Covariance regularity and $$\mathcal {H}$$ -matrix approximation for rough random fields.

In an open, bounded domain $$\mathrm{D}\subset {\mathbb R}^n$$ with smooth boundary $$\partial \mathrm{D}$$ or on a smooth, closed and compact, Riemannian n-manifold $$\mathcal {M}\subset {\mathbb R}^{n+1}$$ , we consider the linear operator equation $$A u = f$$ where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order $$r\in {\mathbb R}$$ with analytic coefficients, 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 $$\mathrm{D}$$ or manifold $$\mathcal {M}$$ , respectively. We prove that the random solution's covariance kernel $$K_u = (A^{-1}\otimes A^{-1})K_f$$ on $$\mathrm{D}\times \mathrm{D}$$ (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 elliptic pseudodifferential operator. As a consequence, numerical $$\mathcal {H}$$ -matrix calculus allows the deterministic approximation of singular covariances $$K_u$$ of the random solution $$u=A^{-1}f \in L^2(\Omega ;H^{t-r})$$ in $$\mathrm{D}\times \mathrm{D}$$ $$(\text {resp. } \mathcal {M} \times \mathcal {M})$$ with work versus accuracy essentially equal to that for the mean field approximation with splines of fixed order $$\mathrm{D}$$ $$(\text {resp. } \mathcal {M} )$$ , overcoming the curse of dimensionality in this case. [ABSTRACT FROM AUTHOR]