Sparse grid quadrature on products of spheres

Details Sparse grid quadrature on products of spheres Sparse grid quadrature on products of spheres

2012-02-26

arxiv.org/abs/1202.5710v1

Mathematics

Numerical Analysis

23 pages, 3 figures, submitted to Journal of Complexity

Scientific paper

This paper examines sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere. We describe a WTP quadrature algorithm based on an algorithm of Hegland, and also formulate a version of Wasilkowski and Wo\'zniakowski's WTP algorithm, here called the WW algorithm. We prove that our algorithm is optimal and therefore lower in cost than the WW algorithm, and therefore both algorithms have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wo\'zniakowski. Even so, the initial rate of convergence can be very slow, if the dimension weights decay slowly enough.

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