Better bases for radial basis function interpolation problems

Radial basis function interpolation involves two stages. The first is fitting, solving a linear system corresponding to the interpolation conditions. The second is evaluation. The systems occurring in fitting problems are often very ill-conditioned. Changing the basis in which the radial basis function space is expressed can greatly improve the conditioning of these systems resulting in improved accuracy, and in the case of iterative methods, improved speed, of solution. The change of basis can also improve the accuracy of evaluation by reducing loss of significance errors. In this paper new bases for the relevant space of approximants, and associated preconditioning schemes are developed which are based on Floater’s mean value coordinates. Positivity results and scale independence results are shown for schemes of a general type. Numerical results show that the given preconditioning scheme usually improves conditioning of polyharmonic spline and multiquadric interpolation problems in R2R2 and R3R3 by several orders of magnitude. The theory indicates that using the new basis elements (evaluated indirectly) for both fitting and evaluation will reduce loss of significance errors on evaluation. Numerical experiments confirm this showing that such an approach can improve overall accuracy by several significant figures.