Learning radial basis functions from the German school: A personal perspective

Radial Basis Functions (RBFs) are a powerful tool for scattered data approximation, interpolation, and the numerical solution of partial differential equations. Over the past decades, a large body of work originating from the so-called German school of RBFs, led by Robert Schaback and collaborators, has made significant contributions to the theoretical understanding and practical development of kernel-based approximation methods. This paper presents a personal perspective on some of the ideas and developments that originated within this community and have influenced subsequent research directions. In particular, we discuss greedy algorithms for center selection, stability issues related to the shape parameter, as well as the choice of the centers inside or along the boundaries of the domain, rational kernel interpolation, and the framework of variably scaled kernels. These topics represent, in the author’s view, the most interesting contributions to the kernels community, made possible by the fruitful relationships with the German school of RBFs that has blossomed from Robert Schaback. This manuscript also tries to illustrate how theoretical insights, gained in kernel approximation, have led to powerful tools in numerical analysis, machine learning, and scientific computing.