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

Abstract

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.MSC 2010 : Primary 47A52; Secondary 26A33, 65J20, 65F22, 65R30,

Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems

“…where I Y is the identity operator in Y. Clearly, we have forG from (24) and G from (8) the identity, (25) (…”

“…This terminology has been frequently used recently for studying convergence rate results, see e.g. [26,25]. Let's first consider the power-type source conditions, i.e.…”

“…The following lemma indicates that function g k can be viewed as a qualification of regularization methods [16,26,25].…”

“…Now, let's consider case (ii). Without loss of generality, we assume that k n increases monotonically with n. Then, by using(25) and the inequality z h,δ k+1 − |z h,δk | = 2(G + A * h A h ) −1 A * h (y δ − A h |z h,δ k |), we deduce together with the positivity ofx † that z h,δ k+1 − x † 2 − z h,δ k − x † 2 ≤ z h,δ k+1 − x † 2 − |z h,δ k | − x † 2 = 2(|z h,δ k | − x † , z h,δ k+1 − |z h,δ k |) + z h,δ k+1 − |z h,δ k | 2 = 4 |z h,δ k | − x † , (G + A * h A h ) −1 A * h (y δ − A h |z h,δ k |) +4 (G + A * h A h ) −1 A * h (y δ − A h |z h,δ k |) 2 = 4 A h (|z h,δ k | − x † ), (G + A h A * h ) −1 (y δ − A h |z h,δ k |) +4 (G + A * h A h ) −1 A * h (y δ − A h |z h,δ k |) 2 = 4 (G + A h A * h ) −1 (y δ − A h |z h,δ k |), y δ − A h x † −4 (G + A h A * h ) −1 (y δ − A h |z h,δ k |),G(G + A h A * h ) −1 (y δ − A h |z h,δ k |) ≤ 4 (G + A h A * h ) −1 (y δ − A h |z h,δ k |) y δ − A h x † −4µ (G + A h A * h ) −1 (y δ − A h |z h,δ k |) 2, which implies together with the inequalityy δ − A h x † ≤ δ + hC † that, (29) z h,δ k+1 − x † 2 ≤ z h,δ k − x † 2 + 4 (G + A h A * h ) −1 (y δ − A h |z h,δ k |) Y • δ + hC † − µ (G + A h A * h ) −1 (y δ − A h |z h,δ k |) Y ,…”

Stochastic asymptotical regularization for linear inverse problems

“…(4), to some classical methods from data assimilation, namely the Kalman-Bucy filter and 3DVAR. Moving in a different direction, the authors in [29,8,28,2] extended (4) to second-order and fractional-order gradient flows. They proved that the developed high-order flows are accelerated optimal regularization methods, i.e.…”