A crucial challenge within snapshot-based POD model order reduction for time-dependent
systems lies in the input dependency. In the ‘offline phase’, the POD basis is computed
from snapshot data obtained by solving the high-fidelity model at several time instances.
If a dynamical structure is not captured by the snapshots, this feature will be missing
in the ROM simulation. Thus, the quality of the POD approximation can only ever be
as good as the input material. In this sense, the accuracy of the POD surrogate solution
is restricted by how well the snapshots represent the underlying dynamical system.
If one restricts the snapshot sampling process to uniform and static discretizations,
this may lead to very fine resolutions and thus large-scale systems which are expensive
to solve or even can not be realized numerically. Therefore, offline adaptation strategies
are introduced which aim to filter out the key dynamics. On the one hand, snapshot
location strategies detect suitable time instances at which the snapshots shall be
generated. On the other hand, adaptivity with respect to space enables us to resolve
important structures within the spatial domain. Motivated from an infinite-dimensional
perspective, we explain how POD in Hilbert spaces can be implemented. The advantage
of this approach is that it only requires the snapshots to lie in a common Hilbert
space. This results in a great flexibility concerning the actual discretization technique,
such that we even can consider r-adaptive snapshots or a blend of snapshots stemming
from different discretization methods. Moreover, in the context of optimal control
problems adaptive strategies are crucial in order to adjust the POD model according
to the current optimization iterate.