ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

We consider an iterated form of Lavrentiev regularization, using a null sequence .k / of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F.x/ D y, where F V 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 k corresponding to inexact data y with ky 􀀀 yk   resulting 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 .k / is weaker than that considered by Bakushinsky and Smirnova.