Speaker: Olivier Le Maitre (Directeur de recherche CNRS, Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, Paris-Saclay)
Title: Polynomial Surrogates for Bayesian Inference (Modèle polynomiaux pour l’inférence bayesienne)
The Bayesian inference is a popular probabilistic method to solve inverse problems, such as the identification of field parameter in a PDE model.
Using observations, the inference rely on the Bayes rule to update the prior density of the sought field to its posterior distribution.
In most cases the posterior distribution has no explicit form and must be sampled for instance using a Markov-Chain Monte Carlo method.
In practice the prior is decomposed (e.g. Karhunen-Loeve decomposition) to recast the problem into the inference of a finite number of coordinates.
Although proved effective in many situations, the Bayesian method faces several difficulties calling for improvements.
First, sampling the posterior can be extremely costly as it requires multiple resolutions of the PDE model for sampled values of the inferred field.
Second, when the observations are not much informative, the posterior can highly depends on the assumed prior whose selection can be to some extend arbitrary.
These issues have motivated the introduction of reduced modeling or surrogate models to approximate the parametrized PDE solution, and hyper-parameters to be determined in the definition of the prior field.
The presentation will focus on recent developments along these two directions: the acceleration of the posterior sampling by means of posterior-adapted polynomial surrogates and the efficient treatment of parametrized covariance functions for the field’s prior.