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Numerical Algorithms
Erschienen in: Numerical Algorithms 1/2021

10.07.2020 | Original Paper

Kernel-based interpolation at approximate Fekete points

verfasst von: Toni Karvonen, Simo Särkkä, Ken’ichiro Tanaka

Erschienen in: Numerical Algorithms | Ausgabe 1/2021

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Abstract

We construct approximate Fekete point sets for kernel-based interpolation by maximising the determinant of a kernel Gram matrix obtained via truncation of an orthonormal expansion of the kernel. Uniform error estimates are proved for kernel interpolants at the resulting points. If the kernel is Gaussian, we show that the approximate Fekete points in one dimension are the solution to a convex optimisation problem and that the interpolants converge with a super-exponential rate. Numerical examples are provided for the Gaussian kernel.
Vorheriger Artikel Derivation and analysis of computational methods for fractional Laplacian equations with absorbing layers
Nächster Artikel Correction to: Kernel-based interpolation at approximate Fekete points

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Fußnoten
1
Wendland [38, p. 192] goes as far as describing these results “almost pointless” for kernels, such as the Gaussian, that are associated with exponential rates of convergence.
 
2
Observe that in this section we begin indexing of the expansion from zero to simplify notation.
 
3
In one dimension, the convergence occurs for any points and most commonly used infinitely smooth radial kernels but in higher dimensions the Gaussian kernel is special in that it is the only known kernel for which convergence to a polynomial interpolant, of minimal degree in a certain sense, occurs for every point set.
 
4
This is the constant C in Theorem 6.1 of [24]. To derive the claimed bound, observe that this constant is given as C = 𝜖B/4 for \(B \leq \min \limits \{(b-a)/6, 1\}\) in their proof of Theorem 4.5. On p. 120, they show that 𝜖 = 1/2 if the kernel is Gaussian.
 
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Metadaten
Titel
Kernel-based interpolation at approximate Fekete points
verfasst von
Toni Karvonen
Simo Särkkä
Ken’ichiro Tanaka
Publikationsdatum
10.07.2020
Verlag
Springer US
Erschienen in
Numerical Algorithms / Ausgabe 1/2021
Print ISSN: 1017-1398
Elektronische ISSN: 1572-9265
DOI
https://doi.org/10.1007/s11075-020-00973-y

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