Multiscale Radial Basis Functions

Abstract

Radial basis functions (RBFs) are a popular meshfree discretisation method for constructing high-order approximation spaces. They are used in various areas comprising, for example, scattered data approximation, computer graphics, machine learning, aeroelasticity and the geosciences.The approximation space is usually formed using the shifts of a fixed basis function. This simple approach makes it easy to construct approximation spaces of arbitrary smoothness and in arbitrary dimensions.Multiscale RBFs employ radial basis functions with compact support. In contrast to classical RBFs they do not only use the shifts of a fixed basis function but also vary the support radius in an orderly fashion. If done correctly, this leads to an extremely versatile and efficient approximation method.This paper will discuss recent developments concerning compactly supported radial basis functions, the basic ideas of multiscale RBFs and the principle ideas of analysing the convergence and stability of an explicit algorithm for the reconstruction of multivariate functions from scattered data. Recent results on data compression and on adaptivity are addressed.