## Ioannis K. Argyros, Santhosh George, Expanding the applicability of an iterative regularization method for ill-posed problems

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DOI: 10.23952/jnva.3.2019.3.03
Volume 3, Issue 3, 1 December 2019, Pages 257-275

Abstract. An iteratively regularized projection method, which converges quadratically, is considered for stable approximate solutions to a nonlinear ill-posed operator equation $F(x)=y$, where $F:D(F)\subseteq X\rightarrow X$ is a nonlinear monotone operator defined on the real Hilbert space $X.$ We assume that only a noisy data $y^\delta$ with $\|y-y^\delta\|\leq \delta$ are available. Under the assumption that the Fréchet derivative $F'$ of $F$ is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that, under a general source condition on $x_0-\hat{x},$ the error $\|x_{n,\alpha}^{h,\delta}-\hat{x}\|$ between the regularized approximation $x_{n,\alpha}^{h,\delta},$ ($x_{0,\alpha}^{h,\delta}:=P_hx_0$, where $P_h$ is an orthogonal projection on to a finite dimensional subspace $X_h$ of $X$) and the solution $\hat{x}$ is of optimal order.