AbstractMahale and Nair [12] considered an iterated
form of Lavrentiev regularization
for obtaining stable approximate solutions for
ill-posed nonlinear equations of the form ${F(x)=y}$, where
${F:D(F)\subseteq X\to X}$ is a nonlinear monotone operator and X is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate
under a general source condition. However, the iterations defined
in [12] require calculation of Fréchet derivatives at each iteration. In this paper,
we consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Fréchet derivative only at the point ${x_{0}}$, i.e., at the initial approximation of the exact solution ${x^{\dagger}}$. Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Fréchet derivative at the point ${x_{0}}$,
instead at the unknown exact solution ${x^{\dagger}}$ as in [12].
Newton Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems in Hilbert Scales
