Publications

2022

• ICU Bed Availability Monitoring and analysis in the Grand Est region of France during the COVID-19 epidemic
  
  • Icubam Consortium
  • Bonnasse-Gahot Laurent
  • Dénès Maxime
  • Dulac-Arnold Gabriel
  • Girgin Sertan
  • Husson François
  • Iovene Valentin
  • Josse Julie
  • Kimmoun Antoine
  • Landes François
  • Nadal Jean-Pierre
  • Primet Romain
  • Quintao Frederico
  • Raverdy Pierre Guillaume
  • Rouvreau Vincent
  • Teboul Olivier
  • Yurchak Roman
  Statistique et Société, Société française de statistique (SFdS), 2022, 10 (1), pp.19-36. Background: Reliable information is an essential component for responding to the COVID-19 epidemic, especially regarding the availability of critical care beds (CCBs). We propose three contributions: a) ICUBAM (ICU Bed Availability Monitor), a tool which both collects and visualizes information on CCB availability entered directly by intensivists. b) An analysis of CCB availability and ICU admissions and outcomes using collected by ICUBAM during a 6-week period in the hard-hit Grand Est region of France, and c) Explanatory and predictive models adapted to CCB availability prediction, and fitted to availability information collected by ICUBAM.Methods: We interact directly with intensivists twice a day, by sending a SMS with a web link to the ICUBAM form where they enter 8 numbers: number of free and occupied CCBs (ventilator-equipped) for both COVID-19 positive and COVID-19- negative patients, the number of COVID-19 related ICU deaths and discharges, the number of ICU refusals, and the number of patients transferred to another region due to bed shortages. The collected data are described using univariate and multivariate methods such as correspondence analysis and then modeled at different scales: a medium and long term prediction using SEIR models, and a short term statistical model to predict the number of CCBs.Results: ICUBAM was brought online March 25, and is currently being used in the Grand-Est region by 109 intensivists representing 40 ICUs (95% of ICUs). ICUBAM allows for the calculation of CCB availability, admission and discharge statistics. Our analysis of data describes the evolution and extent of the COVID-19 health crisis in the Grand-Est region: on April 6th, at maximum bed capacity, 1056 ventilator-equipped CCBs were present, representing 211% of the nominal regional capacity of 501 beds. From March 19th to March 31st, average daily COVID-19 ICU inflow was 68 patients/day, and 314 critical care patients were transferred out of the Grand-Est region. With French lockdown starting on March 17th, a decrease of the daily inflow was found starting on April 1st: 23 patients/day during the first fortnight of April, and 7 patients/day during the last fortnight. However, treatment time for COVID-19 occupied CCBs is long: 15 days after the peak on March 31st, only 20% of ICU beds have been freed (50% after 1 month). Region-wide COVID-19 related in-ICU mortality is evaluated at 31%. Models trained from ICUBAM data are able to describe and predict the evolution of bed usage for the Grand-Estregion.Conclusion: We observe strong uptake of the ICUBAM tool, amongst both physicians and local healthcare stakeholders (health agencies, first responders etc.). We are able to leverage data collected with ICUBAM to better understand the evolution of the COVID-19 epidemic in the Grand Est region. We also present how data ingested by ICUBAM can be used to anticipate CCB shortages and predict future admissions. Most importantly, we demonstrate the importance of having a cross-functional team involving physicians, statisticians and computer scientists working both with first-line medical responders and local health agencies. This allowed us to quickly implement effective tools to assist in critical decision-making processes. (10.1101/2020.05.18.20091264)
  DOI : 10.1101/2020.05.18.20091264
• Monotone discretization of the Monge-Ampère equation of optimal transport
  
  • Bonnet Guillaume
  • Mirebeau Jean-Marie
  , 2022. We design a monotone finite difference discretization of the second boundary value problem for the Monge-Ampère equation, whose main application is optimal transport. We prove the existence of solutions to a class of monotone numerical schemes for degenerate elliptic equations whose sets of solutions are stable by addition of a constant, and we show that the scheme that we introduce for the Monge-Ampère equation belongs to this class. We prove the convergence of this scheme, although only in the setting of quadratic optimal transport. The scheme is based on a reformulation of the Monge-Ampère operator as a maximum of semilinear operators. In dimension two, we recommend to use Selling's formula, a tool originating from low-dimensional lattice geometry, in order to choose the parameters of the discretization. We show that this approach yields a closed-form formula for the maximum that appears in the discretized operator, which allows the scheme to be solved particularly efficiently. We present some numerical results that we obtained by applying the scheme to quadratic optimal transport problems as well as to the far field refractor problem in nonimaging optics.
• Combining supervised deep learning and scientific computing : some contributions and application to computational fluid dynamics
  
  • Novello Paul
  , 2022. Recent innovations in mathematics, computer science, and engineering have enabled more and more sophisticated numerical simulations. However, some simulations remain computationally unaffordable, even for the most powerful supercomputers. Lately, machine learning has proven its ability to improve the state-of-the-art in many fields, notoriously computer vision, language understanding, or robotics. This thesis settles in the high-stakes emerging field of Scientific Machine Learning which studies the application of machine learning to scientific computing. More specifically, we consider the use of deep learning to accelerate numerical simulations.We focus on approximating some components of Partial Differential Equation (PDE) based simulation software by a neural network. This idea boils down to constructing a data set, selecting and training a neural network, and embedding it into the original code, resulting in a hybrid numerical simulation. Although this approach may seem trivial at first glance, the context of numerical simulations comes with several challenges. Since we aim at accelerating codes, the first challenge is to find a trade-off between neural networks’ accuracy and execution time. The second challenge stems from the data-driven process of the training, and more specifically, its lack of mathematical guarantees. Hence, we have to ensure that the hybrid simulation software still yields reliable predictions. To tackle these challenges, we thoroughly study each step of the deep learning methodology while considering the aforementioned constraints. By doing so, we emphasize interplays between numerical simulations and machine learning that can benefit each of these fields.We identify the main steps of the deep learning methodology as the construction of the training data set, the choice of the hyperparameters of the neural network, and its training. For the first step, we leverage the ability to sample training data with the original software to characterize a more efficient training distribution based on the local variation of the function to approximate. We generalize this approach to general machine learning problems by deriving a data weighting methodology called Variance Based Sample Weighting. For the second step, we introduce the use of sensitivity analysis, an approach widely used in scientific computing, to tackle neural network hyperparameter optimization. This approach is based on qualitatively assessing the effect of hyperparameters on the performances of a neural network using Hilbert-Schmidt Independence Criterion. We adapt it to the hyperparameter optimization context and build an interpretable methodology that yields competitive and cost-effective networks. For the third step, we formally define an analogy between the stochastic resolution of PDEs and the optimization process at play when training a neural network. This analogy leads to a PDE-based framework for training neural networks that opens up many possibilities for improving existing optimization algorithms. Finally, we apply these contributions to a computational fluid dynamics simulation coupled with a multi-species chemical equilibrium code. We demonstrate that we can achieve a time factor acceleration of 21 with controlled to no degradation from the initial prediction.
• Maxwell's equations with hypersingularities at a conical plasmonic tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022, 161, pp.70-110. In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical $L^2$ framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle. (10.1016/j.matpur.2022.03.001)
  DOI : 10.1016/j.matpur.2022.03.001
• Software for optimal ecological transition path of a credit portfolio distribution
  
  • Gobet Emmanuel
  • Lage Clara
  , 2022. This software makes use of optimal transport and Gaussian copula modeling to compute portfolio trajectories, prioritizing ecological transition. With customizable scenario analysis, users can make sustainable investment decisions while minimizing risks. Article: https://hal.archives-ouvertes.fr/hal-03423114
• McKean–Vlasov optimal control: The dynamic programming principle
  
  • Djete Mao Fabrice
  • Possamaï Dylan
  • Tan Xiaolu
  The Annals of Probability, Institute of Mathematical Statistics, 2022, 50 (2). (10.1214/21-AOP1548)
  DOI : 10.1214/21-AOP1548
• Decision-making tools for healthcare structures in times of pandemic
  
  • Garaix Thierry
  • Gaubert Stéphane
  • Josse Julie
  • Vayatis Nicolas
  • Véber Amandine
  Anaesthesia Critical Care & Pain Medicine, Elsevier Masson, 2022, 41 (2), pp.101052. The COVID-19 sanitary crisis has shed an unprecedented light on the role of modellers, mathematicians, computer scientists and data scientists in the monitoring and in the understanding of the dynamics of an epidemic. The general audience became rapidly acquainted with the concepts of R0, of quasi-exponential growth of the number of cases, as well as with the burden that such an explosive dynamics puts on medical structures. But mathematical and computational approaches have even more to offer. From the early stages of the pandemic, many organisational issues arose to cope with the wave of patients, which overwhelmed many hospitals and care structures. Emergency call centres faced a sharp increase in the number of calls, which had to be efficiently sorted according to the urgency of the patient’s condition. The saturation of critical care services led to the need to re-orient incoming patients to other hospitals with available beds, or to organise the transfer of patients from a heavily burdened region to a region under a weaker capacity strain. Hospital services had to find humanly manageable ways to deal with the work surplus, in conditions where health workers were not only at risk of physical and psychological exhaustion, but also at risk of being infected with SARS-CoV-2. In this context, already existing collaborations between mathematicians or computer scientists and medical doctors or emergency call centres gave rise to the extraordinarily quick emergence of tools to help operators and medical staff with logistic decisions. Below, we present four such initiatives, which arose during the first COVID-19 wave in France in 2020. They illustrate what close collaborations between medical staff and researchers from the fields of operations research, modelling of complex dynamics or data sciences can bring to the management of critical situations in health services. Beyond the current pandemic, they also pose the question of how to improve our preparedness to future crises. (10.1016/j.accpm.2022.101052)
  DOI : 10.1016/j.accpm.2022.101052
• Random cographs: Brownian graphon limit and asymptotic degree distribution
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Féray Valentin
  • Gerin Lucas
  • Maazoun Mickaël
  • Pierrot Adeline
  Random Structures and Algorithms, Wiley, 2022, 60 (2), pp.166-200. We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order $n$, and converges after normalization to the Lebesgue measure on $[0,1]$. We finally analyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis. (10.1002/rsa.21033)
  DOI : 10.1002/rsa.21033
• A Control Variate Method Driven by Diffusion Approximation
  
  • Garnier Josselin
  • Mertz Laurent
  Communications on Pure and Applied Mathematics, Wiley, 2022, 75 (3), pp.455-492. In this paper we examine a control variate estimator for a quantity that can be expressed as the expectation of a functional of a random process, that is itself the solution of a differential equation driven by fast mean-reverting ergodic forces. The control variate is the expectation of the same functional for the limit diffusion process that approximates the original process when the mean-reversion time goes to zero. To get an efficient control variate estimator, we propose a coupling method to build the original process and the limit diffusion process. We show that the correlation between the two processes indeed goes to one when the mean reversion time goes to zero and we quantify the convergence rate, which makes it possible to characterize the variance reduction of the proposed control variate method. The efficiency of the method is illustrated on a few examples. (10.1002/cpa.21976)
  DOI : 10.1002/cpa.21976
• Feature selection for kernel methods in systems biology
  
  • Brouard Celine
  • Mariette Jérôme J.
  • Flamary Rémi
  • Vialaneix Nathalie
  NAR Genomics and Bioinformatics, Oxford University Press, 2022, 4 (1). The substantial development of high-throughput biotechnologies has rendered large-scale multiomics datasets increasingly available. New challenges have emerged to process and integrate this large volume of information, often obtained from widely heterogeneous sources. Kernel methods have proven successful to handle the analysis of different types of datasets obtained on the same individuals. However, they usually suffer from a lack of interpretability since the original description of the individuals is lost due to the kernel embedding. We propose novel feature selection methods that are adapted to the kernel framework and go beyond the well established work in supervised learning by addressing the more difficult tasks of unsupervised learning and kernel output learning. The method is expressed under the form of a non-convex optimization problem with a 1 penalty, which is solved with a proximal gradient descent approach. It is tested on several systems biology datasets and shows good performances in selecting relevant and less redundant features compared to existing alternatives. It also proved relevant for identifying important governmental measures best explaining the time series of Covid-19 reproducing number evolution during the first months of 2020. The proposed feature selection method is embedded in the R package mixKernel version 0.7, published on CRAN. (10.1093/nargab/lqac014)
  DOI : 10.1093/nargab/lqac014
• A Bilevel Energy Management Strategy for HEVs Under Probabilistic Traffic Conditions
  
  • Le Rhun Arthur
  • Bonnans Joseph Frédéric
  • de Nunzio Giovanni
  • Leroy Thomas
  • Martinon Pierre
  IEEE Transactions on Control Systems Technology, Institute of Electrical and Electronics Engineers, 2022, 30 (2), pp.728-739. This work proposes a new approach for the optimal energy management of a hybrid electric vehicle taking into account traffic conditions. The method is based on a bi-level decomposition. At the microscopic level, the offline part computes cost maps thanks to a stochastic optimization that considers the influence of traffic, in terms of speed/acceleration probability distributions. At the online macroscopic level, a deterministic optimization computes the ideal state of charge at the end of each road segment, using the computed cost maps. The optimal torque split can then be recovered according to the cost maps and this SoC target sequence. Since the high computational cost due to the uncertainty of traffic conditions has been managed offline, the online part should be fast enough for real-time implementation on board the vehicle. Errors due to discretization and computation in the proposed algorithm have been studied. Finally, we present numerical simulations using actual traffic data, and compare the proposed bi-level method to the best possible consumption, obtained by a deterministic optimization with full knowledge of future traffic conditions, as well as to an established solution for the energy management of a hybrid electric vehicle. The solutions show a reasonable over-consumption compared with deterministic optimization, and manageable computational times for both the offline and online part. (10.1109/TCST.2021.3073607)
  DOI : 10.1109/TCST.2021.3073607
• MDA for random forests: inconsistency, and a practical solution via the Sobol-MDA
  
  • Bénard Clément
  • da Veiga Sébastien
  • Scornet Erwan
  Biometrika, Oxford University Press (OUP), 2022, 109, pp.881–900. Variable importance measures are the main tools to analyze the black-box mechanisms of random forests. Although the mean decrease accuracy (MDA) is widely accepted as the most efficient variable importance measure for random forests, little is known about its statistical properties. In fact, the definition of MDA varies across the main random forest software. In this article, our objective is to rigorously analyze the behavior of the main MDA implementations. Consequently, we mathematically formalize the various implemented MDA algorithms, and then establish their limits when the sample size increases. This asymptotic analysis reveals that these MDA versions differ as importance measures, since they converge towards different quantities. More importantly, we break down these limits into three components: the first two terms are related to Sobol indices, which are well-defined measures of a covariate contribution to the response variance, widely used in the sensitivity analysis field, as opposed to the third term, whose value increases with dependence within covariates. Thus, we theoretically demonstrate that the MDA does not target the right quantity to detect influential covariates in a dependent setting, a fact that has already been noticed experimentally. To address this issue, we define a new importance measure for random forests, the Sobol-MDA, which fixes the flaws of the original MDA, and consistently estimates the accuracy decrease of the forest retrained without a given covariate, but with an efficient computational cost. The Sobol-MDA empirically outperforms its competitors on both simulated and real data for variable selection. An open source implementation in R and C++ is available online.
• Deterministic Approximate EM Algorithm; Application to the Riemann Approximation EM and the Tempered EM
  
  • Lartigue Thomas
  • Durrleman Stanley
  • Allassonnière Stéphanie
  Algorithms, MDPI, 2022, 15 (3), pp.78. The Expectation Maximisation (EM) algorithm is widely used to optimise non-convex likelihood functions with latent variables. Many authors modified its simple design to fit more specific situations. For instance, the Expectation (E) step has been replaced by Monte Carlo (MC), Markov Chain Monte Carlo or tempered approximations... Most of the well-studied approximations belong to the stochastic class. By comparison, the literature is lacking when it comes to deterministic approximations. In this paper, we introduce a theoretical framework, with state-of-the-art convergence guarantees, for any deterministic approximation of the E step. We analyse theoretically and empirically several approximations that fit into this framework. First, for intractable E-steps, we introduce a deterministic version of MC-EM using Riemann sums. A straightforward method, not requiring any hyper-parameter fine-tuning, useful when the low dimensionality does not warrant a MC-EM. Then, we consider the tempered approximation, borrowed from the Simulated Annealing literature and used to escape local extrema. We prove that the tempered EM verifies the convergence guarantees for a wider range of temperature profiles than previously considered. We showcase empirically how new non-trivial profiles can more successfully escape adversarial initialisations. Finally, we combine the Riemann and tempered approximations into a method that accomplishes both their purposes. (10.3390/a15030078)
  DOI : 10.3390/a15030078
• Random stable-type minimal factorizations of the n -cycle
  
  • Thévenin Paul
  Advances in Applied Probability, Applied Probability Trust, 2022, 54 (1), pp.1-63. Abstract We investigate random minimal factorizations of the n -cycle, that is, factorizations of the permutation $(1 \, 2 \cdots n)$ into a product of cycles $\tau_1, \ldots, \tau_k$ whose lengths $\ell(\tau_1), \ldots, \ell(\tau_k)$ satisfy the minimality condition $\sum_{i=1}^k(\ell(\tau_i)-1)=n-1$ . By associating to a cycle of the factorization a black polygon inscribed in the unit disk, and reading the cycles one after another, we code a minimal factorization by a process of colored laminations of the disk. These new objects are compact subsets made of red noncrossing chords delimiting faces that are either black or white. Our main result is the convergence of this process as $n \rightarrow \infty$ , when the factorization is randomly chosen according to Boltzmann weights in the domain of attraction of an $\alpha$ -stable law, for some $\alpha \in (1,2]$ . The limiting process interpolates between the unit circle and a colored version of Kortchemski’s $\alpha$ -stable lamination. Our principal tool in the study of this process is a new bijection between minimal factorizations and a model of size-conditioned labeled random trees whose vertices are colored black or white, as well as the investigation of the asymptotic properties of these trees. (10.1017/apr.2021.21)
  DOI : 10.1017/apr.2021.21
• Une régularisation quadratique pour la tarification de contrats d'électricité
  
  • Jacquet Quentin
  • van Ackooij Wim
  • Alasseur Clémence
  • Gaubert Stéphane
  , 2022.
• Parametric Shape Optimization using the Support Function
  
  • Antunes Pedro R S
  • Bogosel Beniamin
  Computational Optimization and Applications, Springer Verlag, 2022. The optimization of functionals depending on shapes which have convexity, diameter or constant width constraints poses difficulties from a numerical point of view. We show how to use the support function in order to approximate solutions to such problems by finite dimensional optimization problems under various constraints. After constructing the numerical framework, we present some applications from the field of convex geometry. We consider the optimization of various functionals depending on the volume, perimeter and Dirichlet Laplace eigenvalues under the constraints presented earlier. In particular we confirm numerically Meissner's conjecture, regarding three dimensional bodies of constant width with minimal volume, by directly solving an optimization problem.
• Scaling limits of random looptrees and bipartite plane maps with prescribed large faces
  
  • Marzouk Cyril
  , 2022. We first rephrase and unify known bijections between bipartite plane maps and labelled trees with the formalism of looptrees, which we argue to be both more relevant and technically simpler since the geometry of a looptree is explicitly encoded by the depth-first walk (or {\L}ukasiewicz path) of the tree, as opposed to the height or contour process for the tree. We then construct continuum analogues associated with any c\`adl\`ag path with no negative jump and derive several invariance principles. We especially focus on uniformly random looptrees and maps with prescribed face degrees and study their scaling limits in the presence of macroscopic faces, which complements a previous work in the case of no large faces. The limits (along subsequences for maps) form new families of random metric measured spaces related to processes with exchangeable increments with no negative jumps and our results generalise previous works which concerned the Brownian and stable L\'evy bridges.
• 6 - AN INTRODUCTION TO STATE SPACE MODELS
  
  • Douc Randal
  • Moulines Éric
  • Stoffer David
  , 2022, pp.215-258. (10.1051/978-2-7598-2741-1.c011)
  DOI : 10.1051/978-2-7598-2741-1.c011
• Variance reduction for additive functionals of Markov chains via martingale representations
  
  • Belomestny D.
  • Moulines E.
  • Samsonov S.
  Statistics and Computing, Springer Verlag (Germany), 2022, 32 (1), pp.16. (10.1007/s11222-021-10073-z)
  DOI : 10.1007/s11222-021-10073-z
• Diffusion bridges vector quantized Variational AutoEncoders
  
  • Cohen Max
  • Quispe Guillaume
  • Le Corff Sylvain
  • Ollion Charles
  • Moulines Eric
  , 2022 (39). Vector Quantized-Variational AutoEncoders (VQ-VAE) are generative models based on discrete latent representations of the data, where inputs are mapped to a finite set of learned embeddings. To generate new samples, an autoregressive prior distribution over the discrete states must be trained separately. This prior is generally very complex and leads to slow generation. In this work, we propose a new model to train the prior and the encoder/decoder networks simultaneously. We build a diffusion bridge between a continuous coded vector and a non-informative prior distribution. The latent discrete states are then given as random functions of these continuous vectors. We show that our model is competitive with the autoregressive prior on the mini-Imagenet and CIFAR dataset and is efficient in both optimization and sampling. Our framework also extends the standard VQ-VAE and enables end-to-end training.
• Multiphysical modelling and simulation of the ignition transient of complete solid rocket motors
  
  • Francois Laurent
  , 2022. Solid rocket motors (SRMs) use the combustion of a solid material, the propellant, as an energy source.A crucial step in the operation of such an engine is its ignition, during which the surface of the propellant must be heated by about 400 degrees to initiate combustion.This is usually done by letting a hot jet impact the surface.The ignition of an SRM involves a wide variety of phenomena, including: combined heat transfer between the igniter gases and the propellant, pyrolysis of the propellant below the surface, release of gaseous species that burn in an intense flame attached to the surface, heating of the propellant by radiation emitted from the gas phase, compressible multiphase flow in the combustion chamber, supersonic flow in the nozzle.The multiphysical nature and the disparities in space-time scales make it impossible to simulate ignition using a single tool that would include a complete modelling of all the phenomena. Typically, the propellant flame is so thin that it cannot be reasonably resolved in a CFD mesh for a complete motor. In addition, it involves stiff and potentially complex chemical kinetics.This is why the classical approach is to use a 1D model of the propellant combustion, at each boundary face of the CFD mesh belonging to the propellant surface. Thus, all the physico-chemical and numerical complexity of solving this combustion is encapsulated in a dynamic boundary condition. However, the existing 1D models are very simplified, impacting the fidelity of the reproduction of ignition in some motors.In this thesis, we choose to use a more advanced 1D approach, which includes a numerical resolution of the flame, able to use complex or global kinetics. Specific attention is paid to the mathematical analysis of the 1D model in steady state, through the study of a travelling combustion wave, clarifying the notion of eigenvalue for the regression speed. To simulate unsteady combustion, a semi-discretisation in space is obtained by the method of lines. The differential-algebraic nature of the resulting system of equations is clearly exposed, allowing for the choice of efficient integration methods to solve the propellant dynamics with high order in time and adaptive time step.In order to ensure an accurate reproduction of the ignition of different propellants, an optimisation process is developed to automatically parameterise the model, allowing for a good agreement between experimental and simulated ignition times.The 1D model is then coupled with the semi-industrial 3D CFD code CEDRE from ONERA, in order to allow for the simulation of ignition in complete motors. The coupling is initially operated at order 1, but techniques are suggested to allow for a higher-order and adaptive coupling.In order to verify the effect of the 1D representation of the flame, a more detailed but more expensive coupling is also developed, where the flame is solved in the CFD code itself.The comparison of the two approaches on an academic 2D configuration allows to verify the consistency and accuracy of the new approach.The coupling between the 1D model and the CFD code developed during this thesis and the interdisciplinary approaches used offer new perspectives both for the development of mathematical tools for adaptive coupling strategies for a wide range of applications, allowing to optimise the accuracy and the cost of the computations, as well as for a better reproduction of ignition in various motors.
• Importance sampling based active learning for parametric seismic fragility curve estimation
  
  • Gauchy Clement
  • Feau Cyril
  • Garnier Josselin
  , 2022. The key elements of seismic probabilistic risk assessment studies are the fragility curves which express the probabilities of failure of structures conditional to a seismic intensity measure. A multitude of procedures is currently available to estimate these curves. For modeling-based approaches which may involve complex and expensive numerical models, the main challenge is to optimize the calls to the numerical codes to reduce the estimation costs. Adaptive techniques can be used for this purpose, but in doing so, taking into account the uncertainties of the estimates (via confidence intervals or ellipsoids related to the size of the samples used) is an arduous task because the samples are no longer independent and possibly not identically distributed. The main contribution of this work is to deal with this question in a mathematical and rigorous way. To this end, we propose and implement an active learning methodology based on adaptive importance sampling for parametric estimations of fragility curves. We prove some theoretical properties (consistency and asymptotic normality) for the estimator of interest. Moreover, we give a convergence criterion in order to use asymptotic confidence ellipsoids. Finally, the performances of the methodology are evaluated on analytical and industrial test cases of increasing complexity.
• Mon Cartable Connecté, du son 3D pour l’enfant hospitalisé
  
  • Aussal Matthieu
  • Chavanne Marie-Françoise
  Interstices, INRIA, 2022. En France, près de 2 millions d’enfants hospitalisés chaque année ne peuvent se rendre physiquement à l’école. Quelle réponse humaine et technologique peut-on apporter pour améliorer l’exigence d’une « présence-connectée » en classe ? Pour tout enfant, l’éducation est un droit qui répond à son besoin d’être instruit, de se former, d’acquérir des compétences tout en grandissant avec des jeunes de son âge. Mais lorsqu’une longue maladie impose au jeune élève une rupture violente avec son école, sa classe et ses amis, il a le sentiment d’avoir perdu sa place. À sa souffrance s’ajoute l’exclusion et la solitude. Le Collectif contre les discriminations a conçu « Mon Cartable Connecté », afin de veiller à ce que le jeune malade hospitalisé garde au sein de sa classe sa place, sa vie sociale, scolaire et affective et prépare son retour. Bien plus qu’une « télé-présence », Mon Cartable Connecté veille à ce que la « présence » de l’élève soit au plus près de ce qui se passe au quotidien dans la classe tant pour apprendre, que pour être avec ses camarades. Voir, observer, écouter, s’exprimer, échanger, réfléchir, interagir, mais aussi rire et bavarder, sont ce qui permet aux enfants d’apprendre avec les autres, ce dont l’élève absent est malheureusement privé. Compte-tenu de notre expérience et de notre attention aux besoins particuliers de ces jeunes malades que nous équipons, et dans le souci d’être au plus près d’une immersion dans leur classe, nous sommes soucieux que la technologie compense les déficits ou les difficultés liées à leurs pathologies. Cela afin de leur offrir les meilleures conditions d’acquisition des compétences du socle commun que l’école se doit de leur garantir. Aspects pédagogiques et acqu
• Longest minimal length partitions
  
  • Bogosel Beniamin
  • Oudet Edouard
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2022, 24 (1), pp.95-135. This article provides numerical evidence that under volume constraint the ball is the set which maximizes the perimeter of the least-perimeter partition into cells with prescribed areas. We introduce a numerical maximization algorithm which performs multiple optimizations steps at each iteration to approximate minimal partitions. Using these partitions we compute perturbations of the domain which increase the minimal perimeter. The initialization of the optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A new algorithm is proposed to identify such diagrams, by computing the gradients of areas and perimeters for the Voronoi cells with respect to the Voronoi points. (10.4171/IFB/468)
  DOI : 10.4171/IFB/468
• Multi-stage Stochastic Alternating Current Optimal Power Flow with Storage: Bounding the Relaxation Gap
  
  • Grangereau Maxime
  • van Ackooij Wim
  • Gaubert Stéphane
  Electric Power Systems Research, Elsevier, 2022, 206, pp.107774. We propose a generic multistage stochastic model for the Alternating Current Optimal Power Flow (AC OPF) problem for radial distribution networks, to account for the random electricity production of renewable energy sources and dynamic constraints of storage systems. We consider single-phase radial networks. Radial three-phase balanced networks (medium-voltage distribution networks typically have this structure) reduce to the former case. This induces a large scale optimization problem, which, given the non-convex nature of the AC OPF, is generally challenging to solve to global optimality. We derive a priori conditions guaranteeing a vanishing relaxation gap for the multi-stage AC OPF problem, which can thus be solved using convex optimization algorithms. We also give an a posteriori upper bound on the relaxation gap. In particular, we show that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available. Then, we discuss the validity of our results when incorporating voltage regulation devices. Finally, we illustrate our results on problems of planning of a realistic distribution feeder with distributed solar production and storage systems. Scenario trees for solar production are constructed from a stochastic model, by a quantile-based algorithm. (10.1016/j.epsr.2022.107774)
  DOI : 10.1016/j.epsr.2022.107774