This short course will introduce participants to the fundamental concepts of probabilistic uncertainty quantification, including model building, characterization, and prediction. The course will include mathematical foundations for random variables and processes, algorithmic and computational procedures for probabilistic model identification. Particular attention will be devoted to the construction and application to Polynomial Chaos approximations and non-parametric representations using random matrix theory.

The detailed syllabus of the course is as follows: each session has a duration of 1.5 hours.

Session 1:

1. Motivation: types of uncertainties in mechanical systems
2. Elements of probability theory: random variables and stochastic processes
3. Parametric and nonparametric models for representing probabilistic uncertainties

Session 2:

4. Karhunen-Loeve expansion
5. Polynomial Chaos expansion
6. Vector space description of random variables and processes
7. Solution of stochastic partial differential equations

Session 3:

8. Computational aspects, preconditionning and reduced-models for stochastic problems.
9. Estimation of probabilistic models: Maximum entropy, maximum likelihood and Bayesian inference methods.

Session 4:

10. Nonparametric probabilistic approach of data and model uncertainties in computational mechanics

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