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

Abstract

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.