Multiscale approximation and reproducing kernel Hilbert space methods
Multiscale approximation and reproducing kernel Hilbert space methods
Dokument öffnen (783.8KB)Datum der Veröffentlichung
2013Verlag / herausgebende Einrichtung
Institut für Numerische Simulation (INS)Verlagsort
BonnMetadaten
Zur Langanzeige
Zitierbarer Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11928
Inhalt
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}.
Schlagwörter
Reproducing kernel Hilbert spaces, multiscale expansion, a priori error estimates, Bernstein estimates
Klassifikation (DDC)
510 Mathematik 518 Numerische AnalysisErstpublikation
https://ins.uni-bonn.de/publication/preprintsZugehörige Publikation(en)
https://doi.org/10.1137/130932144Reihe, Nummer
INS Preprints ; 1312Griebel, Michael; Rieger, Christian; Zwicknagl, Barbara: Multiscale approximation and reproducing kernel Hilbert space methods. In: INS Preprints, 1312.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11928
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11928
@unpublished{handle:20.500.11811/11928,
author = {{Michael Griebel} and {Christian Rieger} and {Barbara Zwicknagl}},
title = {Multiscale approximation and reproducing kernel Hilbert space methods},
publisher = {Institut für Numerische Simulation (INS)},
year = 2013,
INS Preprints},
volume = 1312,
note = {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}.},
url = {https://hdl.handle.net/20.500.11811/11928}
}
author = {{Michael Griebel} and {Christian Rieger} and {Barbara Zwicknagl}},
title = {Multiscale approximation and reproducing kernel Hilbert space methods},
publisher = {Institut für Numerische Simulation (INS)},
year = 2013,
INS Preprints},
volume = 1312,
note = {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}.},
url = {https://hdl.handle.net/20.500.11811/11928}
}
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