  Parametric high-fidelity simulations are of interest for a wide range of
applications. But the restriction of computational resources renders such
models to be inapplicable in a real-time context or in multi-query scenarios.
Model order reduction (MOR) is used to tackle this issue. Recently, MOR is
extended to preserve specific structures of the model throughout the reduction,
e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as
symplectic MOR. It is based on the classical projection-based MOR and uses a
symplectic reduced order basis (ROB). Such a ROB can be derived in a
data-driven manner with the Proper Symplectic Decomposition (PSD) in the form
of a minimization problem. Due to the strong nonlinearity of the minimization
problem, it is unclear how to efficiently find a global optimum. In our paper,
we show that current solution procedures almost exclusively yield suboptimal
solutions by restricting to orthonormal ROBs. As new methodological
contribution, we propose a new method which eliminates this restriction by
generating non-orthonormal ROBs. In the numerical experiments, we examine the
different techniques for a classical linear elasticity problem and observe that
the non-orthonormal technique proposed in this paper shows superior results
with respect to the error introduced by the reduction.
