Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dictionary-based Online-adaptive Structure-preserving Model Order Reduction for Parametric Hamiltonian Systems

Robin Herkert, Patrick Buchfink, Bernard Haasdonk

TL;DR

This work tackles MOR for parametric Hamiltonian systems where slowly decaying $n$-widths hinder projection-based reduction, by introducing a dictionary-based online-adaptive, structure-preserving MOR framework. Central contributions include DB-cSVD for online, dictionary-driven symplectic basis construction and DB-DEIM/DB-SDEIM for efficient, online hyper-reduction of nonlinear terms, all supported by an offline-online decomposition that makes online costs independent of the full state dimension. The authors provide a Hamiltonian-error bound for basis changes and demonstrate substantial speedups and accurate energy behavior on a 2D linear wave equation and a nonlinear Sine-Gordon model, with much smaller average basis sizes than standard POD-based approaches. This dictionary-based approach enables real-time or many-query scenarios for parametric Hamiltonian systems while preserving the energy structure and stability of the reduced models.

Abstract

Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e. the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.

Dictionary-based Online-adaptive Structure-preserving Model Order Reduction for Parametric Hamiltonian Systems

TL;DR

This work tackles MOR for parametric Hamiltonian systems where slowly decaying -widths hinder projection-based reduction, by introducing a dictionary-based online-adaptive, structure-preserving MOR framework. Central contributions include DB-cSVD for online, dictionary-driven symplectic basis construction and DB-DEIM/DB-SDEIM for efficient, online hyper-reduction of nonlinear terms, all supported by an offline-online decomposition that makes online costs independent of the full state dimension. The authors provide a Hamiltonian-error bound for basis changes and demonstrate substantial speedups and accurate energy behavior on a 2D linear wave equation and a nonlinear Sine-Gordon model, with much smaller average basis sizes than standard POD-based approaches. This dictionary-based approach enables real-time or many-query scenarios for parametric Hamiltonian systems while preserving the energy structure and stability of the reduced models.

Abstract

Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e. the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.
Paper Structure (15 sections, 2 theorems, 82 equations, 13 figures, 4 tables, 3 algorithms)

This paper contains 15 sections, 2 theorems, 82 equations, 13 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Essentials
  3. Classical projection-based MOR
  4. Hamiltonian Systems and Symplectic MOR
  5. Dictionary-based symplectic MOR
  6. Workflow
  7. Online-selection
  8. Dictionary-based POD
  9. Dictionary-based DEIM
  10. Dictionary-based Symplectic MOR
  11. Error Analysis
  12. Numerical Experiments
  13. 2D linear wave equation
  14. Sine-Gordon Equation
  15. Conclusion and Outlook

Key Result

Proposition 1

Let $[\hat{{\bm{P}}}_{{\bm{D}}_{\text{P}}},$$\hat{{\bm{\rho}}}^{{\bm{D}}_{\text{P}}}]$$:=$$\texttt{DEIM\_idx}(\hat{{\bm{U}}})$ be the selection matrix and index vector corresponding to the dictionary ${\bm{D}}_{\text{P}}$. Let $[{\bm{P}}, {\bm{\rho}}]= \texttt{DEIM\_idx}({\bm{U}})$, and $[{\bm{P}}_

Figures (13)

  • Figure 1: Workflow sketch for the online-phase of symplectic dictionary-based MOR
  • Figure 2: Linear wave equation: Average relative reduction error over number of basis vectors and online-runtime, reproduction experiment
  • Figure 3: Linear wave equation: Average relative error in Hamiltonian over time-steps, reproduction experiment
  • Figure 4: Linear wave equation: Average relative error in Hamiltonian over time-steps, reproduction experiment
  • Figure 5: Linear wave equation: Average relative reduction error over number of basis vectors and online-runtime, generalization experiment
  • ...and 8 more figures

Theorems & Definitions (5)

  • Proposition 1: Equivalence of DB-DEIM index selection
  • Proof 1
  • Proposition 2: Basis Change Hamiltonian Error Bound
  • Proof 2
  • Remark 1