High-dimensional stochastic problems: reduced basis and greedy algorithms

We will present two numerical techniques to deal with high-dimensional problems in stochastic modelling. The first one is motivated by micro-macro problems, where one typically has to compute many times averages by Monte Carlo methods, these averages being parameterized by a parameter. We will propose a reduced basis technique to perform these computations efficiently. The second problem is related to uncertainty quantification. In this context, one wants to understand how some randomness on the parameters of a deterministic problem propagates to the output. A greedy algorithm may be used to obtain an approximation of the so-called response function (typically a high-dimensional function), which associates to the values of the parameters, the ouput quantities of interest.