Iterated Lavrentiev regularization for nonlinear ill-posed problems

Abstract

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. doi:10.1017/S1446181109000418