Estimating solution smoothness and data noise with Tikhonov regularization

TL;DR

This paper demonstrates that residuals in Tikhonov regularization can reveal unknown solution smoothness and noise levels, enabling parameter estimation from a single noisy data set without prior knowledge.

Contribution

It introduces a method to extract solution smoothness and noise level from residuals in Tikhonov regularization, unifying various parameter choice rules.

Findings

Residuals contain information about solution smoothness and noise level.

Approximate solutions across a range of parameters enable parameter estimation.

Numerical experiments validate the proposed residual-based parameter inference.

Abstract

A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.