High dimensional sparse polynomial approximation of parametric PDE's

Various mathematical problems 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 the context of stochastic-parametric PDE's based on the concept of sparse polynomial approximation.