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) |
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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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