  In PDE-constrained optimization, proper orthogonal decomposition (POD)
provides a surrogate model of a (potentially expensive) PDE discretization, on
which optimization iterations are executed. Because POD models usually provide
good approximation quality only locally, they have to be updated during
optimization. Updating the POD model is usually expensive, however, and
therefore often impossible in a model-predictive control (MPC) context. Thus,
reduced models of mediocre quality might be accepted. We take the view of a
simplified Newton method for solving semilinear evolution equations to derive
an algorithm that can serve as an offline phase to produce a POD model.
Approaches that build the POD model with impulse response snapshots can be
regarded as the first Newton step in this context.
  In particular, POD models that are based on impulse response snapshots are
extended by adding a second simplified Newton step. This procedure improves the
approximation quality of the POD model significantly by introducing a moderate
amount of extra computational costs during optimization or the MPC loop. We
illustrate our findings with an example satisfying our assumptions.
