number of DOFs; see, e.g., [@BKR2000; @MR2556843; @Clever2012; @MR2905006; @MR1887737; @MR2892950] and the references therein.

**Model order reduction for PDE constrained optimization and optimal control.** A more recent approach is the usage of model order reduction (MOR) methods in order to replace the FOM by a surrogate reduced order model (ROM) of possibly very low dimension. MOR is a very active research field that has seen tremendous development in recent years, both from a theoretical and application point of view. For an introduction and overview we refer to the monographs and collections [@BCOW2017; @BOP+2017; @HRS2016; @QMN2016]. A particular promising model reduction approach for parameterized partial differential equations (pPDEs) is the Reduced Basis (RB) method that relies on the approximation of the solution manifold of pPDEs by low dimensional linear approximation spaces that are spanned from suitably selected particular solutions, called snapshots. A posteriori error estimation for solutions of the ROM with respect to the FOM is the basis for efficient Greedy algorithms to select the snapshots in a quasi-optimal way [@BCD+2011; @Haasdonk2013]. Alternatively, construction of reduced bases using proper orthogonal decomposition (POD) may be used [@GV17]. The construction of a reduced basis and the respective projected ROM is generally called 