Applying Lepskij-Balancing in Practice

TL;DR

This paper investigates the practical application of the Lepskij balancing principle in inverse problems, explaining its robustness without noise level knowledge and proposing modifications to enhance its efficiency.

Contribution

It provides a stochastic model explanation for the method's practical performance and introduces a modified algorithm that improves speed and accuracy.

Findings

The Lepskij balancing principle performs well without knowing the noise level.

A small modification enhances the method's speed and accuracy.

Theoretical justification supports the practical robustness of the approach.

Abstract

In a stochastic noise setting the Lepskij balancing principle for choosing the regularization parameter in the regularization of inverse problems is depending on a parameter $\tau$ which in the currently known proofs is depending on the unknown noise level of the input data. However, in practice this parameter seems to be obsolete. We will present an explanation for this behavior by using a stochastic model for noise and initial data. Furthermore, we will prove that a small modification of the algorithm also improves the performance of the method, in both speed and accuracy.