Sparse grid quadrature on products of spheres

TL;DR

This paper analyzes sparse grid quadrature methods on products of spheres, introducing adaptive algorithms with proven optimal convergence rates, and discusses their practical performance in high-dimensional settings.

Contribution

It introduces a dimension adaptive quadrature algorithm for products of spheres and proves its optimality and convergence properties compared to existing methods.

Findings

The adaptive algorithm is optimal in cost and convergence rate.

Numerical results show slow initial convergence when weights decay slowly.

The methods are effective for high-dimensional integration problems.

Abstract

We examine sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere, in the case of worst case quadrature error for rules with arbitrary quadrature weights. We describe a dimension adaptive quadrature algorithm based on an algorithm of Hegland (2003), and also formulate a version of Wasilkowski and Wozniakowski's WTP algorithm (1999), here called the WW algorithm. We prove that the dimension adaptive algorithm is optimal in the sense of Dantzig (1957) and therefore no greater in cost than the WW algorithm. Both algorithms therefore have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wozniakowski (1999). A numerical example shows that, even though the asymptotic convergence rate is optimal, if the dimension weights decay slowly enough, and the dimensionality of the problem is large enough, the initial convergence of the dimension adaptive algorithm can be slow.