TUHH / Institut für Mathematik / Forschungsgebiete / Kernel-based reconstruction methods Englische Flagge

Kernel-based reconstruction methods

Working Groups: Lehrstuhl Angewandte Analysis

Collaborators (MAT): Kristof Albrecht, Prof. Dr. Marko Lindner

Collaborators (External): Armin Iske

Description

Given a data set \(X = \lbrace x_1,...,x_n \rbrace \subset \mathbb{R}^d\) (\(d \geq 2\)) and values \(f_1,...,f_n \in \mathbb{R}\), we want to find a function \(s\) such that

\[ f(x_i) = s(x_i) \qquad \forall i \in \lbrace 1,...,n \rbrace.\]

Since there are no Haar systems on \(\mathbb{R}^d\) ([1],[2]), we cannot simply use polynomials or similar tools from the one-dimensional case. One approach to this problem are positive definite (kernel) functions [3, Chapter 8]

\[K:\mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}.\]

For this type of functions, the interpolation problem has a unique solution if we restrict ourselves to

\[s \in \text{span} \lbrace K(\cdot,x_1),..., K(\cdot,x_n) \rbrace.\]

Our research on kernel-based reconstruction methods can be roughly divided into two areas:

References

[1] J. Mairhuber. On Haar?s theorem concerning Chebysheff problems having unique solutions, Proc. Am. Math. Soc. 7 (1956), 609?615

[2] P.C. Curtis. N-parameter families and bestapproximation, Pacific J. Math.9 (1959, 1013?1027

[3] A. Iske. Approximation Theory and Algorithms for Data Analysis. Springer International Publishing, 2018 (Texts in Applied Mathematics)

[4] H. Wendland. Scattered Data Approximation. Cambridge University Press, 2004 (Cam-bridge Monographs on Applied and Computational Mathematics)

[5] S. De Marchi, A. Iske, G. Santin. Image reconstruction from scattered Radon data by weighted positive definite kernel functions. In: Calcolo 55 (2018), 03

[6] S. Müller, R. Schaback. A Newton basis for Kernel spaces. In: Journal of Approximation Theory 161 (2009), S. 645?655

[7] S. De Marchi, R. Schaback, H. Wendland. Near-optimal data-independent point locations for radial basis function interpolation. In: Adv. Comput. Math. 23 (2005), S. 317? 330