A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data

• Verfasst von: Lang, Jens [VerfasserIn]
• Verfasst von: Scheichl, Robert [VerfasserIn]
• Verfasst von: Silvester, David J. [VerfasserIn]
• Titel: A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data
• Verf.angabe: J. Lang, R. Scheichl, D. Silvester
• E-Jahr: 2020
• Jahr: 1 July 2020
• Umfang: 17 S.
• Fussnoten: Gesehen am 14.10.2020
• Titel Quelle: Enthalten in: Journal of computational physics
• Ort Quelle: Amsterdam : Elsevier, 1961
• Jahr Quelle: 2020
• Band/Heft Quelle: 419(2020) Artikel-Nummer 109692, 17 Seiten
• ISSN Quelle: 1090-2716
• DOI: doi:10.1016/j.jcp.2020.109692
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: Adaptivity
• Sach-SW: High-dimensional approximation
• Sach-SW: Multilevel methods
• Sach-SW: Sparse grids
• Sach-SW: Stochastic collocation
• Sach-SW: Uncertainty quantification
• K10plus-PPN: 1735520519

Abstract

We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.