Kernel Based Quadrature on Spheres and Other Homogeneous Spaces

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.

31 pages, 8 figures

References in corpus (5)

• Measurement of areas on a sphere using Fibonacci and latitude-longitude lattices
• Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant
• Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation
• A Study of Different Modeling Choices For Simulating Platelets Within the Immersed Boundary Method
• Low complexity methods for discretizing manifolds via Riesz energy minimization