Iterative choices of regularization parameters in linear inverse problems

We investigate possibilities of choosing reasonable regularization parameters for the output least squares formulation of linear inverse problems. Based on the Morozov and damped Morozov discrepancy principles, we propose two iterative methods, a quasi-Newton method and a two-parameter model function method, for finding some reasonable regularization parameters in an efficient manner. These discrepancy principles require knowledge of the error level in the data of the considered inverse problems, which is often inaccessible or very expensive to achieve in real applications. We therefore propose an iterative algorithm to estimate the observation errors for linear inverse problems. Numerical experiments for one- and two-dimensional elliptic boundary value problems and an integral equation are presented to illustrate the efficiency of the proposed algorithms.