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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:419

Article

Lang, J.¹  Scheichl, R.^2,3  Silvester, D.⁴ 

[1] Tech Univ Darmstadt, Dept Math, Dolivostr 15, D-64293 Darmstadt, Germany

[2] Heidelberg Univ, Inst Appl Math, Neuenheimer Feld 205, D-69120 Heidelberg, Germany

[3] Heidelberg Univ, Interdisciplinary Ctr Sci Comp, Neuenheimer Feld 205, D-69120 Heidelberg, Germany

[4] Univ Manchester, Dept Math, Manchester M13 9PL, Lancs, England

关键词: Multilevel methods;    Adaptivity;    Stochastic collocation;    Sparse grids;    Uncertainty quantification;    High-dimensional approximation;   

DOI  :  10.1016/j.jcp.2020.109692

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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