Fractional Tikhonov regularization method in Hilbert scales

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

“…In this work, the finite dimensional version of the parameter choice strategy introduced in [16] has been studied. We also obtained the optimal order error estimate for the finite dimensional FTR method in Hilbert scales .…”

“…The proof of the following theorem is analogous to the proof of the theorem in [16], but for sake of completion we give the proof as well.…”

“…One drawback of the FTR method is that it reduces the order of convergence of the solutions. However, to overcome this problem too we study FTR method in the setting of Hilbert scales [16]. Our approach reduces the oversmoothing and gives a more preferable convergence rate as well.…”

“…(1.6) It is known that the fractional power has lower damping at small singular values and that fractional power has a greater effect on the filtering function [26] (also see [16]). While performing numerical calculations in practical applications, x δ α,β being the minimizer of J β α over an infinite dimensional space is not easy to compute.…”

“…However a computable approximation can be obtained using the finite dimensional realization of the infinite dimensional method [18]. The aim of this paper is to study the finite dimensional realization of the parameter choice rule introduced in [16] (motivated reader can refer [19] for finite dimentional fractional Tikhonov regularization).…”