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arXiv · 1812.09734

On fractional asymptotical regularization of linear ill-posed problems in Hilbert spaces

Ye Zhang·Bernd Hofmann
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Abstract

In this paper, we study a fractional-order variant of the asymptotical regularization method, called {\it 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.

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Ye Zhang, Bernd Hofmann. 2018-12-23. On fractional asymptotical regularization of linear ill-posed problems in Hilbert spaces. https://doi.org/10.1515/fca-2019-0039

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