  We propose a model order reduction technique integrating the Shifted Boundary
Method (SBM) with a POD-Galerkin strategy. This approach allows to treat more
complex parametrized domains in an efficient and straightforward way. The
impact of the proposed approach is threefold.
  First, problems involving parametrizations of complex geometrical shapes
and/or large domain deformations can be efficiently solved at full-order by
means of the SBM, an unfitted boundary method that avoids remeshing and the
tedious handling of cut cells by introducing an approximate surrogate boundary.
  Second, the computational effort is further reduced by the development of a
reduced order model (ROM) technique based on a POD-Galerkin approach.
  Third, the SBM provides a smooth mapping from the true to the surrogate
domain, and for this reason, the stability and performance of the reduced order
basis are enhanced. This feature is the net result of the combination of the
proposed ROM approach and the SBM. Similarly, the combination of the SBM with a
projection-based ROM gives the great advantage of an easy and fast to implement
algorithm considering geometrical parametrization with large deformations. The
transformation of each geometry to a reference geometry (morphing) is in fact
not required.
  These combined advantages will allow the solution of PDE problems more
efficiently. We illustrate the performance of this approach on a number of
two-dimensional Stokes flow problems.
