22-24 September 2021
Alpen-Adria-Universität Klagenfurt
Europe/Vienna timezone

Frame Decompositions and Inverse Problems

23 Sep 2021, 10:50
20m
HS 1 (Alpen-Adria-Universität Klagenfurt)

HS 1

Alpen-Adria-Universität Klagenfurt

Speaker

Dr Simon Hubmer (Johann Radon Institute Linz)

Description

The singular-value decomposition (SVD) is an important tool for the analysis and solution of linear ill-posed problems in Hilbert spaces. However, it is often difficult to derive the SVD of a given operator explicitly, which limits its practical usefulness. An alternative in these situations are frame decompositions (FDs), which are a generalization of the SVD based on suitably connected families of functions forming frames. Similar to the SVD, these FDs encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. Crucially though, FDs can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In this talk, we consider various theoretical aspects of FDs such as recipes for their construction and some properties of the reconstruction formulae induced by them. Furthermore, we present convergence and convergence rates results for continuous regularization methods based on FDs under both a-priori and a-posteriori parameter choice rules. Finally, we consider the practical utility of FDs for solving inverse problems by considering two numerical examples from computerized and atmospheric tomography.

Primary author

Dr Simon Hubmer (Johann Radon Institute Linz)

Co-authors

Prof. Ronny Ramlau (Johannes Kepler University Linz) Mr Lukas Weissinger (Johannes Kepler University Linz)

Presentation Materials

There are no materials yet.