Paper Info
Title
Convergence of quasi-optimal sparse-grid approximation of Hilbert-space-valued functions: application to random elliptic PDEs
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.
Year
DOI
Venue
2016
10.1007/s00211-015-0773-y
Numerische Mathematik
Keywords
Field
DocType
sparse grids,uncertainty quantification
Mathematical optimization,Polynomial,Polynomial interpolation,Mathematical analysis,Sparse approximation,Rate of convergence,Knapsack problem,Collocation method,Sparse grid,Grid,Mathematics
Journal
Volume
Issue
ISSN
134
2
0945-3245
Citations 
PageRank 
References 
6
0.54
11
Authors
3
Name
Order
Citations
PageRank
Fabio Nobile133629.63
Lorenzo Tamellini2293.22
Raul Tempone3111.44