  Our aim is to study the backward problem, i.e. recover the initial data from
the terminal observation, of the subdiffusion with time dependent coefficients.
First of all, by using the smoothing property of solution operators and a
perturbation argument of freezing the diffusion coefficients, we show a
stability estimate in Sobolev spaces, under some smallness/largeness condition
on the terminal time. Moreover, in case of noisy observation, we apply a
quasi-boundary value method to regularize the problem and then show the
convergence of the regularization scheme. Finally, to numerically reconstruct
the initial data, we propose a completely discrete scheme by applying the
finite element method in space and backward Euler convolution quadrature in
time. An \textsl{a priori} error estimate is then established. The proof is
heavily built on a perturbation argument dealing with time dependent
coefficients and some nonstandard error estimates for the direct problem. The
error estimate gives a useful guide for balancing discretization parameters,
regularization parameter and noise level. Some numerical experiments are
presented to illustrate our theoretical results.
