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Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion

Harbrecht, Helmut and Schmidlin, Marc. (2017) Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion. Preprints Fachbereich Mathematik, 2017 (07).

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Abstract

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Loève expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Loève expansion entirely determines the regularity of the solution’s dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel collocation and multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution’s mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
12 Special Collections > Preprints Fachbereich Mathematik
UniBasel Contributors:Harbrecht, Helmut and Schmidlin, Marc
Item Type:Preprint
Publisher:Universität Basel
Language:English
Last Modified:20 Apr 2019 15:21
Deposited On:28 Mar 2019 09:51

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