Finite-Dimensional Realization of Lavrentiev Regularization for Nonlinear III-posed Equations

Abstract

A finite-dimensional realization of the two-step Newton method is considered for obtaining an approximate solution (reconstructed signals) for the nonlinear ill-posed equation $F(x) = f$ when the available data (noisy signal) is $f^{\delta }$ with $f - f^{\delta } \le \delta$ and the operator F is monotone. We derived an optimal-order error estimate under a general source condition on $x_{0} - \bar{x}$, where $x_{0}$ is the initial approximation to the actual solution (signal) $\bar{x}.$ The choice of the regularization parameter is made according to the adaptive method considered by Pereverzev and Schock (2005). 2D visualization shows the effectiveness of the proposed method.