A popular example for structured, mathematical models are Hamiltonian systems which are known for the characteristic property to preserve the Hamiltonian function. With symplectic model order reduction (symplectic MOR) [2, 3], the dimension of such systems can be reduced while preserving the underlying symplectic structure. In this talk, we investigate a new basis generation technique for symplectic MOR  motivated by a symplectic analogue to the Gramian matrix. For the generated bases, we can quantify the error of the reduction with respect to the training data analytically. Furthermore, the new method complements existing basis generation techniques as it is able to compute a symplectic, non-orthogonal basis instead of a symplectic, orthogonal basis. A numerical example in an uncertainty quantification setting will demonstrate the superiority of the new basis generation technique over conventional (symplectic) MOR approaches.
 P. Buchfink, A. Bhatt, and B. Haasdonk. Symplectic Model Order Reduction with Non-Orthonormal Bases. Mathematical and Computational Applications, 24(2), 2019.
 B. Maboudi Afkham and J. Hesthaven. Structure Preserving Model Reduction of Parametric Hamiltonian Systems. SIAM Journal on Scientific Computing, 39(6):A2616-A2644, 2017.
 L. Peng and K. Mohseni. Symplectic Model Reduction of Hamiltonian Systems. SIAM Journal on Scientific Computing, 38(1):A1-A27, 2016.