Multiscale approximation and reproducing kernel Hilbert space methods
Multiscale approximation and reproducing kernel Hilbert space methods
| dc.contributor.author | Griebel, Michael | |
| dc.contributor.author | Rieger, Christian | |
| dc.contributor.author | Zwicknagl, Barbara | |
| dc.date.accessioned | 2024-08-23T07:14:18Z | |
| dc.date.available | 2024-08-23T07:14:18Z | |
| dc.date.issued | 2013 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.11811/11928 | |
| dc.description.abstract | We consider reproducing kernels K : Ω x Ω → ℝ in multiscale series expansion form, i.e., kernels of the form K (x, y) = ∑ℓ∈ℕλℓ∑j∈IℓΦℓ(x)Φℓ(y) with weights λℓ and structurally simple basis functions {Φℓ,i}. Here, we deal with basis functions such as polynomials or frame systems, where, for ℓ ∈ ℕ, the index set Iℓ is finite or countable. We derive relations between approximation properties of spaces based on basis functions {Φℓ,j : 1 ≤ ℓ ≤ L,j ∈ Ij} and spaces spanned by translates of the kernel span{K(x1,·), . . . , K(xN,·)} with XN := {x1, . . . ,XN } ⊂ Ω if the truncation index L is appropriately coupled to the discrete set XN . An analysis of a numerically feasible approximation from trial spaces span{KL(x1,·), . . . , KL(xN,·)} based on finitely truncated series kernels of the form KL (x, y) := ∑Lℓ=1λℓ∑j∈IℓΦℓ(x)Φℓ(y) is provided where the truncation index L is chosen sufficiently large depending on the point set XN . Furthermore, Bernstein-type inverse estimates and derivative-free sampling inequalities for kernel based spaces are obtained from estimates for spaces based on the basis functions {Φℓ,j : 1 ≤ ℓ ≤ L,j ∈ Ij}. | en |
| dc.format.extent | 22 | |
| dc.language.iso | eng | |
| dc.relation.ispartofseries | INS Preprints ; 1312 | |
| dc.rights | In Copyright | |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | |
| dc.subject | Reproducing kernel Hilbert spaces | |
| dc.subject | multiscale expansion | |
| dc.subject | a priori error estimates | |
| dc.subject | Bernstein estimates | |
| dc.subject.ddc | 510 Mathematik | |
| dc.subject.ddc | 518 Numerische Analysis | |
| dc.title | Multiscale approximation and reproducing kernel Hilbert space methods | |
| 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/10.1137/130932144 | |
| ulbbn.pubtype | Zweitveröffentlichung | |
| dcterms.bibliographicCitation.url | https://ins.uni-bonn.de/publication/preprints |
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