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Kernel-based interpolation at approximate Fekete points

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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.

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Notes

  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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Funding

T. Karvonen was supported by the Aalto ELEC Doctoral School and the Lloyd’s Register Foundation programme on data-centric engineering at the Alan Turing Institute, UK. This research was partially carried out while he was visiting the University of Tokyo, funded by the Finnish Foundation for Technology Promotion and Oskar Öflunds Stiftelse. S. Särkkä was supported by the Academy of Finland. K. Tanaka was supported by the grant-in-aid of Japan Society of the Promotion of Science with KAKENHI Grant Number 17K14241.

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Authors and Affiliations

  1. Department of Electrical Engineering and Automation, Aalto University, Espoo, Finland

    Toni Karvonen & Simo Särkkä

  2. The Alan Turing Institute, London, UK

    Toni Karvonen

  3. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan

    Ken’ichiro Tanaka

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  1. Toni Karvonen
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  2. Simo Särkkä
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  3. Ken’ichiro Tanaka
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Correspondence to Toni Karvonen.

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The original online version of this article was revised: The corrected equations in Sections 2.1, 2.2, 3.2, 3.3 and 4.3.

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Karvonen, T., Särkkä, S. & Tanaka, K. Kernel-based interpolation at approximate Fekete points. Numer Algor 87, 445–468 (2021). https://doi.org/10.1007/s11075-020-00973-y

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