Réunion

9h00 accueil avec un café

9h20 - 10h50,  session de 2 exposés de 35 min +10 min de questions. 

• Clementine Prieur : On feasible set estimation with Bayesian active learning

The general topic of this talk is Bayesian adaptive learning of excursion sets defined from a costly black-box model. This research field has received many attention in the last decades. During this talk, we will first review Gaussian Process Regression for feasible set estimation in the framework where the set to recover is defined from a numerical model with scalar values. We will exhibit that usual adaptive sampling criteria may lack of robustness, e.g., when the set to recover has several connex components. Then we will address more complex frameworks, such as the presence of uncertainties or the case of numerical models with vector outputs.

• Freedy Bouchet : Rare event simulations for applications in climate and the energy transition

pause café (20 min)

11h10 - 11h55,  session de 1 exposé de 35 min + 10 min de questions. 

• Mathias Rousset and Patrick Héas (joint work with Frédéric Cérou): Adaptive reduced tempering For Bayesian inverse problems and rare event simulation

This talk is about an adaptive sequential Monte Carlo sampling algorithm for solving inverse Bayesian problems in a context where a (costly) likelihood evaluation can be approximated by a surrogate, constructed from previous evaluations of the true likelihood. A rough error estimation of the obtained surrogates is required. The method is based on an adaptive sequential Monte-Carlo (SMC) simulation that jointly adapts the likelihood approximations and a standard tempering scheme of the target posterior distribution. This algorithm is well-suited to cases where the posterior is concentrated in a rare and unknown region of the prior. It is also suitable for solving low-temperature and rare-event simulation problems. The main contribution is to propose an entropy criteria that associates to the accuracy of the current surrogate a maximum inverse temperature for the likelihood approximation. The latter is used to sample a so-called snapshot, perform an exact likelihood evaluation, and update the surrogate and its error quantification. Some consistency results are presented in an idealized framework of the proposed algorithm. Our numerical experiments use in particular a reduced basis approach to construct approximate parametric solutions of a partially observed solution of an elliptic Partial Differential Equation. They demonstrate the convergence of the algorithm and show a significant cost reduction (close to a factor 10) for comparable accuracy.

11h55 - 12h15 : poster blitz session (3min de presentation resumée de chaque poster)

pause repas (1h45)

14h00 - 15h30, session de 2 exposés de 35 min +10 min de questions 

• Benjamin Zanger (joint work with Olivier Zahm, Tiangang Cui, Martin Schreiber): Sequential measure transport as density surrogates

Density estimation is a fundamental task in data science and engineering. Two major challenges are to estimate highly concentrated densities and to sample from the estimated density. Our work investigates and extends recent ideas of combining sequential Monte Carlo with measure transport based methods to overcome these challenges. Based on their components, we name these emerging methods sequential measure transport. We demonstrate these methods in variational density estimation tasks both from unnormalized densities (Bayesian inversion, rare event estimation) as well as from data (unsupervised learning).

• Virginie Ehrlacher (joint work with Mohamed-Raed Blel and Tony Lelièvre) : Greedy algorithms and reduced basis methods for variance reduction in parameter-dependent problems

The main focus of the talk is to provide a mathematical study of the algorithm proposed in [Boyaval, Lelievre et al] where the authors proposed a variance reduction technique for the computation of parameter-dependent expectations using a reduced basis paradigm. We study the effect of Monte-Carlo sampling on the theoretical properties of greedy algorithms. In particular, using concentration inequalities for the empirical measure in Wasserstein distance proved in [Fournier, Guilin, 2015], we provide sufficient conditions on the number of samples used for the computation of empirical variances at each iteration of the greedy procedure to guarantee that the resulting method algorithm is a weak greedy algorithm with high probability. These theoretical results are not fully practical and we therefore propose a heuristic procedure to choose the number of Monte-Carlo samples at each iteration, inspired from this theoretical study, which provides satisfactory results on several numerical test cases. 

pause café (20 min)

15h50 - 16h35, session de 1 exposé de 35 min +10 min de questions

• Elias Fekhari (joint work with Vincent Chabridon, Bertrand Iooss , Joseph Muré):  Bernstein adaptive nonparametric conditional sampling: a new method for rare event probability estimation

In the context of reliability assessment, estimating a failure probability associated to a rare event is a common task. To do so, various techniques have been proposed to overcome traditional crude Monte Carlo which becomes intractable in such a context. Among others, Subset Simulation is a widely used technique which relies on "splitting" the rare event probability into a sequence (i.e., a product) of less rare conditional probabilities associated to nested failure events, easier to estimate. However, this technique relies on simulating samples conditionally to the failure event by means of Monte Carlo Markov chain algorithms. These algorithms enable, at convergence, to simulate according to the target density. However, in practice, it often produces non-independent and identically distributed (i.i.d.) samples due to the correlation between Markov chains. In the present work, we propose another way to sample conditionally to the nested failure events in order to get i.i.d. samples which can be required (e.g., to perform dedicated sensitivity analysis). The proposed algorithm relies on a nonparametric fit of the conditional joint distribution using a combined kernel density estimation for marginals fitting and the Empirical Bernstein Copula (EBC). Thus, this new method presents some similarities with "Nonparametric Adaptive Importance Sampling" but addresses the problem of copula fitting by means of EBC. The proposed algorithm is tested on three toy-cases and its performances are compared with those obtained from Subset Sampling.

16h35 - 17h20, session posters:

• Alexandre Paso (joint work with Anthony Nouy): Surrogate to Poincaré inequalities on manifolds for dimension reduction in nonlinear feature spaces
•  Guillaume CHENNETIER (joint work with Hassane CHRAIB, Anne DUTFOY, Josselin GARNIER): Graph-informed importance sampling for piecewise deterministic Markov processes. Application in reliability assessment. 
• Jason Beh (joint work with Florian Simatos, Jerome Morio): Valeurs propres extremes du modele de matrice de covariance empirique en echantillonnage d’importance adaptatif en grande dimension
• Nicolas GOEMAN (joint work with Pierre-Antoine THOUVENIN Pierre CHAINAIS): Approche hiérarchique pour la résolution de problèmes inverses non-linéaires sujets à plusieurs sources de bruit
• Maxime BOUTON  (joint work with Pierre-Antoine THOUVENIN Audrey REPETTI, Pierre CHAINAIS) Une approche PnP distribuée pour l’échantillonnage de problèmes inverses en grande dimension

17h20 fin