A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data
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| Title: | A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data |
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| Authors: | Lang, Jens, Scheichl, Robert, Silvester, David |
| Source: | Lang, J, Scheichl, R & Silvester, D 2020, 'A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data', Journal of Computational Physics, vol. 419, 109692. https://doi.org/10.1016/j.jcp.2020.109692 |
| Publication Year: | 2020 |
| Collection: | The University of Manchester: Research Explorer - Publications |
| Subject Terms: | Adaptivity, High-dimensional approximation, Multilevel methods, Sparse grids, Stochastic collocation, Uncertainty quantification |
| Description: | 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. |
| Document Type: | article in journal/newspaper |
| File Description: | application/pdf |
| Language: | English |
| DOI: | 10.1016/j.jcp.2020.109692 |
| Availability: | https://research.manchester.ac.uk/en/publications/c08deffa-734b-475f-a4f8-ac606656201f https://doi.org/10.1016/j.jcp.2020.109692 https://pure.manchester.ac.uk/ws/files/172960384/AMLUQ_LSS2020_final.pdf http://www.scopus.com/inward/record.url?scp=85087588525&partnerID=8YFLogxK |
| Rights: | info:eu-repo/semantics/openAccess |
| Accession Number: | edsbas.1029C3D1 |
| Database: | BASE |
Nájsť tento článok vo Web of Science | 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. |
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| DOI: | 10.1016/j.jcp.2020.109692 |