In a first part we quickly recall some previous work in collaboration
with F. Santambrogio related to the functional relaxation of the
irrigation cost. We establish a $\Gamma$-convergence of Modica and
Mortola's type and illustrate its efficiency from a numerical point of
view by computing optimal networks associated to simple sources-sinks
configurations. We also present more evolved situations with non Dirac
sinks in which a fractal behavior of the optimal network is expected.

In the second part of the talk we restrict our study to the euclidean
Steiner's tree problem. We recall recent numerical approach which have
been developed the last five years to approximate optimal trees:
partitioning formulation, relaxation with geodesic distance terms and
energetic constraints. We describe the first results obtained in
collaboration with A. Massaccesi and B. Velichkov to certify the
optimality of a given tree. With our discrete parametrization of
generalized calibration, we are able to recover the theoretical optimal
matrix fields which certify the optimality of simple trees associated to
the vertices of regular polygons.

Finally, we focus on the delicate problem of the identification of the
optimal structure. Based on a recent approach obtained in collaboration
with G. Orlandi and M. Bonafini, we describe the first convexification
framework associated to the Euclidean Steiner tree problem which
provide relevant tools from a numerical point of view.
http://cvgmt.sns.it/seminar/580/