In this work we propose tailored model order reduction for varying boundary
optimal control problems governed by parametric partial differential equations.
With varying boundary control, we mean that a specific parameter changes where
the boundary control acts on the system. This peculiar formulation might
benefit from model order reduction. Indeed, fast and reliable simulations of
this model can be of utmost usefulness in many applied fields, such as
geophysics and energy engineering. However, varying boundary control features
very complicated and diversified parametric behaviour for the state and adjoint
variables. The state solution, for example, changing the boundary control
parameter, might feature transport phenomena. Moreover, the problem loses its
affine structure. It is well known that classical model order reduction
techniques fail in this setting, both in accuracy and in efficiency. Thus, we
propose reduced approaches inspired by the ones used when dealing with
wave-like phenomena. Indeed, we compare standard proper orthogonal
decomposition with two tailored strategies: geometric recasting and local
proper orthogonal decomposition. Geometric recasting solves the optimization
system in a reference domain simplifying the problem at hand avoiding
hyper-reduction, while local proper orthogonal decomposition builds local bases
to increase the accuracy of the reduced solution in very general settings
(where geometric recasting is unfeasible). We compare the various approaches on
two different numerical experiments based on geometries of increasing
complexity