In this paper, an inverse source problem for the time-fractional diffusion equation
is investigated. The observational data is on the final time and the source term
is assumed to be temporally independent and with a sparse structure. Here the sparsity
is understood with respect to the pixel basis, i.e., the source has a small support.
By an elastic-net regularization method, this inverse source problem is formulated into
an optimization problem and a semismooth Newton (SSN) algorithm is developed
to solve it. A discretization strategy is applied in the numerical realization. Several
one and two dimensional numerical examples illustrate the efficiency of the proposed