Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs
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| Title: | Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs |
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| Authors: | Nobile, Fabio, Tamellini, Lorenzo, Tempone, Raul |
| Publisher Information: | Springer |
| Publication Year: | 2014 |
| Collection: | Ecole Polytechnique Fédérale Lausanne (EPFL): Infoscience |
| Subject Terms: | Uncertainty Quantification, random PDEs, linear elliptic equations, multivariate polynomial approximation, best M-terms polynomial approximation, Smolyak approximation, Sparse grids, Stochastic Collocation method |
| Description: | In this work we provide a convergence analysis for the quasi-optimal version of the sparse-grids stochastic collocation method we presented in a previous work: “On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods” (Beck et al., Math Models Methods Appl Sci 22(09), 2012). The construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid. The convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits. This is a very general argument that can be applied to sparse grids built with any uni-variate family of points, both nested and non-nested. As an example, we apply such quasi-optimal sparse grids to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the “inclusions problem”: we detail the convergence estimates obtained in this case using polynomial interpolation on either nested (Clenshaw-Curtis) or non-nested (Gauss-Legendre) abscissas, verify their sharpness numerically, and compare the performance of the resulting quasi-optimal grids with a few alternative sparse-grid construction schemes recently proposed in the literature. ; CSQI |
| Document Type: | article in journal/newspaper |
| Language: | unknown |
| ISSN: | 0029-599X 0945-3245 |
| Relation: | https://infoscience.epfl.ch/record/196966/files/2016_Nobile_Tamellini_Tempone_NM_Convergence.pdf; Numerische Mathematik; #PLACEHOLDER_PARENT_METADATA_VALUE#; https://infoscience.epfl.ch/handle/20.500.14299/100963; WOS:000382146600005 |
| DOI: | 10.1007/s00211-015-0773-y |
| Availability: | https://doi.org/10.1007/s00211-015-0773-y https://infoscience.epfl.ch/handle/20.500.14299/100963 https://hdl.handle.net/20.500.14299/100963 |
| Accession Number: | edsbas.5FEE5760 |
| Database: | BASE |
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Nájsť tento článok vo Web of Science | Abstract: | In this work we provide a convergence analysis for the quasi-optimal version of the sparse-grids stochastic collocation method we presented in a previous work: “On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods” (Beck et al., Math Models Methods Appl Sci 22(09), 2012). The construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid. The convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits. This is a very general argument that can be applied to sparse grids built with any uni-variate family of points, both nested and non-nested. As an example, we apply such quasi-optimal sparse grids to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the “inclusions problem”: we detail the convergence estimates obtained in this case using polynomial interpolation on either nested (Clenshaw-Curtis) or non-nested (Gauss-Legendre) abscissas, verify their sharpness numerically, and compare the performance of the resulting quasi-optimal grids with a few alternative sparse-grid construction schemes recently proposed in the literature. ; CSQI |
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| ISSN: | 0029599X 09453245 |
| DOI: | 10.1007/s00211-015-0773-y |