## Abstract

Classical model reduction methods disregard the special symplectic structure associated with Hamiltonian systems. A key challenge in projection-based approaches is to construct a symplectic basis that captures the essential system information. This necessitates the computation of a so-called proper symplectic decomposition (PSD) of a given sample data set. The PSD problem allows for a canonical formulation as an optimization problem on the symplectic

Stiefel manifold. However, as with their Euclidean counterparts, symplectic projectors only depend on the underlying symplectic subspaces and not on the selected symplectic bases. This motivates to tackle the PSD problem as a Riemannian optimization problem on the symplectic Grassmann manifold, i.e., the matrix manifold of symplectic projectors. Initial investigations on this manifold feature in a recent preprint of the authors. In this work, we investigate

the feasibility and performance of this approach on two academic numerical examples. More precisely, we calculate an optimized PSD for snapshot matrices that stem from solving the one-dimensional linear wave equation and the one-dimensional nonlinear Schrödinger equation.

Stiefel manifold. However, as with their Euclidean counterparts, symplectic projectors only depend on the underlying symplectic subspaces and not on the selected symplectic bases. This motivates to tackle the PSD problem as a Riemannian optimization problem on the symplectic Grassmann manifold, i.e., the matrix manifold of symplectic projectors. Initial investigations on this manifold feature in a recent preprint of the authors. In this work, we investigate

the feasibility and performance of this approach on two academic numerical examples. More precisely, we calculate an optimized PSD for snapshot matrices that stem from solving the one-dimensional linear wave equation and the one-dimensional nonlinear Schrödinger equation.

Original language | English |
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Title of host publication | 10th Vienna International Conference on Mathematical Modelling : MATHMOD 2022 |

Volume | 55 |

Publisher | Elsevier |

Publication date | 2022 |

Edition | 20 |

Pages | 457-462 |

DOIs | |

Publication status | Published - 2022 |

Event | 10th Vienna International Conference on Mathematical Modelling: MATHMOD 2022 - Wien, Austria Duration: 27. Jul 2022 → 29. Jul 2022 Conference number: 10 |

### Conference

Conference | 10th Vienna International Conference on Mathematical Modelling |
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Number | 10 |

Country/Territory | Austria |

City | Wien |

Period | 27/07/2022 → 29/07/2022 |

Series | IFAC-PapersOnLine |
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ISSN | 2405-8963 |

## Keywords

- Hamiltonian Systems
- Model Reduction
- Proper Symplectic Decomposition
- Riemannian Optimization
- Symplectic Grassmann Manifold
- Symplectic Stiefel Manifold