Structure-Preserving Model Reduction of Forced Hamiltonian Systems

TL;DR

This paper introduces a structure-preserving model reduction method for forced Hamiltonian systems that maintains the system's symplectic and forced structures, leading to efficient and stable reduced models.

Contribution

It develops two equivalent approaches for symplectic model reduction based on d'Alembert's principle, incorporating energy preservation and stability in reduced forced Hamiltonian systems.

Findings

Achieves computational savings for large-scale systems.

Automatically preserves dissipativity, boundedness, and stability.

Demonstrates effectiveness through numerical simulations.

Abstract

This paper reports a development in the proper symplectic decomposition (PSD) for model reduction of forced Hamiltonian systems. As an analogy to the proper orthogonal decomposition (POD), PSD is designed to build a symplectic subspace to fit empirical data. Our aim is two-fold. First, to achieve computational savings for large-scale Hamiltonian systems with external forces. Second, to simultaneously preserve the symplectic structure and the forced structure of the original system. We first reformulate d'Alembert's principle in the Hamiltonian form. Corresponding to the integral and local forms of d'Alembert's principle, we propose two different structure-preserving model reduction approaches to reconstruct low-dimensional systems, based on the variational principle and on the structure-preserving projection, respectively. These two approaches are proven to yield the same reduced system. Moreover, by incorporating the vector field into the data ensemble, we provided several algorithms for energy preservation. In a special case when the external force is described by the Rayleigh dissipative function, the proposed method automatically preserves the dissipativity, boundedness, and stability of the original system. The stability, accuracy, and efficiency of the proposed method are illustrated through numerical simulations of a dissipative wave equation.