  While reduced-order models (ROMs) have been popular for efficiently solving
large systems of differential equations, the stability of reduced models over
long-time integration is of present challenges. We present a greedy approach
for ROM generation of parametric Hamiltonian systems that captures the
symplectic structure of Hamiltonian systems to ensure stability of the reduced
model. Through the greedy selection of basis vectors, two new vectors are added
at each iteration to the linear vector space to increase the accuracy of the
reduced basis. We use the error in the Hamiltonian due to model reduction as an
error indicator to search the parameter space and identify the next best basis
vectors. Under natural assumptions on the set of all solutions of the
Hamiltonian system under variation of the parameters, we show that the greedy
algorithm converges with exponential rate. Moreover, we demonstrate that
combining the greedy basis with the discrete empirical interpolation method
also preserves the symplectic structure. This enables the reduction of the
computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy,
and stability of this model reduction technique is illustrated through
simulations of the parametric wave equation and the parametric Schrodinger
equation.
