# Symplectic Model Reduction of Hamiltonian Systems

Source: [Symplectic Model Reduction of Hamiltonian Systems](<https://www.sota2.com/research/paper/68fa519e1e209131ce974eae>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-06-05T05:00:26.567Z

Authors: Liqian Peng, Kamran Mohseni

Published: 2014-07-23T07:25:33.000Z

## Abstract

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon: the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Reconstruction on Crystal Lattice System](<https://www.sota2.com/research/sota/reconstruction-on-crystal-lattice-system.md>): Reconstruction; Crystal Lattice System; Absolute Error: 2.0306.

- [Full Prediction on Two-Stream System](<https://www.sota2.com/research/sota/full-prediction-on-two-stream-system.md>): Full Prediction; Two-Stream System; Absolute Error: 4.8566.

- [Full Prediction on Tokamak System](<https://www.sota2.com/research/sota/full-prediction-on-tokamak-system.md>): Full Prediction; Tokamak System; Absolute Error: 1.0956.

- [Reconstruction on Two-Stream System](<https://www.sota2.com/research/sota/reconstruction-on-two-stream-system.md>): Reconstruction; Two-Stream System; Absolute Error: 1.2481.

- [Reconstruction on Tokamak System](<https://www.sota2.com/research/sota/reconstruction-on-tokamak-system.md>): Reconstruction; Tokamak System; Absolute Error: 3.635.

- [Trajectory Prediction on Crystal Lattice System](<https://www.sota2.com/research/sota/trajectory-prediction-on-crystal-lattice-system.md>): Trajectory Prediction; Crystal Lattice System; Absolute Error: 3.7905.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1407.6118>)

[PDF](<https://arxiv.org/pdf/1407.6118.pdf>)