Sparse grid quadrature on products of spheres

This paper examines sparse grid quadrature on weighted tensor products (wtp) of reproducing kernel Hilbert spaces on products of the unit sphere S2 . We describe a wtp quadrature algorithm based on an algorithm of Hegland [1], and also formulate a version of Wasilkowski and Wo´zniakowski’s wtp algorithm [2], 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 [2]. Even so, the initial rate of convergence can be very slow, if the dimension weights decay slowly enough.

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Markus Hegland, Paul Leopardi
• Keywords: Asymptotic Rate, CORR 2012, Education, Kernel Hilbert Spaces, Unit Sphere