Iterated Lavrentiev regularization for nonlinear ILL-posed problems

We consider an iterated form of Lavrentiev regularization, using a null sequence (αk) 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)≤X→ is a nonlinear operator andX is a Hilbert space. Recently, Bakushinsky and Smirnova [Iterative regularization and generalized discrepancy principle for monotone operator equations, Numer. Funct. Anal. Optim. 28 (2007) 13-25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with

y-δ

≥δresulting in the convergence of the method as 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 (αk) is weaker than that considered by Bakushinsky and Smirnova. © 2010 Australian Mathematical Society.