A Simplified Gauss–Newton Iterative Scheme with an a Posteriori Parameter Choice Rule for Solving Nonlinear Ill-Posed Problems

Abstract

Nonlinear inverse problems occur in many applications. Finding solution to such problems are mathematically challenging and interesting due to the fact that in most of the situations, they are unstable under data perturbations. Classical numerical schemes proposed in literature require many assumptions. Therefore, refinement as well as development of efficient numerical schemes for solving these problems are under continuous development. In this paper, we are concerned with two things. Firstly, we consider a simplified Gauss–Newton iterative scheme that use minimal and weaker assumptions. Secondly, we propose an order optimal a posteriori parameter choice rule to choose the regularization parameter. The convergence analysis and error estimates are derived by choosing the regularization parameter according to both a priori and a posteriori methods. We prove that the method achieves \(O(\delta ^{2/3})\) as the order of converge rate, where \(\delta \) is the noise level in the data. The iterative scheme is stopped using an a posteriori stopping rule. The salient features of our proposed scheme are: (i) Convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in standard scheme in literature; (ii) Consideration of an adaptive, numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; (iii) The regularization parameter computed using the discrepancy principle is of the order \(O(\delta ^{2/3})\). We supply the numerical results to illustrate the above features. Further, we compare the numerical result of the proposed method with the standard approach and it demonstrates that our scheme is a stable approach and achieves good computational output.