Abstract

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 discuss the particular example of elliptic partial differential equations with diffusion coefficients of lognormal form, that is, of the form exp(b) where b is a gaussian random field. The numerical strategy consists in searching for a sparse polynomial approximation by best n-term truncation of tensorized Hermite expansions in stochastic variables which represent the gaussian fields. One interesting conclusion from our analysis is that in certain relevant cases, the often used Karhunen–Loeve representation might not be the best choice in terms of the resulting sparsity and approximability of Hermite expansion.