What are 'good' points for local interpolation by radial basis functions?

Abstract : Radial basis function interpolation has an advantage over other methods in that the interpolation matrix is nonsingular under very weak conditions on the location of the interpolation points. However, we show that point location can have a significant effect on the performance of an approximation in certain cases. Specifically, we consider multiquadric and thin plate spline interpolation to small data sets where derivative estimates are required. Approximations of this type are important in the motion of unsteady interfaces in fluid dynamics. For data points in the plane, it is shown that interpolation to data on a circle can be related to the polynomial case. For scattered data on the sphere, a comparison is made with the results of Sloan and Womersley.