A new interpretation of (Tikhonov) regularization

TL;DR

This paper introduces a new interpretative framework for Tikhonov regularization, linking it to approximate source conditions and broadening understanding of convergence, noise handling, and oversmoothing, with implications for iterative methods.

Contribution

The paper presents a novel strategy based on approximate source conditions that simplifies convergence analysis and extends to iterative regularization methods like Landweber iteration.

Findings

Provides a concise derivation of convergence results

Establishes a connection between Tikhonov regularization and iterative methods

Demonstrates the applicability to oversmoothing regularization

Abstract

Tikhonov regularization with square-norm penalty for linear forward operators has been studied extensively in the literature. However, the results on convergence theory are based on technical proofs and difficult to interpret. It is also often not clear how those results translate into the discrete, numerical setting. In this paper we present a new strategy to study the properties of a regularization method on the example of Tikhonov regularization. The technique is based on the observation that Tikhonov regularization approximates the unknown exact solution in the range of the adjoint of the forward operator. This is closely related to the concept of approximate source conditions, which we generalize to describe not only the approximation of the unknown solution, but also noise-free and noisy data; all from the same source space. Combining these three approximation results we derive the well-known convergence results in a concise way and improve the understanding by tightening the relation between concepts such as convergence rates, parameter choice, and saturation. The new technique is not limited to Tikhonov regularization, it can be applied also to iterative regularization, which we demonstrate by relating Tikhonov regularization and Landweber iteration. Because the Tikhonov functional is no longer the centrepiece of the analysis, we can show that Tikhonov regularization can be used for oversmoothing regularization. All results are accompanied by numerical examples.