Piece-Wise Symplectic Model Reduction on Quadratically Embedded Manifolds

Silke Glas*, Hongliang Mu

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

Abstract

In this work, we present a piece-wise symplectic model order reduction (MOR) method for Hamiltonian systems on quadratically embedded manifolds. For Hamiltonian systems, which suffer from slowly decaying Kolmogorov N-widths, linear-subspace reduced-order models (ROMs) of low dimension can have insufficient accuracy. The recently proposed symplectic manifold Galerkin projection combined with the quadratic manifold cotangent lift approximation (QMCL) is a symplectic MOR method that can achieve higher accuracy than linear-subspace symplectic MOR methods. In this paper, we improve the online computational complexity and energy-preserving ability of the QMCL by proposing a piece-wise symplectic MOR approach. First, the QMCL map is approximated by a linear symplectic map on each discrete time-interval. Then, the symplectic Galerkin projection is applied to obtain a sequence of reduced-order Hamiltonian systems. In case that the Hamiltonian of the full-order model is a polynomial, the sequence of the Hamiltonians of the ROMs can be preserved up to a multiple of a pre-given tolerance used in the Newton iteration. In the numerical example, we investigate the approximation quality and the energy-preservation of the proposed algorithm.

Original languageEnglish
Title of host publicationNumerical Mathematics and Advanced Applications ENUMATH 2023
EditorsAdélia Sequeira, Ana Silvestre, Svilen S. Valtchev, João Janela
PublisherSpringer
Pages355-364
Number of pages10
Volume1 - European Conference
ISBN (Electronic)978-3-031-86173-4
ISBN (Print)978-3-031-86172-7, 978-3-031-86175-8
DOIs
Publication statusPublished - 2025
EventEuropean Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2023 - Lisbon, Portugal
Duration: 4 Sept 20238 Sept 2023

Publication series

NameLecture Notes in Computational Science and Engineering
PublisherSpringer
Volume153
ISSN (Print)1439-7358
ISSN (Electronic)2197-7100

Conference

ConferenceEuropean Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2023
Abbreviated titleENUMATH 2023
Country/TerritoryPortugal
CityLisbon
Period4/09/238/09/23

Keywords

  • 2025 OA procedure

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