  Inverse problems are ubiquitous in science and engineering. Many of these are
naturally formulated as a PDE-constrained optimization problem. These
non-linear, large-scale, constrained optimization problems know many
challenges, of which the inherent non-linearity of the problem is an important
one. As an alternative to this physics-driven approach, data-driven methods
have been proposed. These methods come with their own set of challenges, and it
appears that, ideally, one would devise hybrid methods that combine the best of
both worlds. In this paper, we propose one way of combining PDE-constrained
optimization with recently proposed data-driven reduced-order models. Starting
from an infinite-dimensional formulation of the inverse problem with discrete
data, we propose a general framework for the analysis and discretisation of
such problems. The proposed approach is based on a relaxed formulation of the
PDE-constrained optimization problem, which reduces to a weighted non-linear
least-squares problem. The weight matrix turns out to be the Gram matrix of
solutions of the PDE, and it can be estimated directly from the measurements.
We provide a number of representative case studies and numerical examples.
