Covariance regularity and H-matrix approximation for rough random fields

In an open, bounded domain D ⊂ R n with smooth boundary ∂ D or on a smooth, closed and compact, Riemannian n -manifold M ⊂ R n +1 , 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 2 ( Ω ; H t ) is an H t -valued random field with finite second moments, with H t denoting the (isotropic) Sobolev space of (not necessarily integer) order t modelled on the domain D or manifold M , respectively. We prove that the random solution’s covariance kernel K u = ( A − 1 ⊗ A − 1 ) K f on D × D (resp. M×M ) is an asymptotically smooth function provided that the covariance function K f 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 K u of the random solution u = A − 1 f ∈ L 2 ( Ω ; H t − 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.