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MATHICSE Technical Report : Convergence of quasi-optimal sparse grid approximation of Hilbert-valued functions: application to random elliptic PDEs

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Title: MATHICSE Technical Report : Convergence of quasi-optimal sparse grid approximation of Hilbert-valued functions: application to random elliptic PDEs
Authors: Nobile, Fabio, Tamellini, Lorenzo, Tempone, Raúl
Contributors: MATHICSE-Group
Publisher Information: MATHICSE
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Publication Year: 2019
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 Stochastic Sparse Grid Collocation method we had presented in our previous work \On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods" [6]. Here the construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hi- erarchical 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 argument is very gen- eral and 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 grid to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the \inclusions problem": we detail the conver- gence estimate obtained in this case, using polynomial interpolation on either nested (Clenshaw{Curtis) or non-nested (Gauss{Legendre) abscissas, verify its sharpness numerically, and compare the performance of the resulting quasi- optimal grids with a few alternative sparse grids construction schemes recently proposed in literature. ; CSQI ; MATHICSE Technical Report Nr. 12.2014 February 2014 (NEW: 09.2014)
Document Type: report
Language: unknown
Relation: https://infoscience.epfl.ch/record/263221/files/12-2014NEW_FN-LT-RT.pdf; #PLACEHOLDER_PARENT_METADATA_VALUE#; https://infoscience.epfl.ch/handle/20.500.14299/153701
DOI: 10.5075/epfl-MATHICSE-263221
Availability: https://doi.org/10.5075/epfl-MATHICSE-263221
https://infoscience.epfl.ch/handle/20.500.14299/153701
https://hdl.handle.net/20.500.14299/153701
Accession Number: edsbas.71A2987F
Database: BASE
Description
Abstract:In this work we provide a convergence analysis for the quasi-optimal version of the Stochastic Sparse Grid Collocation method we had presented in our previous work \On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods" [6]. Here the construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hi- erarchical 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 argument is very gen- eral and 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 grid to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the \inclusions problem": we detail the conver- gence estimate obtained in this case, using polynomial interpolation on either nested (Clenshaw{Curtis) or non-nested (Gauss{Legendre) abscissas, verify its sharpness numerically, and compare the performance of the resulting quasi- optimal grids with a few alternative sparse grids construction schemes recently proposed in literature. ; CSQI ; MATHICSE Technical Report Nr. 12.2014 February 2014 (NEW: 09.2014)
DOI:10.5075/epfl-MATHICSE-263221