Nonlinear Tikhonov regularization in Hilbert scales with oversmoothing penalty: inspecting balancing principles

TL;DR

This paper investigates nonlinear Tikhonov regularization in Hilbert scales with oversmoothing penalties, focusing on error analysis and parameter selection via balancing principles to achieve optimal reconstruction in inverse problems.

Contribution

It introduces an error decomposition applicable to various smoothness levels and analyzes the effectiveness of balancing principles for parameter choice in oversmoothing scenarios.

Findings

Error decomposition into smoothness-dependent and noise-dependent parts.

Balancing principles can yield order optimal reconstruction under certain conditions.

Numerical case study demonstrates practical applicability.

Abstract

The analysis of Tikhonov regularization for nonlinear ill-posed equations with smoothness promoting penalties is an important topic in inverse problem theory. With focus on Hilbert scale models, the case of oversmoothing penalties, i.e., when the penalty takes an infinite value at the true solution gained increasing interest. The considered nonlinearity structure is as in the study B. Hofmann and P. Math\'{e}. Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales. Inverse Problems, 2018. Such analysis can address two fundamental questions. When is it possible to achieve order optimal reconstruction? How to select the regularization parameter? The present study complements previous ones by two main facets. First, an error decomposition into a smoothness dependent and a (smoothness independent) noise propagation term is derived, covering a large range of smoothness conditions. Secondly, parameter selection by balancing principles is presented. A detailed discussion, covering some history and variations of the parameter choice by balancing shows under which conditions such balancing principles yield order optimal reconstruction. A numerical case study, based on some exponential growth model, provides additional insights.