  This paper addresses a backward heat conduction problem with fractional
Laplacian and time-dependent coefficient in an unbounded domain. The problem
models generalized diffusion processes and is well-known to be severely
ill-posed. We investigate a simple and powerful variational regularization
technique based on mollification. Under classical Sobolev smoothness
conditions, we derive order-optimal convergence rates between the exact
solution and regularized approximation in the practical case where both the
data and the operator are noisy. Moreover, we propose an order-optimal
a-posteriori parameter choice rule based on the Morozov principle. Finally, we
illustrate the robustness and efficiency of the regularization technique by
some numerical examples including image deblurring.
