On Fractional Asymptotical Regularization of Linear Ill-Posed Problems in Hilbert Spaces

• Published in: Fractional calculus & applied analysis Vol. 22; no. 3; pp. 699 - 721
• Main Authors: Zhang, Ye, Hofmann, Bernd
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 26.06.2019; De Gruyter; Walter de Gruyter GmbH
• Subjects: 65F22; 65J20; 65R30; Abstract Harmonic Analysis; Acceleration; Analysis; Asymptotic properties; asymptotical regularization; convergence rates; fractional derivatives; Functional Analysis; Hilbert space; Ill posed problems; Integral Transforms; Iterative methods; linear ill-posed operator equation; Mathematics; Operational Calculus; Primary 47A52; Secondary 26A33; Regularization; Regularization methods; Research Paper; Smoothness; source conditions; stopping rules; 65R30; linear ill-posed operator equation; fractional derivatives; Primary 47A52; 65F22; Secondary 26A33; stopping rules; source conditions; 65J20; asymptotical regularization; convergence rates, acceleration
• ISSN: 1311-0454; 1314-2224; 1314-2224
• DOI: 10.1515/fca-2019-0039

In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR) , for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams- Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.