This work proposes a hyper-reduction method for nonlinear parametric
dynamical systems characterized by gradient fields such as Hamiltonian systems
and gradient flows. The gradient structure is associated with conservation of
invariants or with dissipation and hence plays a crucial role in the
description of the physical properties of the system. Traditional
hyper-reduction of nonlinear gradient fields yields efficient approximations
that, however, lack the gradient structure. We focus on Hamiltonian gradients
and we propose to first decompose the nonlinear part of the Hamiltonian, mapped
into a suitable reduced space, into the sum of d terms, each characterized by a
sparse dependence on the system state. Then, the hyper-reduced approximation is
obtained via discrete empirical interpolation (DEIM) of the Jacobian of the
derived d-valued nonlinear function. The resulting hyper-reduced model retains
the gradient structure and its computationally complexity is independent of the
size of the full model. Moreover, a priori error estimates show that the
hyper-reduced model converges to the reduced model and the Hamiltonian is
asymptotically preserved. Whenever the nonlinear Hamiltonian gradient is not
globally reducible, i.e. its evolution requires high-dimensional DEIM
approximation spaces, an adaptive strategy is performed. This consists in
updating the hyper-reduced Hamiltonian via a low-rank correction of the DEIM
basis. Numerical tests demonstrate the applicability of the proposed approach
to general nonlinear operators and runtime speedups compared to the full and
the reduced models