The Multilevel Kernel-method for Compactly Supported RBFs and its Application in High-dimensional Approximation

Speaker: Rüdiger Kempf, University of Basel

Abstract: Reproducing kernel Hilbert spaces (RKHSs) and the closely related kernel methods are well-established and well-studied tools in classical approximation
theory. More recently, they see many uses in other problems in applied and numerical analysis. In machine learning, support vector machines heavily rely on RKHSs. For
neural networks Barron spaces are connected to certain RKHSs and offer apossibility for a theoretical analysis of these networks. Another application of RKHSs is in high(er)-dimensional approximation. For instance in the field of quasi Monte-Carlo methods, kernel-techniques are used to derive an error analysis for high-dimensional quadrature rules. We also developed a novel kernel-based approximation method for higher-dimensional meshfree function reconstruction, based on Smolyak operators. In this talk I will provide an introduction into the theory of RKHSs, their kernels and associated kernel methods. In particular, I will focus on a multiscale approximation scheme for rescaled radial basis functions. This method will then be used to derive the new tensor product multilevel method for higher- dimensional meshfree approximation, which I will discuss in detail.

Biography: Rüdiger Kempf studied Technomathematik at the University of Bayreuth. Following the completion of his Master’s program, he started his PhD research at the chair of Prof. Holger Wendland in Bayreuth. In 2023, he successfully defended his doctoral thesis and continued his research in Bayreuth. As of February 2024, he is conducting research at the University of Basel, collaborating with Prof. Helmut Harbrecht during a dedicated research semester.

Host: Prof. Michael Multerer