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Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation. (English) Zbl 1436.31015

Summary: It is known that discrete sets of uniformly distributed points on the hypersphere \(\mathbb{S}^d\subset \mathbb{R}^{d+1}\) can be obtained from minimizing the energy functional corresponding to Riesz \(s\)-kernels \(k_s(\boldsymbol{x},\boldsymbol{y})=\lVert \boldsymbol{x}-\boldsymbol{y}\rVert^{-s}\) \((s > 0)\) or the logarithmic kernel \(k_{\log }(\boldsymbol{x},\boldsymbol{y})=-\log \lVert \boldsymbol{x}-\boldsymbol{y}\rVert +\log 2\). We prove the same for the kernel \(k_{\operatorname{log}}(\boldsymbol{x},\boldsymbol{y})=\lVert \boldsymbol{x}-\boldsymbol{y}\rVert (\log{\frac{\lVert \boldsymbol{x}-\boldsymbol{y}\rVert }{2}}-1)+2\) which is a front-extension of the sequence of derivatives \(k_{\log }, k_1, k_2, k_3, \dots \), up to sign and constants. The boundedness of the kernel simplifies the classical potential-theoretical proof of the asymptotic uniformity of the point distributions. Still, the property of a singular derivative for \(x \rightarrow y\) is preserved, with the physical interpretation of infinite repulsive forces for touching particles. The quality of the resulting point distributions is exemplary compared with that of Riesz- and classical logarithmic point sets, and found to be competitive. Originally motivated by problems of high-dimensional data, the applicability of LOG-optimal point sets with a novel concentric interpolation and differentiation scheme is demonstrated. The method is significantly optimized by the introduction of symmetrized kernels for both the generation of the minimum energy points and the spherical basis functions. Both the point generation and the Concentric Interpolation software are available as Open Source software and selected point sets are provided.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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