Recent research at Cambridge on radial basis functions

Abstract

Much of the research at Cambridge on radial basis functions during the last four years has addressed the solution of the thin plate spline interpolation equations in two dimensions when the number of interpolation points, n say, is very large. It has provided some techniques that will be surveyed because they allow values of n up to 10⁵, even when the positions of the points are general. A close relation between these techniques and Newton’s interpolation method is explained. Another subject of current research is a new way of calculating the global minimum of a function of several variables. It is described briefly, because it employs a semi-norm of a large space of radial basis functions. Further, it is shown that radial basis function interpolation minimizes this semi-norm in a way that is a generalisation of the well-known variational property of thin plate spline interpolation in two dimensions. The final subject is the deterioration in accuracy of thin plate spline interpolation near the edges of finite grids. Several numerical experiments in the one-dimensional case are reported that suggest some interesting conjectures that are still under investigation.