29 Sep 2023
11:00  - 12:00

Seminar in Numerical Analysis: Rüdiger Kempf (U Bayreuth)

A Kernel-based Multilevel Approximation Method for Higher-Dimensional Problems on Sparse Grids

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 a possibility 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.


For further information about the seminar, please visit this webpage.

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