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

Vinicius  Albani; Adriano De  Cezaro; Jorge P. Zubelli

doi:10.3934/ipi.2016.10.1

Article Contents

2016, Volume 10, Issue 1: 1-25. Doi: 10.3934/ipi.2016.10.1

This issue Previous Article Next Article A fractional-order derivative based variational framework for image denoising

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

1.
Computational Science Center, University of Vienna, Oskar Morgenstern-Platz 1, 1090 Vienna
2.
Institute of Mathematics, Statistics and Physics, Federal University of Rio Grande, Av. Italia km 8, 96201-900 Rio Grande
3.
Instituto Nacional de Matemática Pura e Aplicada, Rio do Janeiro, RJ 22460-320

Received: October 2014

Revised: September 2015

Published online: February 16, 2016

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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.

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Mathematics Subject Classification: 35R30, 65J22 and 47J06.

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