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 S n , oblate spheroids for instance. The weights are obtained by solving a single linear system. For 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.

A Galerkin Radial Basis Function Method for Nonlocal Diffusion

“…where k D R '.kx x k k/dx and Á l D R p l .x/dx; see the paper [7] for further details. This choice of quadrature scheme has important practical considerations; see Sect.…”

“…A practical quadrature method is realized by a slight modification of a recent scheme proposed by Fuselier, Hangelbroek, Narcowich, Ward, and Wright [7]. The necessary quadrature weights w j , see (8), can be constructed by solving the linear system…”

“…There is evidence that the Lagrange functions for the thin plate splines can be replaced with local Lagrange functions, which are cheaper to construct than the Lagrange functions. The local property has been recently proven for spheres, and current work investigates these methods for domains in R n [7,8,10]. The local property is due to results on both the decay of the Lagrange functions away from their centers and the decay of the coefficients of the Lagrange functions.…”

“…In this note, we propose a Galerkin radial basis function (RBF) method to numerically solve (2). We also exploit a recently developed Lagrange function quadrature scheme [7] for the necessary integrations. Consequently, our proposed method requires only information at the radial basis function nodes and also yields a straight forward assembly of a sparse stiffness matrix.…”

RBF-vortex methods for the barotropic vorticity equation on a sphere

“…With M windows, an O (N 3 ) global problem is replaced by M subproblems of size O (N/M) at a cost that is O (N 3 /M 2 ). Fuselier et al[35][36][37] and Cavoretto and De Rossi[19] have made a good start at applying PUM, but there are many practicalities still in flux. What is the optimum subdomain size?…”