Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space

TL;DR

This paper develops a unified parameter selection strategy for Newton-Landweber iterations in Banach spaces, ensuring convergence, convergence rates, and improved efficiency, even without additional regularity assumptions.

Contribution

It introduces a new, more general parameter choice method that guarantees both convergence and convergence rates, and demonstrates enhanced efficiency over previous approaches.

Findings

Method achieves strong convergence as noise tends to zero.

Numerical tests show improved efficiency.

Applicable without additional regularity assumptions.

Abstract

This paper is a close follow-up of Kaltenbacher and Tomba 2013 and Jin 2012, where Newton-Landweber iterations have been shown to converge either (unconditionally) without rates or (under an additional regularity assumption) with rates. The choice of the parameters in the method were different in each of these two cases. We now found a unified and more general strategy for choosing these parameters that enables both convergence and convergence rates. Moreover, as opposed to the previous one, this choice yields strong convergence as the noise level tends to zero, also in the case of no additional regularity. Additionally, the resulting method appears to be more efficient than the one from Kaltenbacher and Tomba 2013, as our numerical tests show.