Logarithmic convergence rates of tikhonov regularization for nonlinear ill-posed problems

Abstract

In this paper, the problem of reconstruction of the solution of nonlinear ill-posed problem F(x)= y by Tikhonov regularization method is considered, where instead of y noisy data yδ with ∥y − yδ∥≤ δ are given and F:D(F)⊂X→Y is a nonlinear operator. A priori parameter choice rule and two a posteriori parameter choice rules are suggested. The order optimal convergence rate is O((−logδ)−p) under logarithmic-type source conditions. Moreover, we apply this method to an inverse boundary identification problem for verifying some of the theoretical results.