When building a mathematical model to describe the behavior of a physical system, one has often to face a certain level of uncertainty in the proper characterization of the model parameters and input data.  The increasing computer power and the need for reliable predictions have pushed researchers to include uncertainty models, often in a probabilistic setting, for the input parameters of otherwise deterministic mathematical models.
The course will  focus on mathematical models based on Partial Differential Equations with random parameters, and presents numerical techniques for forward uncertainty propagation, including Monte Carlo, Multilevel Monte Carlo,  polynomial chaos and rational approximation techniques; inverse uncertainty analysis in a Bayesian framework and Markov Chain Monte Carlo methods; optimal control under uncertainty.
Particular attention is devoted to addressing the case of a large (even infinite) number of input parameters thus leasing to High Dimensional Approximation problems and presenting recent results such as the "Cohen-Devore" theory on polynomial approximation in infinite dimensions.