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Covariance regularity and H-matrix approximation for rough random fields

by J. Dölz and H. Harbrecht and Ch. Schwab

(Report number 2014-19)

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 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 \(\mathcal{H}\)-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 \({\mathrm D}\times {\mathrm D}\) with work versus accuracy essentially equal to that for the mean field approximation in \({\mathrm D}\), overcoming the curse of dimensionality in this case.

Keywords: Operator Equations, Covariance Kernels, Tensor-Operators

@Techreport{DHS14_569,
  author = {J. D\"olz and H. Harbrecht and Ch. Schwab},
  title = {Covariance regularity and H-matrix approximation for rough random fields},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-19.pdf },
  year = {2014}
}

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