Abstract

Lecture: Monday 10 July 2023, 16:00 – 17h00Title: Probabilistic and statistical tools 1Presenter: Christian SOIZE 1. Types of representation for stochastic modeling. Why a probability distribution cannot arbitrarily be chosen for a stochastic modeling? Impact of an arbitrary stochastic modeling of the uncertain parameter. What is important in UQ? : the stochastic modeling of uncertainties (step 1). Direct approach for constructing the probability measure.- Indirect approach for representing the probability measure. 2. Maximum Entropy (MaxEnt) principle from Information Theory as a direct approach for constructing a priorprobability model. Entropy as a measure of uncertainties for a vector-valued random variable. Maximum entropy principle. Reformulation of the optimization problem by using Lagrange’s multipliers.  Existence and uniqueness of the principle. Analytical examples of classical probability distributions deduced from the MaxEnt principle. MaxEnt as a numerical tool for probability measure in any dimension. 3. Random Matrix Theory for uncertainty quantification in computational mechanics. A few words on fundamentals of the random matrix theory. Ensembles of random matrices for uncertainty quantification. Volume element and probability density function for random matrices. The Shannon entropy as a measure of uncertainties for a symmetric real random matrix and MaxEnt principle. Ensemble of positive-definite random matrices with a unit mean value. Ensemble of positive-definite random matrices with a unit mean value and a positive-definite lower bound.- Ensemble of positive-definite random matrices with a given mean value and a positive-definite lower bound.