Iterated fractional Tikhonov regularization. (15th April 2015)

• Record Type: Journal Article
• Authors: Bianchi, Davide; Buccini, Alessandro; Donatelli, Marco; Serra-Capizzano, Stefano
• Is Part Of: Inverse problems. Volume 31:Number 5(2015:May)
• Publication Date: 2015-04-15
• Subjects: Tikhonov regularization -- filter function -- iterated Tikhonov -- fractional Tikhonov -- weighted Tikhonov; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/5/055005
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.