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→ X is a nonlinear operator and X 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 y with ‖y − y‖ ≤ δ resulting in the convergence of the method as δ→ 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. 2000 Mathematics subject classification: primary 47A52; secondary 65F22, 65J15, 65J22, 65M30.