  Inverse problems involve making inference about unknown parameters of a
physical process using observational data. This paper investigates an important
class of inverse problems -- the estimation of the initial condition of a
spatio-temporal advection-diffusion process using spatially sparse data
streams. Three spatial sampling schemes are considered, including irregular,
non-uniform and shifted uniform sampling. The irregular sampling scheme is the
general scenario, while computationally efficient solutions are available in
the spectral domain for non-uniform and shifted uniform sampling. For each
sampling scheme, the inverse problem is formulated as a regularized convex
optimization problem that minimizes the distance between forward model outputs
and observations. The optimization problem is solved by the Alternating
Direction Method of Multipliers algorithm, which also handles the situation
when a linear inequality constraint (e.g., non-negativity) is imposed on the
model output. Numerical examples are presented, code is made available on
GitHub, and discussions are provided to generate some useful insights of the
proposed inverse modeling approaches.
