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Covariance regularity and H-matrix approximation for rough random fields

Dölz, Jürgen and Harbrecht, Helmut and Schwab, Christoph. (2017) Covariance regularity and H-matrix approximation for rough random fields. Numerische Mathematik, 135 (4). pp. 1045-1071.

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Abstract

In an open, bounded domain D ⊂ R n with smooth boundary ∂ D or on a smooth, closed and compact, Riemannian n -manifold M ⊂ R n +1 , we consider the linear operator equation Au = f where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order r ∈ R with analytic coefficients, covering all linear, second order elliptic PDEs as well as their boundary reductions. Here, f ∈ L 2 ( Ω ; 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 M , respectively. We prove that the random solution’s covariance kernel K u = ( A − 1 ⊗ A − 1 ) K f on D × D (resp. M×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 H -matrix calculus allows deterministic approximation of singular covariances K u of the random solution u = A − 1 f ∈ L 2 ( Ω ; H t − r ) in D × D with work versus accuracy essentially equal to that for the mean field approximation in D, overcoming the curse of dimensionality in this case.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
UniBasel Contributors:Harbrecht, Helmut and Dölz, Jürgen
Item Type:Article, refereed
Article Subtype:Research Article
ISSN:0029-599X
Note:Publication type according to Uni Basel Research Database: Journal article
Language:English
Identification Number:
Last Modified:02 Oct 2017 12:26
Deposited On:02 Oct 2017 12:26

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