ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

Abstract

Abstract We consider an iterated form of Lavrentiev regularization, using a null sequence ( α 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 )⊆ 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 δ corresponding to inexact data y δ with $\|y-y^\d \|\leq \d $ 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 ( α k ) is weaker than that considered by Bakushinsky and Smirnova.