Covariance regularity and H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}-matrix approxi

In an open, bounded domain D ⊂ R with smooth boundary ∂D or on a smooth, closed and compact, Riemannian n-manifold M⊂ R, we consider the linear operator equation Au = f where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order r ∈ R with analytic coefficients, covering all linear, second order elliptic PDEs as well as their boundary reductions. Here, f ∈ L(Ω;H) is an H-valued random field with finite second moments, with H denoting the (isotropic) Sobolev space of (not necessarily integer) order t modelled on the domain D or manifoldM, respectively. We prove that the random solution’s covariance kernelKu = (A ⊗A)Kf on D×D (resp.M×M) is an asymptotically smooth function provided that the covariance function Kf of the random data is a Schwartz distributional kernel of an elliptic pseudodifferential operator. As a consequence, numerical H-matrix calculus allows deterministic approximation of singular covariances Ku of the random solution u = A −1f ∈ L2(Ω;Ht−r) in D × D with work versus accuracy essentially equal to that for the mean field approximation in D, overcoming the curse of dimensionality in this case.