Seminar in Numerical Analysis: Christian Rieger (Universität Bonn)

Reproducing kernel methods

In this talk, we will briefly discuss a general methodology of approximation algorithms based on reproducing kernels and their associated Hilbert spaces. We will outline how reproducing kernels naturally arise in many reconstruction problems.

Furthermore, we will present a deterministic a priori (often exponential) convergence analysis via sampling inequalities which can be employed to analyze a large class of regularized reconstruction schemes.

Such an analysis enables us to derive a priori couplings of various discretization and regularization parameters. Such parameters can range from iteration numbers in numerical linear algebra, numerical evaluation of input parameters to rounding errors.

An important issue is the choice of the reproducing kernel. We will discuss some implications of such choices and address the problem of approximating the solution of a parametric partial differential equation using problem adapted kernels.

This is partly based on joint work with M. Griebel and B. Zwicknagl (both Bonn University).

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