A Data Driven Approach to Multiscale Approximation with Anisotropically Scaled Radial Basis Functions

We study elliptic basis functions, which are a generalization of the class of well-known radial basis functions. We derive a purely data-driven way to determine the scaling matrix by using the principle component analysis method and combine the elliptic basis functions with a multilevel approximation scheme. We derive some results on the native spaces of such functions and the stability of the interpolation process. We give some numerical examples, showing that multilevel methods using elliptic basis functions can be superior to standard multilevel methods.