Josef Dick, Martin Ehler, Manuel Gräf and Christian Krattenthaler

Spectral decomposition of discrepancy kernels on the Euclidean ball, the special orthogonal group, and the Grassmannian manifold

(32 pages)

Abstract. To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of Rd. For restrictions to the Euclidean ball in odd dimensions, to the rotation group SO(3), and to the Grassmannian manifold G2,4, we compute the kernels' Fourier coefficients and determine their asymptotics. The L2-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 G2,4, the transform is derived here. We also provide numerical experiments for SO(3) and G2,4.

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