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).