  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.
