Singular value decomposition versus sparse grids: Refined complexity estimates
Singular value decomposition versus sparse grids: Refined complexity estimates
View/Date of Publication
2017Publisher
Institut für Numerische Simulation (INS)Publisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11834
Abstract
We compare the cost complexities of two approximation schemes for functions which live on the product domain Ω1 × Ω2 of sufficiently smooth domains Ω1 ⊂ Rn1 and Ω2 ⊂ Rn2 , namely the singular value / Karhunen-Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders r1 and r2 of accuracy are given on the domains Ω1 and Ω2, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from Griebel & Harbrecht (2014). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.
Subjects
singular value decomposition, sparse grids, complexity
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1093/imanum/dry039Part of Series
INS Preprints ; 1708Griebel, Michael; Harbrecht, Helmut: Singular value decomposition versus sparse grids: Refined complexity estimates. In: INS Preprints, 1708.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11834
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11834
@unpublished{handle:20.500.11811/11834,
author = {{Michael Griebel} and {Helmut Harbrecht}},
title = {Singular value decomposition versus sparse grids: Refined complexity estimates},
publisher = {Institut für Numerische Simulation (INS)},
year = 2017,
INS Preprints},
volume = 1708,
note = {We compare the cost complexities of two approximation schemes for functions which live on the product domain Ω1 × Ω2 of sufficiently smooth domains Ω1 ⊂ Rn1 and Ω2 ⊂ Rn2 , namely the singular value / Karhunen-Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders r1 and r2 of accuracy are given on the domains Ω1 and Ω2, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from Griebel & Harbrecht (2014). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.},
url = {https://hdl.handle.net/20.500.11811/11834}
}
author = {{Michael Griebel} and {Helmut Harbrecht}},
title = {Singular value decomposition versus sparse grids: Refined complexity estimates},
publisher = {Institut für Numerische Simulation (INS)},
year = 2017,
INS Preprints},
volume = 1708,
note = {We compare the cost complexities of two approximation schemes for functions which live on the product domain Ω1 × Ω2 of sufficiently smooth domains Ω1 ⊂ Rn1 and Ω2 ⊂ Rn2 , namely the singular value / Karhunen-Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders r1 and r2 of accuracy are given on the domains Ω1 and Ω2, respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from Griebel & Harbrecht (2014). Especially, we consider the approximation of functions from generalized isotropic and anisotropic Sobolev spaces.},
url = {https://hdl.handle.net/20.500.11811/11834}
}
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