On Nonstationary Iterated Tikhonov Methods for Ill-Posed Equations in Banach Spaces

Abstract

In this article we propose a novel nonstationary iterated Tikhonov (nIT) type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Banach spaces. We propose a novel a posteriori strategy for choosing the sequence of regularization parameters (or, equivalently, the Lagrange multipliers) for the nIT iteration, aiming to obtain a fast decay of the residual. Numerical experiments are presented for a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space), and for a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space).