Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 419, Issue -, Pages -Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2020.109692
Keywords
Multilevel methods; Adaptivity; Stochastic collocation; Sparse grids; Uncertainty quantification; High-dimensional approximation
Funding
- Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [TRR154, 239904186, TRR154/2-2018]
- Graduate School of Excellence Computational Engineering [DFG GSC233]
- Graduate School of Excellence Energy Science and Engineering [DFG GSC1070]
- UK Engineering and Physical Sciences Research Council (EPSRC) [EP/K031368/1]
- Romberg visiting scholarship at the University of Heidelberg in 2019
- Isaac Newton Institute for Mathematical Sciences, Cambridge
- EPSRC [EP/K031368/1, EP/P013317/1] Funding Source: UKRI
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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.
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