Grady Wright

Edward Fuselier, Grady Wright

In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a smooth, compact embedded submanifold $\M\subset \R^d$. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on $\M$. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in $\R^3$ and a two-dimensional torus.

Heather Wilber, Alex Townsend, Grady Wright

A collection of algorithms is described for numerically computing with smooth functions defined on the unit disk. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier sphere method with a structure-preserving variant of iterative Gaussian elimination that is shown to converge geometrically for certain analytic functions. This adaptive procedure is near-optimal in its sampling strategy, producing approximants that are stable for differentiation and facilitate the use of FFT-based algorithms in both variables. The low rank form of the approximants is especially useful for operations such as integration and differentiation, reducing them to essentially 1D procedures, and it is also exploited to formulate a new fast disk Poisson solver that computes low rank approximations to solutions. This work complements a companion paper (Part I) on computing with functions on the surface of the unit sphere.