A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems (Q426556)

Assume, a problem is modeled by a nonlinear operator equation \(F(x)= y\) in Hilbert spaces, and that the problem is ill-posed: The available data stems from a measurement process. Due to measurement errors, we have to deal with noise data \(y^\delta\) and we have to solve an equation \(F(x)= y^\delta\). The minimizer \(x^\delta_\alpha\) of the Tikhonov function NEWLINE\[NEWLINEJ_\alpha(x,\overline x)=\| y^\delta- F(x)\|^2+ \alpha\| x-\overline x\|^2NEWLINE\]NEWLINE is regarded as a regularized solution of the disturbed equation. In this paper, under certain assumptions a nonlinear implicit iterative scheme is proposed: NEWLINE\[NEWLINEx^\delta_{k+1}= x^\delta_k+{1\over\alpha_k} F'(x^\delta_k)^*(y^\delta- F(x^\delta_k)).NEWLINE\]NEWLINE This is the classical Landweber iterative method as \(\alpha_k= 1\). Under the restriction that \(\alpha_k\) is appropriate large, the monotonicity of iterative errors and the convergence and stability of the iterative sequence is proved.NEWLINENEWLINE The convergence and stability of the proposed method is also analyzed when the numbers \(\alpha_k\) are determined by the Hanke criterion. Numerical tests (for a two-point boundary value problem) show, that the method under consideration for nonlinear ill-posed problems is efficient.