Regularized Versions of Continuous Newton's Method and Continuous Modified Newton's Method Under General Source Conditions

Abstract

Regularized versions of continuous analogues of Newton's method and modified Newton's method for obtaining approximate solutions to a nonlinear ill-posed operator equation of the form F(u) = f, where F is a monotone operator defined from a Hilbert space H into itself, have been studied in the literature. For such methods, error estimates are available only under Hölder-type source conditions on the solution. In this paper, presenting the background materials systematically, we derive error estimates under a general source condition. For the special case of the regularized modified Newton's method under a Hölder-type source condition, we also carry out error analysis by replacing the monotonicity of F by a weaker assumption. This analysis facilitates inclusion of certain examples of parameter identification problems, which was not possible otherwise. Moreover, an a priori stopping rule is considered when we have a noisy data f δ instead of f. This rule yields not only convergence of the regularized approximations to the exact solution as the noise level δ tends to zero but also provides convergence rates that are optimal under the source conditions considered.