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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Fast evaluation of quadrature formulae on the sphere


Authors: Jens Keiner and Daniel Potts
Journal: Math. Comp. 77 (2008), 397-419
MSC (2000): Primary 65T99, 33C55, 42C10, 65T50
Published electronically: June 20, 2007
MathSciNet review: 2353959
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Abstract: Recently, a fast approximate algorithm for the evaluation of expansions in terms of standard $ \mathrm{L}^2\left(\mathbb{S}^2\right)$-orthonormal spherical harmonics at arbitrary nodes on the sphere $ \mathbb{S}^2$ has been proposed in [S. Kunis and D. Potts. Fast spherical Fourier algorithms. J. Comput. Appl. Math., 161:75-98, 2003]. The aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules.

We give a formulation in matrix-vector notation and an explicit factorisation of the spherical Fourier matrix based on the former algorithm. Starting from this, we obtain the corresponding factorisation of the adjoint spherical Fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere. We provide results of numerical tests showing the stability of the obtained algorithm using as examples classical Gauß-Legendre and Clenshaw-Curtis quadrature rules as well as the HEALPix pixelation scheme and an equidistribution.


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Additional Information

Jens Keiner
Affiliation: Institute of Mathematics, University of Lübeck, Wallstraße 40, 23560 Lübeck, Germany
Email: keiner@math.uni-luebeck.de

Daniel Potts
Affiliation: Department of Mathematics, Chemnitz University of Technology, Reichenhainer Straße 39, 09107 Chemnitz, Germany
Email: potts@mathematik.tu-chemnitz.de

DOI: http://dx.doi.org/10.1090/S0025-5718-07-02029-7
PII: S 0025-5718(07)02029-7
Keywords: Two-sphere, quadrature, nonequispaced fast spherical Fourier transform, NFFT, FFT
Received by editor(s): June 1, 2006
Published electronically: June 20, 2007
Article copyright: © Copyright 2007 American Mathematical Society