CS: Gravimetric Detection by Compressed Sensing

Here is an inverse problem using sparsity as a constraint alone: Gravimetric Detection by Compressed Sensing by Marina Meila, Caren Marzban, Ulvi Yurtsever. The beginning of the article goes with:

This paper sets forth a novel approach for solving geospatial inversion problems using cutting edge techniques from statistics. Assume for instance that we want to obtain the mass density function in a given domain of interest from gravitational field measurements on the boundary of that domain. This is a well-studied and difficult problem. In all except a few special cases, the inverse problem has multiple solutions, and additional constraints (physical, or problem related) are needed to regularize it and to select a single, plausible solution.