Iterated Lavrentiev regularization for nonlinear ill-posed problems

Pallavi Mahale, M. Thamban Nair


We consider an iterated form of Lavrentiev regularization, using a null sequence $(\alpha_k)$ of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form $F(x)=y$, where $F:D(F)\subseteq X\to X$ is a nonlinear operator and $X$ is a Hilbert space. Recently, Bakushinsky and Smirnova [``Iterative regularization and generalized discrepancy prinicple for monotone operator equations", Numer. Funct. Anal. Optim. 28 (2007) 13-25] considered an a posteriori strategy to find a stopping index $k_{\delta}$ corresponding to inexact data $y^{\delta}$ with $\|y-y^{\delta} \| \leq \delta$ resulting in the convergence of the method as $\delta \to 0$. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on $(\alpha_k)$ is weaker than that considered by Bakushinsky and Smirnova.



Lavrentiev regularization, nonlinear ill-posed problems, monotone operator Frechet derivative, regularization parameter, stopping index, Gauss--Newton method, source function, general source condition, order optimal result


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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.