On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy

TL;DR

This paper proposes a discrepancy-based strategy for selecting the regularization parameter and discretization level in Tikhonov regularization, ensuring proper regularization of inverse problems with noisy data.

Contribution

It introduces a relaxed discrepancy principle to jointly determine the regularization parameter and discretization level, with proven existence and regularizing properties.

Findings

Existence of suitable regularization parameter and discretization level satisfying the discrepancy principle.

Proven regularizing properties of the Tikhonov minimizers under the proposed strategy.

The approach effectively balances regularization and discretization for inverse problems.

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.