High Dimensional Sparse Approximation
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 some cases based on important interconnected concepts: (i) variable reduction, (ii) anisotropic smoothness and (iii) sparse tensor product approximation. A sample result will be given in the context of parametric elliptic PDE's, and we shall discuss some first numerical results and remaining challenges.