Iterative Regularization Theory for Nonlinear Ill-Posed Problems

Abstract

In science and engineering many practical problems can be formulated using mathematical modelling and can be classified as nonlinear ill-posed problems. Here we consider those ill-posed equations involving m-accretive operators in Banach spaces. Using a general H¨older type source condition we were able to obtain an optimal order error estimate. For nonlinear problems, obtaining a closed form solution is possible only in rare cases, so most of the methods considered for approximating the solution of nonlinear problems are iterative. Four different types of iterative schemes are being discussed in this thesis. Firstly, we consider a derivative and inverse free method and obtained second order convergence. Then, we produced an extended Newton-type iterative scheme that converges cubically to the solution which uses assumptions only on the first Fr´echet derivative of the operator. Afterwards, we studied Newton-Kantorovich regularization method and obtained second order convergence with weak assumptions. Finally, we examined Secant-type iteration and proved that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on first Fr´echet derivative of the operator. Through out the work, for choosing the regularization parameter we have taken the adaptive parameter choice strategy given by Pereverzev and Schock (2005).