Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations
Volume 41 / 2014
Abstract
We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations $KF(x)=y.$ It is assumed that the available data is $y^\delta $ with $\| y-y^\delta \| \leq \delta ,$ $ K:Z\rightarrow Y$ is a bounded linear operator and $ F:X\rightarrow Z $ is a nonlinear operator where $X,Y,Z$ are Hilbert spaces. Two cases of $F$ are considered: where $F'(x_0)^{-1}$ exists ($F'(x_0)$ is the Fréchet derivative of $F$ at an initial guess $x_0$) and where $F$ is a monotone operator. The parameter choice using an a priori and an adaptive choice under a general source condition are of optimal order. The computational results provided confirm the reliability and effectiveness of our method.
