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This work proposes a hyper-reduction method for nonlinear parametricdynamical systems characterized by gradient fields such as Hamiltonian systemsand gradient flows. The gradient structure is associated with conservation ofinvariants or with dissipation and hence plays a crucial role in thedescription of the physical properties of the system. Traditionalhyper-reduction of nonlinear gradient fields yields efficient approximationsthat, however, lack the gradient structure. We focus on Hamiltonian gradientsand we propose to first decompose the nonlinear part of the Hamiltonian, mappedinto a suitable reduced space, into the sum of d terms, each characterized by asparse dependence on the system state. Then, the hyper-reduced approximation isobtained via discrete empirical interpolation (DEIM) of the Jacobian of thederived d-valued nonlinear function. The resulting hyper-reduced model retainsthe gradient structure and its computationally complexity is independent of thesize of the full model. Moreover, a priori error estimates show that thehyper-reduced model converges to the reduced model and the Hamiltonian isasymptotically preserved. Whenever the nonlinear Hamiltonian gradient is notglobally reducible, i.e. its evolution requires high-dimensional DEIMapproximation spaces, an adaptive strategy is performed. This consists inupdating the hyper-reduced Hamiltonian via a low-rank correction of the DEIMbasis. Numerical tests demonstrate the applicability of the proposed approachto general nonlinear operators and runtime speedups compared to the full andthe reduced models.

Gradient-preserving hyper-reduction of nonlinear dynamical systems via discrete empirical interpolation