ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

E-ISSN： 1446-8735|51|2|191-217

ISSN： 1446-1811

Source： ANZIAM Journal, Vol.51, Iss.2, 2009-10, pp. : 191-217

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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.