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This work proposes an adaptive structure-preserving model order reductionmethod for finite-dimensional parametrized Hamiltonian systems modelingnon-dissipative phenomena. To overcome the slowly decaying Kolmogorov widthtypical of transport problems, the full model is approximated on local reducedspaces that are adapted in time using dynamical low-rank approximationtechniques. The reduced dynamics is prescribed by approximating the symplecticprojection of the Hamiltonian vector field in the tangent space to the localreduced space. This ensures that the canonical symplectic structure of theHamiltonian dynamics is preserved during the reduction. In addition, accurateapproximations with low-rank reduced solutions are obtained by allowing thedimension of the reduced space to change during the time evolution. Wheneverthe quality of the reduced solution, assessed via an error indicator, is notsatisfactory, the reduced basis is augmented in the parameter direction that isworst approximated by the current basis. Extensive numerical tests involvingwave interactions, nonlinear transport problems, and the Vlasov equationdemonstrate the superior stability properties and considerable runtime speedupsof the proposed method as compared to global and traditional reduced basisapproaches.

Rank-adaptive structure-preserving model order reduction of Hamiltonian systems