  We present a reduced-order model (ROM) methodology for inverse scattering
problems in which the reduced-order models are data-driven, i.e. they are
constructed directly from data gathered by sensors. Moreover, the entries of
the ROM contain localised information about the coefficients of the wave
equation.
  We solve the inverse problem by embedding the ROM in physical space. Such an
approach is also followed in the theory of ``optimal grids,'' where the ROMs
are interpreted as two-point finite-difference discretisations of an underlying
set of equations of a first-order continuous system on this special grid. Here,
we extend this line of work to wave equations and introduce a new embedding
technique, which we call Krein embedding, since it is inspired by Krein's
seminal work on vibrations of a string. In this embedding approach, an adaptive
grid and a set of medium parameters can be directly extracted from a ROM and we
show that several limitations of optimal grid embeddings can be avoided.
Furthermore, we show how Krein embedding is connected to classical optimal grid
embedding and that convergence results for optimal grids can be extended to
this novel embedding approach. Finally, we also briefly discuss Krein embedding
for open domains, that is, semi-infinite domains that extend to infinity in one
direction.
