Symplectic Model Reduction of Hamiltonian Systems

TL;DR

This paper introduces a symplectic model reduction method called proper symplectic decomposition (PSD) for Hamiltonian systems, which preserves energy and stability while reducing computational costs, especially for long-term simulations.

Contribution

The paper proposes a novel PSD technique with three algorithms that maintains symplectic structure and stability, improving upon classical methods for Hamiltonian system reduction.

Findings

Preserves system energy and stability.

Reduces computational cost for nonlinear systems.

Effective for long-time integration of wave equations.

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.