  A model order reduction technique is combined with an embedded boundary
finite element method with a POD-Galerkin strategy. The proposed methodology is
applied to parametrized heat transfer problems and we rely on a sufficiently
refined shape-regular background mesh to account for parametrized geometries.
In particular, the employed embedded boundary element method is the Shifted
Boundary Method (SBM) recently proposed. This approach is based on the idea of
shifting the location of true boundary conditions to a surrogate boundary, with
the goal of avoiding cut cells near the boundary of the computational domain.
This combination of methodologies has multiple advantages. In the first place,
since the Shifted Boundary Method always relies on the same background mesh,
there is no need to update the discretized parametric domain. Secondly, we
avoid the treatment of cut cell elements, which usually need particular
attention. Thirdly, since the whole background mesh is considered in the
reduced basis construction, the SBM allows for a smooth transition of the
reduced modes across the immersed domain boundary. The performances of the
method are verified in two dimensional heat transfer numerical examples.
