A parameter choice rule for Tikhonov regularization based on predictive risk. (18th May 2020)

• Record Type: Journal Article
• Authors: Benvenuto, Federico; Jin, Bangti
• Is Part Of: Inverse problems. Volume 36:Number 6(2020)
• Publication Date: 2020-05-18
• Subjects: Tikhonov regularization -- regularization parameter -- predictive risk optimization; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ab6d58
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: In this work, we propose a new criterion for choosing the regularization parameter in Tikhonov regularization when the noise is white Gaussian. The criterion minimizes a lower bound of the predictive risk, when both data norm and noise variance are known, and the parameter choice involves minimizing a function whose solution depends only on the signal-to-noise ratio. Moreover, when neither noise variance nor data norm is given, we propose an iterative algorithm which alternates between a minimization step of finding the regularization parameter and an estimation step of estimating signal-to-noise ratio. Simulation studies on both small- and large-scale datasets suggest that the approach can provide very accurate and stable regularized inverse solutions and, for small sized samples, it outperforms discrepancy principle, balancing principle, unbiased predictive risk estimator, L-curve method, generalized cross validation, and quasi-optimality criterion, and achieves excellent stability hitherto unavailable.