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Model order reduction provides low-complexity high-fidelity surrogate modelsthat allow rapid and accurate solutions of parametric differential equations.The development of reduced order models for parametric nonlinear Hamiltoniansystems is challenged by several factors: (i) the geometric structure encodingthe physical properties of the dynamics; (ii) the slowly decaying Kolmogorovn-width of conservative dynamics; (iii) the gradient structure of the nonlinearflow velocity; (iv) high variations in the numerical rank of the state as afunction of time and parameters. We propose to address these aspects via astructure-preserving adaptive approach that combines symplectic dynamicallow-rank approximation with adaptive gradient-preserving hyper-reduction andparameters sampling. Additionally, we propose to vary in time the dimensions ofboth the reduced basis space and the hyper-reduction space by monitoring thequality of the reduced solution via an error indicator related to theprojection error of the Hamiltonian vector field. The resulting adaptivehyper-reduced models preserve the geometric structure of the Hamiltonian flow,do not rely on prior information on the dynamics, and can be solved at a costthat is linear in the dimension of the full order model and linear in thenumber of test parameters. Numerical experiments demonstrate the improvedperformances of the fully adaptive models compared to the original and reducedmodels.

Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems