The Runge phenomenon and spatially variable shape parameters in RBF interpolation

Many studies, mostly empirical, have been devoted to finding an optimal shape parameter for radial basis functions (RBF). When exploring the underlying factors that determine what is a good such choice, we are led to consider the Runge phenomenon (RP; best known in cases of high order polynomial interpolation) as a key error mechanism. This observation suggests that it can be advantageous to let the shape parameter vary spatially, rather than assigning a single value to it. Benefits typically include improvements in both accuracy and numerical conditioning. Still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas. This idea is routinely used when working with splines or finite element methods. However, local refinement with RBFs may cause RP-type errors unless we use a spatially variable shape paremeter. With this enhancement, RBF approximations combine freedom from meshes with spectral accuracy on irregular domains, and furthermore permit local node clustering to improve the resolution wherever this might be needed.