  In the recent years, considerable attention has been paid to preserving
structures and invariants in reduced basis methods, in order to enhance the
stability and robustness of the reduced system. In the context of Hamiltonian
systems, symplectic model reduction seeks to construct a reduced system that
preserves the symplectic symmetry of Hamiltonian systems. However, symplectic
methods are based on the standard Euclidean inner products and are not suitable
for problems equipped with a more general inner product. In this paper, we
generalize symplectic model reduction to allow for the norms and inner products
that are most appropriate to the problem while preserving the symplectic
symmetry of the Hamiltonian systems. To construct a reduced basis and
accelerate the evaluation of nonlinear terms, a greedy generation of a
symplectic basis is proposed. Furthermore, it is shown that the greedy approach
yields a norm-bounded reduced basis. The accuracy and the stability of this
model reduction technique are illustrated through the development of reduced
models for a vibrating elastic beam and the sine-Gordon equation.
