TY - INPR A1 - Ebner, Andrea A1 - Frikel, Jürgen A1 - Lorenz, Dirk A1 - Schwab, Johannes A1 - Haltmeier, Markus T1 - Regularization of inverse problems by filtered diagonal frame decomposition N2 - The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis. Y1 - 2020 UR - https://arxiv.org/pdf/2008.06219.pdf UR - https://applied-math.uibk.ac.at/images/Preprints/2021/pp_76_2021.pdf ER - TY - JOUR A1 - Ebner, Andrea A1 - Frikel, Jürgen A1 - Lorenz, D. A1 - Schwab, J. A1 - Haltmeier, Markus T1 - Regularization of inverse problems by filtered diagonal frame decomposition JF - Applied and Computational Harmonic Analysis N2 - Inverse problems are at the heart of many practical problems such as image reconstruction or nondestructive testing. A characteristic feature is their instability with respect to data perturbations. To stabilize the inversion process, regularization methods must be developed and applied. In this paper, we introduce the concept of filtered diagonal frame decomposition, which extends the classical filtered SVD to the case of frames. The use of frames as generalized singular systems allows a better match to a given class of potential solutions and is also beneficial for problems where the SVD is not analytically available. We show that filtered diagonal frame decompositions yield convergent regularization methods, derive convergence rates under source conditions and prove order optimality. Our analysis applies to bounded and unbounded forward operators. As a practical application of our tools, we study filtered diagonal frame decompositions for inverting the Radon transform as an unbounded operator on L2(R2). Y1 - 2023 U6 - https://doi.org/10.1016/j.acha.2022.08.005 N1 - Preprint unter: https://arxiv.org/pdf/2008.06219.pdf OPUS-Einrag unter: https://opus4.kobv.de/opus4-oth-regensburg/admin/document/edit/id/2290 VL - 62 IS - January SP - 66 EP - 83 PB - Elsevier ER -