An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations

Abstract

Study of inverse problems are interesting and mathematically challenging due to the fact that in most of the situation they are unstable with respect to perturbations of the data. In this paper to solve such operator equations, we propose a modified form of Gauss-Newton method combined with an a posteriori parameter choice strategy with the inexact data. Convergence and the convergence rate results are proven. We consider both a-priori and a-posteriori choice rule of parameter that guarantees the scheme converges to the exact solution. The theoretical results are illustrated through numerical examples and compared with the standard scheme to demonstrate that the scheme is stable and achieves good computational output. The salient features of our proposed scheme are: 1) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature; 2) consideration of an adaptive and numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; 3) computation of an optimal order regularisation parameter of the order O(δ2/3) using a discrepancy principle.