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Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces. (English) Zbl 1459.65020

Summary: We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields \(n\) points at one sitting for a given integer \(n\). In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the \(P\)-greedy algorithm, which is known to provide nearly optimal points.

MSC:

65D05 Numerical interpolation
41A05 Interpolation in approximation theory
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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