Taking into account input uncertainties in the Bayesian calibration of time-consuming simulators

Perrin, Guillaume; Durantin, Cédric

Journal de la société française de statistique, Tome 160 (2019) no. 2, pp. 24-46.

La prise en compte des incertitudes expérimentales est un élément clé de la quantification des incertitudes et de la prévision par la simulation. Bien qu’une attention particulière soit accordée aux incertitudes expérimentales sur les sorties de simulation, peu de travaux s’intéressent aux incertitudes concernant les entrées de simulation, sous pretexte qu’elles sont négligeables ou suffisamment petites pour être agrégées avec les incertitudes sur les sorties par développement de Taylor. Toutefois, ces incertitudes sur les entrées ne sont pas toujours faibles et, selon la structure du code, la linéarisation autour de celles-ci n’est pas toujours possible. L’objectif de ce travail est donc double. Premièrement, il introduit un cadre bayésien général permettant l’intégration des incertitudes sur les entrées pour le calage de paramètres du code. Il propose ensuite plusieurs approches pour résoudre efficacement ce problème d’inférence, en fonction de la régularité du code et du type d’entrées considérées. Les avantages et les inconvénients des différentes méthodes sont finalement illustrés sur un exemple analytique, ainsi que sur un problème balistique.

The consideration of experimental uncertainties is a key element in the quantification of uncertainties and prediction by simulation. While particular attention is paid to experimental uncertainties on simulation outputs, little work is done on uncertainties on simulation inputs, arguing that they are negligible or small enough to be aggregated with uncertainties on outputs via Taylor development. However, these uncertainties on inputs are not always low and, depending on the structure of the code, linearization around them is not always possible. The objective of this work is therefore twofold. First, it introduces a general Bayesian framework for integrating input uncertainties into the calibration of code parameters. It then proposes several approaches to effectively solve this inference problem, depending on the regularity of the code and the type of inputs considered. The advantages and disadvantages of the different methods are finally illustrated on an analytical example, as well as on a ballistic problem.

Keywords: Bayesian framework, uncertainty quantification, statistical inference, input uncertainty, kernel density estimation
Mot clés : calibration bayésienne, quantification des incertitudes, inférence statistique, méthode des noyaux

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