Iterated fractional Tikhonov regularization

TL;DR

This paper introduces new iterated fractional Tikhonov regularization methods that improve convergence rates and overcome saturation limits, with theoretical analysis and numerical validation demonstrating their effectiveness.

Contribution

It proposes novel iterated fractional Tikhonov regularization techniques that are of optimal order and surpass previous saturation results, including nonstationary variants with proven convergence.

Findings

Iterated fractional Tikhonov methods achieve optimal convergence rates.

New methods overcome previous saturation limitations.

Numerical experiments confirm improved regularization performance.

Abstract

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.