  This thesis is interested in the application of statistical physics methods
and inference to sparse linear estimation problems. The main tools are the
graphical models and approximate message-passing algorithm together with the
cavity method. We will also use the replica method of statistical physics of
disordered systems which allows to associate to the studied problems a cost
function referred as the potential of free entropy in physics. It allows to
predict the different phases of typical complexity of the problem as a function
of external parameters such as the noise level or the number of measurements
one has about the signal: the inference can be typically easy, hard or
impossible. We will see that the hard phase corresponds to a regime of
coexistence of the actual solution together with another unwanted solution of
the message passing equations. In this phase, it represents a metastable state
which is not the true equilibrium solution. This phenomenon can be linked to
supercooled water blocked in the liquid state below its freezing critical
temperature. We will use a method that allows to overcome the metastability
mimicing the strategy adopted by nature itself for supercooled water: the
nucleation and spatial coupling. In supercooled water, a weak localized
perturbation is enough to create a crystal nucleus that will propagate in all
the medium thanks to the physical couplings between closeby atoms. The same
process will help the algorithm to find the signal, thanks to the introduction
of a nucleus containing local information about the signal. It will then spread
as a "reconstruction wave" similar to the crystal in the water. After an
introduction to statistical inference and sparse linear estimation, we will
introduce the necessary tools. Then we will move to applications of these
notions to signal processing and coding theory problems.
