The approximation with positive definite kernels is used in many fields of science and engineering. Furthermore, these kernels play an important role in machine learning, for the solution of partial differential equations and in geostatistics. Especially the approximation of data given on Riemannian manifolds has gained enormous importance in recent years, for example for global weather models. Strictly positive definite kernels allow the construction of approximations from arbitrarily distributed measurement data on the surface of the manifold. However, the theoretical basis of these kernels has not yet been sufficiently studied. This project aims to characterise the strictly positive definite kernels on Riemannian manifolds and to investigate their approximation properties. In later parts of the project, conditionally positive definite kernels will be characterised and investigated. Then the approximant is not only formed by linear combinations of the kernels but the addition of certain functions from finite-dimensional spaces is allowed. Another very promising goal is the study of kernels that are invariant with respect to certain subgroups of the isometry group of the Riemannian manifold, for example, reflections or rotations. Together, the results should allow users to identify or construct more suitable kernels and furthermore existing theories on specific manifolds are unified and extended.
DFG Programme
Research Grants
International Connection
Canada, United Kingdom
