Convergence results for variational regularization and Landweber iteration under heuristic rules

Abstract

We present general convergence results on the variational regularization and the Landweber iteration for inverse problems. First, we consider the variational regularization for inverse problems in a general form. We propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level and is therefore purely data driven. Under variational source conditions, we obtain a posteriori error estimates. A variant of the same heuristic rule is also proposed for the Landweber iteration for solving linear and as well as nonlinear inverse problems in Banach spaces. We present a new convergence result which requires neither the Gateaux differentiability of the forward operator nor the reflexivity of the image space. To address the slow convergence, we also present a new convergence result under the heuristic rule for an accelerated Landweber iteration for linear inverse problems in Hilbert spaces. According to the Bakushinskii veto, convergence in the worst case scenario can not be expected in general. However, by imposing certain conditions on the random noise, we establish convergence results for the heuristic rule. Applications of the results are addressed and numerical simulations are reported.