Underdetermined linear systems of equations are ubiquitous in many fields ranging from biology, to information technology. In recent years, different techniques and approximations have been proposed. In this presentation, I will mostly concentrate on Expectation Propagation (EP) as a general tool for dealing with this problem under a most general structure of the priors. I will start by introducing the formalism with 3 concrete applications:

-sampling metabolic networks
-tomography
-inference of brain activity from eeg data

I will then outline the analytical derivation of the algorithm.

Finally I will discuss some results on metabolic sampling.