Structure-Preserving Galerkin POD Reduced-Order Modeling of Hamiltonian Systems

TL;DR

This paper introduces a structure-preserving Galerkin POD reduced-order model for Hamiltonian systems that maintains the system's Hamiltonian structure and improves approximation accuracy using shifted snapshots.

Contribution

It develops a novel ROM framework that preserves Hamiltonian structure and enhances Hamiltonian approximation, with rigorous error estimates and broad applicability.

Findings

Preserves Hamiltonian structure in reduced models

Improves Hamiltonian function approximation with shifted snapshots

Demonstrates effectiveness through numerical examples

Abstract

The proper orthogonal decomposition reduced-order models (POD-ROMs) have been widely used as a computationally efficient surrogate models in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy its Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduce-order modeling approach for the Hamiltonian system, which uses the traditional framework of Galerkin projection-based model reduction but modifies the ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use the POD basis from shifted snapshots to improve the Hamiltonian function approximation. We further derive a rigorous a priori error estimate of the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems etc.