Weighted least-squares approximation of elliptic PDEs with lognormal diffusion coefficients

Abstract

We consider the weighted least-squares approximation of the solution to elliptic PDEs whose diffusion coefficient is the exponential function of a Gaussian random field. The chosen weighted least-squares estimator is stable and accurate, in the sense that its approximation error is comparable to the best approximation error. The construction of this estimator uses pointwise evaluations of the target function at specific points in the multivariate parameter domain. The number of evaluations required to ensure stability and accuracy is only linearly proportional to the dimension of the underlying approximation space. The evaluation points are independent and identically distributed according to a specific multivariate probability density, that differs from the natural Gaussian density, and that is not of product type. In this paper sampling algorithms are described for the efficient generation of independent random samples from such a multivariate density, in the particular setting of lognormal PDEs. Finally some estimates on the computational cost of these algorithms are presented.