Inverse Problems and Imaging (IPI)

On the choice of the Tikhonov regularization parameter and the discretization level: A discrepancy-based strategy

Pages: 1 - 25, Volume 10, Issue 1, February 2016      doi:10.3934/ipi.2016.10.1

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Vinicius Albani - Computational Science Center, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna, Austria (email)
Adriano De Cezaro - Institute of Mathematics, Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande, Brazil (email)
Jorge P. Zubelli - Instituto Nacional de Matemática Pura e Aplicada, Rio do Janeiro, RJ 22460-320, Brazil (email)

Abstract: We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice for these quantities by applying a relaxed version of Morozov's discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers. We conclude by presenting some numerical examples of interest.

Keywords:  Tikhonov regularization, discrete setting, regularization convergence rates, discrepancy principles.
Mathematics Subject Classification:  35R30, 65J22 and 47J06.

Received: October 2014;      Revised: September 2015;      Available Online: February 16 2016.