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

In an open, bounded domain $$\mathrm{D}\subset {\mathbb R}^n$$D⊂Rn with smooth boundary $$\partial \mathrm{D}$$∂D or on a smooth, closed and compact, Riemannian n-manifold $$\mathcal {M}\subset {\mathbb R}^{n+1}$$M⊂Rn+1, we consider the linear operator equation $$A u = f$$Au=f where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order $$r\in {\mathbb R}$$r∈R with analytic coefficients, covering all linear, second order elliptic PDEs as well as their boundary reductions. Here, $$f\in L^2(\Omega ;H^t)$$f∈L2(Ω;Ht) is an $$H^t$$Ht-valued random field with finite second moments, with $$H^t$$Ht denoting the (isotropic) Sobolev space of (not necessarily integer) order t modelled on the domain $$\mathrm{D}$$D or manifold $$\mathcal {M}$$M, respectively. We prove that the random solution’s covariance kernel $$K_u = (A^{-1}\otimes A^{-1})K_f$$Ku=(A-1⊗A-1)Kf on $$\mathrm{D}\times \mathrm{D}$$D×D (resp. $$\mathcal {M} \times \mathcal {M}$$M×M) is an asymptotically smooth function provided that the covariance function $$K_f$$Kf of the random data is a Schwartz distributional kernel of an elliptic pseudodifferential operator. As a consequence, numerical $$\mathcal {H}$$H-matrix calculus allows the deterministic approximation of singular covariances $$K_u$$Ku of the random solution $$u=A^{-1}f \in L^2(\Omega ;H^{t-r})$$u=A-1f∈L2(Ω;Ht-r) in $$\mathrm{D}\times \mathrm{D}$$D×D$$(\text {resp. } \mathcal {M} \times \mathcal {M})$$(resp.M×M) with work versus accuracy essentially equal to that for the mean field approximation with splines of fixed order $$\mathrm{D}$$D$$(\text {resp. } \mathcal {M} )$$(resp.M), overcoming the curse of dimensionality in this case.