%0 Journal Article %1 kunc2019b %A Kunc, Oliver %A Fritzen, Felix %D 2019 %J Advances in Computational Mathematics %K emma exc310 heisenberg myown %N 5-6 %P 3012-3065 %R 10.1007/s10444-019-09726-5 %T Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation %U https://rdcu.be/b0Fj2 %V 45 %X It is known that discrete sets of uniformly distributed points on the hypersphere Sd⊂ℝd+1\$\backslashmathbb \S\^\d\\backslashsubset \backslashmathbb \R\^\d+1\\$ can be obtained from minimizing the energy functional corresponding to Riesz s-kernels ks(x,y)=∥x−y∥−s\$k\_s(\backslashboldsymbol \x\,\backslashboldsymbol \y\)=\backslashlVert \backslashboldsymbol \x\-\backslashboldsymbol \y\\backslashrVert ^\-s\\$ (s >þinspace0) or the logarithmic kernel klog(x,y)=−log∥x−y∥+log2\$k\_\\backslashlog \(\backslashboldsymbol \x\,\backslashboldsymbol \y\)=-\backslashlog \backslashlVert \backslashboldsymbol \x\-\backslashboldsymbol \y\\backslashrVert +\backslashlog 2\$. We prove the same for the kernel klog(x,y)=∥x−y∥(log∥x−y∥2−1)+2\$k\_\\backslashlog \(\backslashboldsymbol \x\,\backslashboldsymbol \y\)=\backslashlVert \backslashboldsymbol \x\-\backslashboldsymbol \y\\backslashrVert (\backslashlog \\backslashfrac \\backslashlVert \backslashboldsymbol \x\-\backslashboldsymbol \y\\backslashrVert \\2\\-1)+2\$ which is a front-extension of the sequence of derivatives klog,k1,k2,k3,łdots\$k\_\\backslashlog \, k\_\1\, k\_\2\, k\_\3\, \backslashdots \$, 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 → 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\$\backslashlog \$-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.

@article{kunc2019b, abstract = {It is known that discrete sets of uniformly distributed points on the hypersphere Sd⊂ℝd+1{\$}{\backslash}mathbb {\{}S{\}}^{\{}d{\}}{\backslash}subset {\backslash}mathbb {\{}R{\}}^{\{}d+1{\}}{\$} can be obtained from minimizing the energy functional corresponding to Riesz s-kernels ks(x,y)=∥x−y∥−s{\$}k{\_}s({\backslash}boldsymbol {\{}x{\}},{\backslash}boldsymbol {\{}y{\}})={\backslash}lVert {\backslash}boldsymbol {\{}x{\}}-{\backslash}boldsymbol {\{}y{\}}{\backslash}rVert ^{\{}-s{\}}{\$} (s >{\thinspace}0) or the logarithmic kernel klog(x,y)=−log∥x−y∥+log2{\$}k{\_}{\{}{\backslash}log {\}}({\backslash}boldsymbol {\{}x{\}},{\backslash}boldsymbol {\{}y{\}})=-{\backslash}log {\backslash}lVert {\backslash}boldsymbol {\{}x{\}}-{\backslash}boldsymbol {\{}y{\}}{\backslash}rVert +{\backslash}log 2{\$}. We prove the same for the kernel klog(x,y)=∥x−y∥(log∥x−y∥2−1)+2{\$}k{\_}{\{}{\backslash}log {\}}({\backslash}boldsymbol {\{}x{\}},{\backslash}boldsymbol {\{}y{\}})={\backslash}lVert {\backslash}boldsymbol {\{}x{\}}-{\backslash}boldsymbol {\{}y{\}}{\backslash}rVert ({\backslash}log {\{}{\backslash}frac {\{}{\backslash}lVert {\backslash}boldsymbol {\{}x{\}}-{\backslash}boldsymbol {\{}y{\}}{\backslash}rVert {\}}{\{}2{\}}{\}}-1)+2{\$} which is a front-extension of the sequence of derivatives klog,k1,k2,k3,{\ldots}{\$}k{\_}{\{}{\backslash}log {\}}, k{\_}{\{}1{\}}, k{\_}{\{}2{\}}, k{\_}{\{}3{\}}, {\backslash}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 {\textrightarrow} 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{\$}{\backslash}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.}, added-at = {2020-01-22T13:22:21.000+0100}, author = {Kunc, Oliver and Fritzen, Felix}, biburl = {https://puma.ub.uni-stuttgart.de/bibtex/275ad649e7559b0a59106353c6bbb4ae8/ffritzen}, doi = {10.1007/s10444-019-09726-5}, interhash = {214d9184a691ecfd408804dc32e1c85b}, intrahash = {75ad649e7559b0a59106353c6bbb4ae8}, journal = {Advances in Computational Mathematics}, keywords = {emma exc310 heisenberg myown}, note = {Springer ShareIt Link https://rdcu.be/b0Fj2}, number = {5-6}, pages = {3012-3065}, timestamp = {2020-01-23T08:18:35.000+0100}, title = {Generation of energy-minimizing point sets on spheres and their application in mesh-free interpolation and differentiation}, url = {https://rdcu.be/b0Fj2}, volume = 45, year = 2019 }