"Frame-based regularization of inverse problems"
Frikel, JürgenIn this talk, we introduce and analyze the concept of filtered diagonal frame decomposition of operators. This new regularization framework extends the classical filter methods that are based on the singular value decomposition. As one main advantage, we explain that the use of frames (as generalized singular systems) allows for a better adaption to a given class of potential solutions of the inverse problem. This is also beneficial for problems where the SVD is not available analytically. We show that filtered diagonal frame decompositions provide convergent regularization methods and derive convergence rates under source conditions and prove order optimality when the frame under consideration is a Riesz basis. Our analysis applies to unbounded and bounded 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).