Geometric optimization for structure-preserving model reduction of Hamiltonian systems

Thomas Bendokat, Ralf Zimmermann

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

56 Downloads (Pure)

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.
Original languageEnglish
Title of host publication10th Vienna International Conference on Mathematical Modelling : MATHMOD 2022
Volume55
PublisherElsevier
Publication date2022
Edition20
Pages457-462
DOIs
Publication statusPublished - 2022
Event10th Vienna International Conference on Mathematical Modelling: MATHMOD 2022 - Wien, Austria
Duration: 27. Jul 202229. Jul 2022
Conference number: 10

Conference

Conference10th Vienna International Conference on Mathematical Modelling
Number10
Country/TerritoryAustria
CityWien
Period27/07/202229/07/2022
SeriesIFAC-PapersOnLine
ISSN2405-8963

Keywords

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

Fingerprint

Dive into the research topics of 'Geometric optimization for structure-preserving model reduction of Hamiltonian systems'. Together they form a unique fingerprint.

Cite this