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Variationally consistent Hamiltonian model reduction

Anthony Gruber, Irina Tezaur

TL;DR

The paper addresses the difficulty of preserving Hamiltonian structure in projection-based ROMs by introducing a variationally consistent Petrov-Galerkin framework that yields reduced Hamiltonian dynamics for any linear basis. It achieves energy conservation and a valid reduced symplectic form through the reduced inverse-transpose of the projected Symplectic matrix, and extends this to nonintrusive OpInf via re-projection and centering to recover intrusive performance. The authors provide an interpretable error decomposition into basis projection error and deviation from canonicity, and demonstrate superior long-time accuracy and stability in 3D linear elasticity problems compared to previous methods. The approach integrates with existing ROM techniques (centering, re-projection, OpInf) to deliver robust, structure-preserving reduced models applicable to large-scale conservative systems with realistic material parameters.

Abstract

Though ubiquitous as first-principles models for conservative phenomena, Hamiltonian systems present numerous challenges for model reduction even in relatively simple, linear cases. Here, we present a method for the projection-based model reduction of canonical Hamiltonian systems that is variationally consistent for any choice of linear reduced basis: Hamiltonian models project to Hamiltonian models. Applicable in both intrusive and nonintrusive settings, the proposed method is energy-conserving and symplectic, with error provably decomposable into a data projection term and a term measuring deviation from canonical form. Examples from linear elasticity with realistic material parameters are used to demonstrate the advantages of a variationally consistent approach, highlighting the steady convergence exhibited by consistent models where previous methods reliant on inconsistent techniques or specially designed bases exhibit unacceptably large errors.

Variationally consistent Hamiltonian model reduction

TL;DR

The paper addresses the difficulty of preserving Hamiltonian structure in projection-based ROMs by introducing a variationally consistent Petrov-Galerkin framework that yields reduced Hamiltonian dynamics for any linear basis. It achieves energy conservation and a valid reduced symplectic form through the reduced inverse-transpose of the projected Symplectic matrix, and extends this to nonintrusive OpInf via re-projection and centering to recover intrusive performance. The authors provide an interpretable error decomposition into basis projection error and deviation from canonicity, and demonstrate superior long-time accuracy and stability in 3D linear elasticity problems compared to previous methods. The approach integrates with existing ROM techniques (centering, re-projection, OpInf) to deliver robust, structure-preserving reduced models applicable to large-scale conservative systems with realistic material parameters.

Abstract

Though ubiquitous as first-principles models for conservative phenomena, Hamiltonian systems present numerous challenges for model reduction even in relatively simple, linear cases. Here, we present a method for the projection-based model reduction of canonical Hamiltonian systems that is variationally consistent for any choice of linear reduced basis: Hamiltonian models project to Hamiltonian models. Applicable in both intrusive and nonintrusive settings, the proposed method is energy-conserving and symplectic, with error provably decomposable into a data projection term and a term measuring deviation from canonical form. Examples from linear elasticity with realistic material parameters are used to demonstrate the advantages of a variationally consistent approach, highlighting the steady convergence exhibited by consistent models where previous methods reliant on inconsistent techniques or specially designed bases exhibit unacceptably large errors.
Paper Structure (14 sections, 4 theorems, 49 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 4 theorems, 49 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Variationally Consistent Intrusive Hamiltonian ROM
  5. Variationally consistent Petrov-Galerkin projection
  6. Intrusive centering
  7. Variationally Consistent Hamiltonian Operator Inference
  8. OpInf for canonical Hamiltonian systems
  9. Re-projection
  10. Nonintrusive centering
  11. Variationally consistent Hamiltonian OpInf
  12. Analysis
  13. Numerical Examples
  14. Toy Example: One-dimensional (1D) Wave

Key Result

Proposition 2.1

The reduced-order symplectic matrix $\hat{~J}=~U^\intercal~J~U\in\mathbb{R}^{2m\times 2m}$ is invertible for almost every column-orthonormal reduced basis $~U\in\mathbb{R}^{2M\times 2m}, ~U^\intercal~U=~I$.

Figures (1)

  • Figure 1: A diagram illustrating the behavior of Hamiltonian/Lagrangian FOMs under Galerkin projection. While (canonical) Hamiltonian and Lagrangian systems which are "regular" are equivalent under the Legendre transformation (L.T.) at both the FOM and ROM levels, this equivalence does not commute with Galerkin projection onto a general reduced basis $~U\in\mathbb{R}^{N\times n}$. Conversely, the variationally consistent ROM presented here projects Hamiltonian FOMs to Hamiltonian ROMs in every case.

Theorems & Definitions (16)

  • Remark 1.1
  • Remark 2.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 4.1
  • ...and 6 more