Estimating Solution Smoothness and Data Noise with Tikhonov Regularization. (2nd January 2022)

• Record Type: Journal Article
• Authors: Gerth, Daniel; Ramlau, Ronny
• Is Part Of: Numerical functional analysis and optimization. Volume 43:Number 1(2022)
• Page Start: 88
• Page End: 115
• Publication Date: 2022-01-02
• Subjects: Ill-posed problem -- noise level estimation -- source condition -- Tikhonov regularization; 65J20 -- 47A52; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.2007948
• Languages: English
• ISSNs: 0163-0563

Abstract

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.