@article{EbnerFrikelLorenzetal., author = {Ebner, Andrea and Frikel, J{\"u}rgen and Lorenz, D. and Schwab, J. and Haltmeier, Markus}, title = {Regularization of inverse problems by filtered diagonal frame decomposition}, series = {Applied and Computational Harmonic Analysis}, volume = {62}, journal = {Applied and Computational Harmonic Analysis}, number = {January}, publisher = {Elsevier}, doi = {10.1016/j.acha.2022.08.005}, pages = {66 -- 83}, abstract = {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).}, language = {en} }