Steepest Descent Type Methods for Nonlinear Ill-Posed Operator Equations

Abstract

In this thesis, we consider steepest descent method and minimal error method for approximating a solution of the nonlinear ill-posed operator equation F(x) = y, where F : D(F) ⊆ X → Y is nonlinear Fr´echet differentiable operator between the Hilbert spaces X and Y. In practical application, we have only noisy data yδ with ∥y − yδ∥ ≤ δ. To our knowledge, convergence rate result for the steepest descent method and minimal error method with noisy data are not known. We provide error estimate for these methods with noisy data. We modified these methods with less computational cost. Error estimate for steepest descent method and minimal error method is not known under H¨older-type source condition. We provide an error estimate for these methods under H¨older-type source condition and also with noisy data. We also studied the regularized version of steepest descent method and regularization parameter in this regularized version is selected through the adaptive scheme of Pereverzev and Schock (2005).