Error Estimates for the Iteratively Regularized Gradient Method for Ill-Posed Optimization Problems in a Hilbert Space

Abstract

We study the iteratively regularized gradient method applied to the ill-posed problem of minimizing a general smooth functional on a Hilbert space. Under a source condition on the solution, we establish error estimates for this method when the objective functional is known exactly or only approximately. For these estimates we provide two proofs based on substantially different ideas and compare the assumptions and conclusions of the corresponding theorems. In addition, we present an improved proof of an error estimate for the Tikhonov method applied to ill-posed optimization problems with an exactly known functional. We also prove a theorem on the existence of a global minimizer of the Tikhonov functional without assuming weak lower semicontinuity of the original functional.