Abstract

We start by deriving a new variant of the iteratively regularized Landweber iteration for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which forms the core of the new iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. In the second part of the talk we study the solution of linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be implemented without making use of the forward operator. Convergence and stability of the regularized solutions are studied in view of a famous non-convergence statement of Seidman. We show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform. This is joint work with A. Aspri, S. Banert, L. Frischauf, Y. Korolev, O. Öktem.