A derivative free iterative method for the implementation of Lavrentiev regularization method for ill-posed equations (Q2256953)

The authors propose a derivative free iterative method for solving an optimization problem arising in Lavrentiev regularization for nonlinear inverse problems with a monotone operator. Derivative free methods are interesting since they do not require computing the derivatives. In the context of inverse problems, \textit{P. Kügler} [Inverse Probl. 19, No. 6, 1407--1426 (2003; Zbl 1050.65106)] proposed a derivative Landweber iteration. A quadratic convergence of the algorithm was shown, and optimal-order error estimates were provided, under a general source condition. The adaptive rule due to \textit{S. Pereverzev} and \textit{E. Schock} [SIAM J. Numer. Anal. 43, No. 5, 2060--2076 (2005; Zbl 1103.65058)] is adopted for choosing an appropriate regularization parameter. Some numerical results are given. However, the quadratic convergence of the method is not verified numerically.