High-order numerical integration on the sphere and extremal point systems

Hesse, K and Sloan, I H (2004) High-order numerical integration on the sphere and extremal point systems. Journal of Computational Technologies, 9. pp. 4-12. ISSN 1560-7534

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Abstract

This paper reviews some recent developments in interpolation, interpolatory cubature, and high-order numerical integration on the sphere $S^2$. In interpolatory (polynomial) cubature the integrand is approximated by the interpolating polynomial (in the space $\\mathbb{P}_n$ of all spherical polynomials up to a fixed degree $n$) with respect to an appropriate pointset, and the integral of the interpolating polynomial is then evaluated exactly. Formulated in terms of Lagrange polynomials, this leads to a cubature rule with the weights given by integrals over the respective Lagrange polynomials. The quality of such cubature rules depends heavily on the pointset. In this paper we discuss so-called extremal pointsets, which are pointsets for which the determinant of the interpolation matrix for any basis of $\\mathbb{P}_n$ is maximal. Such extremal pointsets have very nice geometrical properties: the points are always well separated, and there are no large `holes' in the pointset, provided that the cubature weights turn out to be positive, which according to recent numerical experiments seems always to be the case. Finally, an asymptotic estimate for the worst-case error $e(H^s;Q_n)$ of a sequence $(Q_n)_{n\\in\\mathbb{N}}$ of cubature rules $Q_n$ in the Sobolev space $H^s(S^2)$, where $s&gt;1$, is discussed. The cubature rules are assumed only to have the properties that $Q_n$ is exact for spherical polynomials up to degree $n$ and that the weights are positive. The worst-case error $e(H^s;Q_n)$ in that situation is shown in recent work to be of the order $O(n^{-s})$. In particular, the result applies to positive weight cubature rules based on extremal systems.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Kerstin Hesse 06 Feb 2012 19:58 04 Apr 2012 11:24 http://sro.sussex.ac.uk/id/eprint/23278