Adaptive thinning of centers for approximation of a large data set by radial functions
Given a bounded set \(D\subset \mathbb{R}^d\), \(d\in \mathbb N\), and a (large) set of centers or points \(\Xi\subset D\), the authors investigate the problem of how to find a small subset \(Y_J\) of \(\Xi\) so that the least squares radial function approximation will be within a given tolerance. The solution the authors offer is based on an adaptive thinning strategy which removes in a greedy way less significant centers in order to minimize the anticipated error. Instead of using the anticipated error, the authors replace it by the new concept of ``predicting functional''. This is a functional which chooses with high probability the same centers as an anticipated error. Several such predicting functionals are derived for specific radial functions. Numerical tests supporting the theoretical findings are also presented.
scientific article; zbMATH DE number 6485851