Covariance regularity and H-matrix approximation for rough random fields

In an open, bounded domain $D\subset \mathbb{R}^n$ with smooth boundary $\partial D$ or on a smooth, closed and compact, Riemannian n-manifold $\mathcal{M}\subset \mathbb{R}^{n+1}$, we consider the linear operator equation Au = f where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order $r\in \mathbb{R}$ with analytic coe cients, covering all linear, second order elliptic PDEs as well as their boundary reductions. Here, $f\in L^2(\Omega; H^t)$ 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 $\mathcal{M}$, respectively. We prove that the random solution’s covariance kernel $K_u = (A^{-1}\otimes A^{-1}) K_f$ on $D\times D$ (resp. $\mathcal{M}\times\mathcal{M}$) is an asymptotically smooth function provided that the covariance function $K_f$ of the random data is a Schwartz distributional kernel of an analytic, elliptic pseudodifferential operator and that A is a strongly elliptic, analytic (pseudo-) differential operator, including in particular second order, elliptic differential operators with analytic coefficients, and their Calderón-projectors on analytic surfaces (resp. analytic surface pieces). As a consequence, numerical H-matrix calculus allows deterministic approximation of singular covariances $K_u$ of the random solution $u = A^{-1}f \in L^2(\Omega; H^{t-r})$ in $D\times D$ with work versus accuracy essentially equal to that for the mean field approximation in D, overcoming the curse of dimensionality in this case.