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SUMMARY:Frame Decompositions and Inverse Problems
DTSTART;VALUE=DATE-TIME:20210923T085000Z
DTEND;VALUE=DATE-TIME:20210923T091000Z
DTSTAMP;VALUE=DATE-TIME:20211201T192328Z
UID:indico-contribution-846@conference2.aau.at
DESCRIPTION:Speakers: Simon Hubmer (Johann Radon Institute Linz)\nThe sing
ular-value decomposition (SVD) is an important tool for the analysis and s
olution of linear ill-posed problems in Hilbert spaces. However\, it is of
ten difficult to derive the SVD of a given operator explicitly\, which lim
its its practical usefulness. An alternative in these situations are frame
decompositions (FDs)\, which are a generalization of the SVD based on sui
tably connected families of functions forming frames. Similar to the SVD\,
these FDs encode information on the structure and ill-posedness of the pr
oblem and can be used as the basis for the design and implementation of ef
ficient numerical solution methods. Crucially though\, FDs can be derived
explicitly for a wide class of operators\, in particular for those satisfy
ing a certain stability condition. In this talk\, we consider various theo
retical aspects of FDs such as recipes for their construction and some pro
perties of the reconstruction formulae induced by them. Furthermore\, we p
resent convergence and convergence rates results for continuous regulariza
tion methods based on FDs under both a-priori and a-posteriori parameter c
hoice rules. Finally\, we consider the practical utility of FDs for solvin
g inverse problems by considering two numerical examples from computerized
and atmospheric tomography.\n\nhttps://conference2.aau.at/event/62/contri
butions/846/
LOCATION:Alpen-Adria-Universität Klagenfurt HS 1
URL:https://conference2.aau.at/event/62/contributions/846/
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