Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold.

Autor: Dick J; School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052 Australia., Ehler M; Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria., Gräf M; Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany., Krattenthaler C; Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Jazyk: angličtina
Zdroj: Constructive approximation [Constr Approx] 2023; Vol. 57 (3), pp. 983-1026. Date of Electronic Publication: 2023 Apr 07.
DOI: 10.1007/s00365-023-09638-0
Abstrakt: To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of R d . For restrictions to the Euclidean ball in odd dimensions, to the rotation group SO ( 3 ) , and to the Grassmannian manifold G 2 , 4 , we compute the kernels' Fourier coefficients and determine their asymptotics. The L 2 -discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For SO ( 3 ) , the nonequispaced fast Fourier transform is publicly available, and, for G 2 , 4 , the transform is derived here. We also provide numerical experiments for SO ( 3 ) and G 2 , 4 .
(© The Author(s) 2023.)
Databáze: MEDLINE
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