Kernel Based Quadrature on Spheres and Other Homogeneous Spaces

TL;DR

This paper develops stable, accurate, coordinate-independent quadrature formulas on spheres and homogeneous manifolds using invariant kernels, enabling efficient computation with large node sets.

Contribution

It introduces new stable quadrature formulas based on positive definite kernels, applicable to spheres and similar manifolds, with efficient computation methods for large node sets.

Findings

Quadrature formulas are accurate and stable with increasing nodes.

Stability results are new for all cases studied.

Efficient computation for large node sets on $ ext{S}^2$ using iterative techniques.

Abstract

Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate -- optimally so in many cases -- and stable under an increasing number of nodes and in the presence of noise, provided the set X of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to $\mathbb{S}^n$, oblate spheroids for instance. The weights are obtained by solving a single linear system. For $\mathbb{S}^2$, and the restricted thin plate spline kernel $r^2 \log r$, these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.