Approximating Functions in High Dimensions

This talk will discuss mathematical problems which are challenged by the fact they involve functions of a very large number of variables. Such problems arise naturally in learning theory, partial differential equations or numerical models depending on parametric or stochastic variables. They typically result in numerical difficulties due to the so-called ''curse of dimensionality''. We shall explain how these difficulties may be handled in various contexts, based on two important concepts: (i) variable reduction and (ii) sparse approximation.