Prof. Ian H. Sloan

Fellow of the Australian Academy of Science
Scientia Professor, University of New South Wales
Chair Professor, Hong Kong Polytechnic University
Past President, International Council for Industrial and Applied Mathematics

Photos

DLB802, David C. Lam Building,
Shaw Campus, Hong Kong Baptist University

Many researchers have discussed approximation by radial basis functions on a sphere, using scattered data. Usually there is no polynomial component in such approximations if, as here, the kernel that generates the radial functions is (strictly) positive definite. On the other hand, the utility of polynomials for approximating slowly varying components is well known - an extreme case is the NASA model of the earth's gravitational potential, which represents the potential by a purely polynomial approximation of high degree. In this joint work with Alvise Sommariva we consider a hybrid approximation, in which there is a radial basis functions component to handle the rapidly varying and localised aspects, but also a polynomial component to handle the more slowly varying and global parts. The convergence theory (including a doubled rate of convergence for sufficiently smooth functions) make use of the "native space" associated with the positive definite kernel (with no polynomial involvement in the definition). A numerical experiment for a simple model with a geophysical flavour establishes the potential value of the hybrid approach.