In this talk, 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. In comparison with existing acceleration regularization methods, FAR is an accelerated method without saturation. A further acceleration of FAR is obtained in Hilbert scales. For numerical realization of FAR, a novel iterative regularization scheme is developed based on the generalized Adams method.

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