A unified framework for the regularization of final value time-fractional diffusion equation

TL;DR

This paper introduces a unified, computationally simple regularization framework for the ill-posed backward time-fractional diffusion problem, with proven error estimates and validated through numerical experiments.

Contribution

It presents a novel unified regularization approach that avoids high-frequency truncation and provides order-optimal error estimates and parameter choice rules.

Findings

The proposed method is computationally simple.

It avoids undesirable oscillations in solutions.

Numerical experiments confirm theoretical convergence rates.

Abstract

This paper focuses on the regularization of backward time-fractional diffusion problem on unbounded domain. This problem is well-known to be ill-posed, whence the need of a regularization method in order to recover stable approximate solution. For the problem under consideration, we present a unified framework of regularization which covers some techniques such as Fourier regularization [19], mollification [12] and approximate-inverse [7]. We investigate a regularization technique with two major advantages: the simplicity of computation of the regularized solution and the avoid of truncation of high frequency components (so as to avoid undesirable oscillation on the resulting approximate-solution). Under classical Sobolev-smoothness conditions, we derive order-optimal error estimates between the approximate solution and the exact solution in the case where both the data and the model are only approximately known. In addition, an order-optimal a-posteriori parameter choice rule based on the Morozov principle is given. Finally, via some numerical experiments in two-dimensional space, we illustrate the efficiency of our regularization approach and we numerically confirm the theoretical convergence rates established in the paper.