos (PC) surrogates substitute the CFD predictions in the definition of the Bayesian posterior, in order to alleviate the computational burden of solving multiple CFD problems. We rely on synthetic data i.e., generated numerically, to assess the potential of expansion flow experiments. Using synthetic data prevents experimental bias, enables the control of model errors (thermodynamic and flow models) and permits the measurement of quantities in conditions that would be hardly achievable in practice. We test three expansion flows with increasing non-ideal effects. Our analyses reveal that the considered experiments have limited potential for the inference of the thermodynamic coefficients. Measuring the temperature, in addition to pressure, improves the posterior knowledge of the specific heat ratio, but other parameters remain highly uncertain. Also, the selection of an expansion condition yielding higher non-ideal effects somehow improves the inference, but the trend is limited, and experimenting with these conditions may be challenging. Our work also supports the use of Bayesian analysis with synthetic data to investigate, analyze, and design new experiments in the future. (10.1016/j.compfluid.2020.104550)
  DOI : 10.1016/j.compfluid.2020.104550
• Convergence Analysis of Riemannian Stochastic Approximation Schemes
  
  • Durmus Alain
  • Jiménez Pablo
  • Moulines Eric
  • Said Salem
  • Wai Hoi-To
  , 2020. This paper analyzes the convergence for a large class of Riemannian stochastic approximation (SA) schemes, which aim at tackling stochastic optimization problems. In particular, the recursions we study use either the exponential map of the considered manifold (geodesic schemes) or more general retraction functions (retraction schemes) used as a proxy for the exponential map. Such approximations are of great interest since they are low complexity alternatives to geodesic schemes. Under the assumption that the mean field of the SA is correlated with the gradient of a smooth Lyapunov function (possibly non-convex), we show that the above Riemannian SA schemes find an O(b ∞ + log n/ √ n)-stationary point (in expectation) within O(n) iterations, where b ∞ ≥ 0 is the asymptotic bias. Compared to previous works, the conditions we derive are considerably milder. First, all our analysis are global as we do not assume iterates to be a-priori bounded. Second, we study biased SA schemes. To be more specific, we consider the case where the mean-field function can only be estimated up to a small bias, and/or the case in which the samples are drawn from a controlled Markov chain. Third, the conditions on retractions required to ensure convergence of the related SA schemes are weak and hold for well-known examples. We illustrate our results on three machine learning problems.
• Cell differentiation, stem cell regulation and impact of the mutations : a stochastic approach
  
  • Bonnet Celine
  , 2020. This thesis focuses on understanding the mechanisms of stem cell differentiation leading to the production of red blood cells (a mechanism called erythropoiesis). To this end, we have developed different mathematical modelling leading to an understanding at different levels. Firstly, we have built and calibrated a model with 8 ordinary differential equations to describe the dynamics of 6 populations of cells in steady-state and stress erythropoiesis. The study of in vivo experimental data, realized by our collaborators St´ephane Giraudier (hematologist) and Evelyne Lauret (INSERM), showed the need of two equations to model erythropoiesis regulations. Modeling calibration was performed using biological data and a stochastic optimization algorithm called CMA-ES. This model highlighted the importance of the self-renewal capacity of the erythropoietic cells in the production of red blood cells. The development of a 3-dimensional probabilistic model then allowed us to understand the dynamic consequences of this capacity on the production of red blood cells. The study of this model required changes of scale in size and time revealing a so-called slow/fast system. Using averaging methods, we described the large population approximation of the number of each cell type. We have also mathematically quantified the large fluctuations in the number of red blood cells, biologically observed. Finally, we constructed a model to understand the influence of long periods of inactivity of mutant stem cells in the production of red blood cells. Mutant stem cells, which are in low numbers in the organism compared to healthy cells, randomly switch between an active and an inactive state. The different size scale between the cell populations led us to study the dynamics of a 4-dimensional piecewise deterministic Markov process. We showed the existence of a unique invariant probability measure towards which the process converges in total variation, and we identified this limits.