Abstract. Iterative regularization methods for nonlinear ill-posed equations of the form F (x) = y, where F : D(F ) ⊂ X → Y is an operator between Hilbert spaces X and Y , usually involve calculation of the Fréchet derivatives of F at each iterate and at the unknown solution x † . In this paper, we suggest a modified form of the generalized Gauss-Newton method which requires the Fréchet derivative of F only at an initial approximation x 0 of the solution x † . The error analysis for this method is done under a general source condition which also involves the Fréchet derivative only at x 0 . The conditions under which the results of this paper hold are weaker than those considered by Kaltenbacher (1998) for an analogous situation for a special case of the source condition.
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 δ ≤ 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 (α k ) is weaker than that considered by Bakushinsky and Smirnova.2000 Mathematics subject classification: primary 47A52; secondary 65F22, 65J15, 65J22, 65M30.
Tikhonov regularization is one of the widely used procedures for the regularization of nonlinear as well as linear ill-posed problems. The error analysis carried out in most of the works that appeared in last few years on Tikhonov regularization of nonlinear ill-posed problems are under Hölder type source conditions on the unknown solution which is known to be applicable only for mildly ill-posed problems. In this paper we consider Tikhonov regularization of nonlinear ill-posed problems and derive order optimal error estimate under a general source condition together with an a posteriori parameter rule proposed by Scherzer et al., which is applicable for severely ill-posed problems as well.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.