Approximation of two-variate functions: singular value decomposition versus sparse grids
Approximation of two-variate functions: singular value decomposition versus sparse grids
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dc.contributor.author | Griebel, Michael | |
dc.contributor.author | Harbrecht, Helmut | |
dc.date.accessioned | 2024-08-26T13:43:15Z | |
dc.date.available | 2024-08-26T13:43:15Z | |
dc.date.issued | 2011 | |
dc.identifier.uri | https://hdl.handle.net/20.500.11811/11958 | |
dc.description.abstract | We compare the cost complexities of two approximation schemes for functions f ∈ Hp(Ω1 × Ω2) 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. Here we assume that suitable finite element methods with associated fixed order r of accuracy are given on the domains Ω1 and Ω2. Then, the sparse grid approximation essentially needs only O(ε−q) with q = max{n1,n2}/r unknowns to reach a prescribed accuracy ε provided that the smoothness of f satisfies p ≥ rn1+n2/max{n1,n2} , which is an almost optimal rate. The singular value decomposition produces this rate only if f is analytical since otherwise the decay of the singular values is not fast enough. If p < rn1+n2/max{n1,n2} , then the sparse grid approach gives essentially the rate O(ε−q) with q = n1,n2/p while, for the singular value decomposition, we can only prove the rate O(ε−q) with q = 2min{r,p}min{n1,n2}+2pmax{n1,n2}/ (2p−min{n1,n2})min{r,p} . We derive the resulting complexities, compare the two approaches and present numerical results which demonstrate that these rates are also achieved in numerical practice. | en |
dc.format.extent | 28 | |
dc.language.iso | eng | |
dc.relation.ispartofseries | INS Preprints ; 1109 | |
dc.rights | In Copyright | |
dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | |
dc.subject | singular value decomposition | |
dc.subject | sparse grids | |
dc.subject | complexity | |
dc.subject.ddc | 510 Mathematik | |
dc.subject.ddc | 518 Numerische Analysis | |
dc.title | Approximation of two-variate functions: singular value decomposition versus sparse grids | |
dc.type | Preprint | |
dc.publisher.name | Institut für Numerische Simulation (INS) | |
dc.publisher.location | Bonn | |
dc.rights.accessRights | openAccess | |
dc.relation.doi | https://doi.org/10.1093/imanum/drs047 | |
ulbbn.pubtype | Zweitveröffentlichung | |
dcterms.bibliographicCitation.url | https://ins.uni-bonn.de/publication/preprints |
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