  We consider the problem of identifying a sparse initial source condition to
achieve a given state distribution of a diffusion-advection partial
differential equation after a given final time. The initial condition is
assumed to be a finite combination of Dirac measures. The locations and
intensities of this initial condition are required to be identified. This
problem is known to be exponentially ill-posed because of the strong diffusive
and smoothing effects. We propose a two-stage numerical approach to treat this
problem. At the first stage, to obtain a sparse initial condition with the
desire of achieving the given state subject to a certain tolerance, we propose
an optimal control problem involving sparsity-promoting and
ill-posedness-avoiding terms in the cost functional, and introduce a
generalized primal-dual algorithm for this optimal control problem. At the
second stage, the initial condition obtained from the optimal control problem
is further enhanced by identifying its locations and intensities in its
representation of the combination of Dirac measures. This two-stage numerical
approach is shown to be easily implementable and its efficiency in short time
horizons is promisingly validated by the results of numerical experiments. Some
discussions on long time horizons are also included.
