English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider elliptic diffusion problems with a random anisotropic diffusioncoefficient, where, in a notable direction given by a random vector field, thediffusion strength differs from the diffusion strength perpendicular to thisnotable direction. The Karhunen-Lo\`eve expansion then yields a parametrisationof the random vector field and, therefore, also of the solution of the ellipticdiffusion problem. We show that, given regularity of the elliptic diffusionproblem, the decay of the Karhunen-Lo\`eve expansion entirely determines theregularity of the solution's dependence on the random parameter, also whenconsidering this higher spatial regularity. This result then implies thatmultilevel collocation and multilevel quadrature methods may be used to lessenthe computation complexity when approximating quantities of interest, like thesolution's mean or its second moment, while still yielding the expected ratesof convergence. Numerical examples in three spatial dimensions are provided tovalidate the presented theory.

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