Kristof Albrecht

Kernel-based approximation methods for mathematical data analysis

Positive definite kernel functions are powerful tools for many relevant applications, including multivariate data fitting, numerical solutions of PDEs, or more recently for machine learning. Most of these kernel-based approaches are built on generalized approximation methods within the native reproducing kernel Hilbert space (RKHS) of the selected kernel function. A naive application of kernel-based approximations, however, often leads to ill-conditioned linear systems, which then dooms the approximation method to fail, mainly for the lack of their numerical stability. Therefore, stabilization techniques and preconditioners for kernel matrices are essentially required, especially in application scenarios where the data are critical, e.g., high-dimensional, heterogeneous, uncertain and/or noisy data.

In this project, we will first analyze kernel-based approximation methods and their generalizations to obtain more flexible and more powerful kernel methods for mathematical data analysis. As regards numerical stability, we will work on stabilization techniques. To this end, we will design orthonormal systems in the RKHS, which are relying on data-dependent greedy point removal schemes. Moreover, we will include problem-adapted preconditioners, where we will collaborate with and benefit from the RTG’s project S2. Finally, our results will take impact on the progress of the RTG’s project S1, which is inspired by specific medical applications in Magnetic Particle Imaging (MPI).

Advisor: Prof. Dr. Armin Iske

Co-Advisor: Prof. Dr. Marko Lindner

26.2.-03.03.2023
SIAM Conference on Computational Science and Engineering (CSE23), Amsterdam (NL)

21.-24.03.2022
YRM & CSE Workshop on Modeling, Simulation & Optimization of Fluid Dynamic Applications, Groß Schwansee (GER): "Greedy methods for kernel-based generalized interpolation and their application to computerized tomography"

22.-25.03.2021
Workshop on Modeling, Simulation & Optimization of Fluid Dynamic Applications  (Online) "Image reconstruction from scattered Radon data by weighted kernel functions"

15.-19.03.2021
GAMM Annual Conference 2020@21 (Online): "Image reconstruction from scattered Radon data by weighted kernel functions"

1.  Albrecht, K.,  Entzian, J., and  Iske, A.: "Anisotropic kernels for particle simulation". In: Modeling, Simulation and Optimization of Fluid Dynamic Application, pages 57-76, A. Iske and T. Rung (eds.), Springer, Heidelberg, 2023

2. Albrecht, K., Iske, A.: "Greedy algorithms for image approximation from scattered Radon data", PAMM 21, 2021