  In this paper we propose a bilevel optimization approach for the placement of
space and time observations in variational data assimilation problems. Within
the framework of supervised learning, we consider a bilevel problem where the
lower-level task is the variational reconstruction of the initial condition of
a semilinear system, and the upper-level problem solves the optimal placement
with help of a sparsity inducing function. Due to the pointwise nature of the
observations, an optimality system with regular Borel measures is obtained as
necessary optimality condition for the lower-level problem. The latter is then
considered as constraint for the upper-level instance, yielding an optimization
problem constrained by a multi-state system with measures. We demonstrate
existence of Lagrange multipliers and derive a necessary optimality system
characterizing the optimal solution of the bilevel problem. The numerical
solution is carried out also on two levels. The lower-level problem is solved
using a standard BFGS method, while the upper-level one is solved by means of a
projected BFGS algorithm based on the estimation of $\epsilon$-active sets.
Finally some numerical experiments are presented to illustrate the main
features of our approach.
