  In this work, the dual-weighted residual (DWR) method is applied to obtain a
certified incremental proper orthogonal decomposition (POD) based reduced order
model. A novel approach called MORe DWR (Model Order Rduction with
Dual-Weighted Residual error estimates) is being introduced. It marries
tensor-product space-time reduced-order modeling with time slabbing and an
incremental POD basis generation with goal-oriented error control based on
dual-weighted residual estimates. The error in the goal functional is being
estimated during the simulation and the POD basis is being updated if the
estimate exceeds a given threshold. This allows an adaptive enrichment of the
POD basis in case of unforeseen changes in the solution behavior which is of
high interest in many real-world applications. Consequently, the offline phase
can be skipped, the reduced-order model is being solved directly with the POD
basis extracted from the solution on the first time slab and -- if necessary --
the POD basis is being enriched on-the-fly during the simulation with
high-fidelity finite element solutions. Therefore, the full-order model solves
can be reduced to a minimum, which is demonstrated on numerical tests for the
heat equation and elastodynamics.
