Energy functionals, numerical integration and asymptotic equidistribution on the sphere.
type: Item
instance of: scholarly article
title: Energy functionals, numerical integration and asymptotic equidistribution on the sphere.
zbMATH Open document ID: 1043.65049
DOI: 10.1016/S0885-064X(02)00006-7
author: S. B. Damelin
author: Peter J. Grabner
published in: Journal of Complexity
publication date: 2003-08-19T00:00:00Z
review text: The authors study equally weighted quadrature rules for the numerical integration of continuous functions on \(d\)-spheres \(S^d\). The quadrature knots are choosen to minimize some energy functional \[ E(g, Z_N)= {1\over N} \sum^N_{\substack{ i,j=1\\ i\neq j}} g(\langle x_i, x_j\rangle) \] with respect to the associated point set \(Z_N\). A point set for which the minimal energy is attained, is called a \(g\)-minimal enery point set. Here the function \(g\) has to fulfill some admissibility conditions and typical admissible functions are \[ g^0_L(t)= {1\over 2}\log{1\over 1-t}- {1\over 2}\log 2 \] and \[ g^s_R(t)= {1\over 2^{s/2}(1-t)^{s/2}} \quad (s> 0) \] for the logarithmic energy and the energy corresponding to the Riesz potential \({1\over r^s}\), respectively. The authors prove some quadrature error estimates involving \(E(g, Z_N)\) for arbitrary point distributions \(Z_N\). For \(0\leq s< d\), they conclude that the \(g^s_R\)-minimal energy point sets are asymptotically equidistributed as \(N\to\infty\) and mention that this result is well known. For \(s= d\) they present explicit rates of convergence which had been open up to now and prove that \(g^d_R\)-minimal energy points are asymptotically equidistributed with rate \(1/\sqrt{\log N}\). [An estimate in theorem 4 is corrected.]
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 1967205
zbMATH Keywords: numerical integration
zbMATH Keywords: spherical cap discrepancy
zbMATH Keywords: equidistribution on spheres
zbMATH Keywords: minimal energy point sets on spheres
zbMATH Keywords: Riesz potential
zbMATH Keywords: point configurations on spheres
zbMATH Keywords: weighted quadrature rules
zbMATH Keywords: error estimates
zbMATH Keywords: convergence
reviewed by: Gabriele Steidl
MaRDI profile type: Publication
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full work available at URL: s0885-064x(02)00006-7
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