  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.
