Linguistics

Philosophy

Well-posed problem

Pure mathematics

Applied mathematics

Regularization (linguistics)

Hilbert space

Computer science

Bounded function

Backus–Gilbert method

Mathematics

Mathematical analysis

Artificial intelligence

Regularization perspectives on support vector machines

Tikhonov regularization

Inverse problem

Linear operators

Fractional Calculus and Applied Analysis

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.

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