Error estimates for the simplified iteratively regularized Gauss–Newton method under a general source condition

Abstract

Recently George (J Inv Ill-posed Prob 18:133–146, 2010), Pradeep and Rajan (Int J Appl Comput Math 2:97–112, 2016) considered a two steps modified Gauss–Newton method for obtaining stable approximate solution for nonlinear ill-posed operator equation $F(x)= y.$ They have obtained order optimal error estimates under suitable choices of stopping rules with appropriate choices of regularization parameters. One drawback of the regularization algorithms considered in the above mentioned references is their two steps nature, which requires both a suitable stopping rule and a suitable choice of continuous regularization parameter. To overcome this issue, we consider a simplified iteratively regularized Gauss–Newton method in which the sequence of iterates $\{x_k^\delta \}$ is defined as

$$\begin{aligned} x_{k+1}^\delta = x_{k}^\delta - (\alpha _{k}I + A_{0}^*A_0)^{-1}(A_{0}^*(F(x_{k}^\delta ) - y^\delta ) + \alpha _{k}(x_{k}^\delta - x_0)), \end{aligned}$$

where $\{\alpha _k\}$ is the decreasing sequence of positive real numbers such that $\displaystyle \lim _{k \rightarrow \infty }\alpha _k = 0,$ $y^\delta$ is a perturbed data, $A_0 = F'(x_0)$ is the Fréchet derivative of F at $x_0 \in D(F),$ $A_{0}^*= F'(x_0)^*$ is the adjoint of $F'(x_0)$ and $x_{0}^\delta = x_0 \in D(F)$ is an initial guess. In this paper, we study the above defined method under a suitable a-posteriori stopping rule which yields an optimal error estimate with general form of source condition.