Finite Dimensional Realization of a Guass-Newton Method for Ill-Posed Hammerstein Type Operator Equations

Abstract

AbstractFinite dimensional realization of an iterative regularization method for approximately solving the non-linear ill-posed Hammerstein type operator equations KF(x) = f, is considered. The proposed method is a combination of the Tikhonov regularization and Guass-Newton method. The advantage of the proposed method is that, we use the Fr\(\acute{e}\)chet derivative of F only at one point in each iteration. We derive the error estimate under a general source condition and the regularization parameter is chosen according to balancing principle of Pereverzev and Schock (2005). The derived error estimate is of optimal order and the numerical example provided proves the efficiency of the proposed method.KeywordsNewton’s methodTikhonov regularizationill-posed Hammerstein operatorBalancing principleMonotone operatorRegularizationAMS Subject Classification47J0647A5265N2065J20