Structure Preserving Reduced-Order Models of Hamiltonian Systems

Large-scale dynamical systems are expensive to simulate due to the computational cost accrued y the substantial number of degrees of freedom. To accelerate repeated numerical simulations of the systems, proper orthogonal decomposition reduced order models (POD-ROMs) have been developed. When applied to Hamiltonian systems, however, special care must be taken when performing the reduced order modeling to keep their energy-preserving nature. This work presents a survey of several structure-preserving reduced order models (SP-ROMs). In addition, this work employs the discrete empirical interpolation method (DEIM) and develops an SP-DEIM model for nonlinear Hamiltonian systems. The wave equation is considered as a test bed for the proposed SP-ROMs. Results show that a model using shifted snapshots to generate a POD basis (SP-ROM-2) is able to produce an exact Hamiltonian. In addition, it is shown that the SP-DEIM model is able to produce similar results with a nonlinear system as the SP-ROM-1 given a linear system