Abstract

Lecture: Tuesday 11 July 2023, 10:00 – 11:00Title: Probabilistic and statistical tools 2Presenter: Christian SOIZE 1. Statistical reduction (PCA and KL expansion). Principal component analysis (PCA) of a random vector. Numerical construction from a training dataset using a SVD .- Spectral representation (Karhunen-Loeve expansion) of the random field. 2. Brief overview of stochastic solvers for the propagation of uncertainties. Why a stochastic solver is not a stochastic modeling process. First class of techniques related to Galerkin-type projections.- Second class of techniques related to the direct simulation methods. 3. Brief overview on the fundamental tools for statistical inverse problems. Description of the problem. First method: a least square method or the maximum likelihood method. Second method: the Bayesian inference in the nonparametric framework. Identification scheme with available data coming from experiments. Scheme to identify model-parameter uncertainties from experiments. Least-square method. Maximum likelihood method for estimating the hyperparameters. Asymptotic probability measure of the maximum likelihood estimator. Bayesian inference in a nonparametric framework. The output-prediction-error method with an additive Gaussian noise.