Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

TL;DR

This paper introduces a new strategy for selecting regularization parameters in a nonstationary iterated Tikhonov method, improving stability and convergence in solving ill-posed linear operator equations with practical applications.

Contribution

A novel geometrically-informed approach for choosing Lagrange multipliers in the NIT method, with proven convergence and demonstrated effectiveness in inverse problems and image deblurring.

Findings

Enhanced stability and convergence demonstrated in numerical experiments.

Outperforms standard NIT implementations with geometrical parameter selection.

Validated on inverse potential and image deblurring problems.

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 Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: I) A 2D elliptic parameter identification problem (Inverse Potential Problem); II) An image deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).