Optimal Bases for Symplectic Model Order Reduction of Canonizable Linear Hamiltonian Systems

Symplectic model order reduction is a structure-preserving reduction technique for Hamiltonian systems. Apart from theoretical results like the preservation of stability, it has been demonstrated to give improved numerical results compared to classical MOR techniques. A key element in this procedure is the choice of a good symplectic reduced order basis (ROB). In our work, we introduce so-called canonizable Hamiltonian systems in energy coordinates. For such systems with the assumption of a periodic solution, we derive a globally optimal symplectic ROB in the sense of the proper symplectic decomposition (PSD). To this end, we show that the proper orthogonal decomposition (POD) of a canonizable Hamiltonian system is automatically symplectic from which we deduce optimality of the PSD. To verify our findings numerically, we consider a reproduction experiment for the linear wave equation.