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Error Analysis of Randomized Symplectic Model Order Reduction for Hamiltonian systems

Robin Herkert, Patrick Buchfink, Bernard Haasdonk, Johannes Rettberg, Jörg Fehr

TL;DR

This paper tackles preserving Hamiltonian structure in model order reduction by analyzing randomized symplectic basis generation. It develops two error bounds for rcSVD that depend on hyperparameters and proves quasi-optimality within the set of ortho-symplectic bases, showing rcSVD’s projection error is at most a constant factor worse than the optimal cSVD under suitable choices. A refined real-arithmetic formulation and extensive numerical experiments on a large 2D wave problem demonstrate that rcSVD, especially with nonzero power iterations, achieves near-equivalent accuracy to cSVD while offering significant offline-speedups. The work provides practical guidance for hyperparameter selection and lays groundwork for adaptive error estimation and broader randomized, structure-preserving MOR approaches for Hamiltonian systems.

Abstract

Solving high-dimensional dynamical systems in multi-query or real-time applications requires efficient surrogate modelling techniques, as e.g., achieved via model order reduction (MOR). If these systems are Hamiltonian systems their physical structure should be preserved during the reduction, which can be ensured by applying symplectic basis generation techniques such as the complex SVD (cSVD). Recently, randomized symplectic methods such as the randomized complex singular value decomposition (rcSVD) have been developed for a more efficient computation of symplectic bases that preserve the Hamiltonian structure during MOR. In the current paper, we present two error bounds for the rcSVD basis depending on the choice of hyperparameters and show that with a proper choice of hyperparameters, the projection error of rcSVD is at most a constant factor worse than the projection error of cSVD. We provide numerical experiments that demonstrate the efficiency of randomized symplectic basis generation and compare the bounds numerically.

Error Analysis of Randomized Symplectic Model Order Reduction for Hamiltonian systems

TL;DR

This paper tackles preserving Hamiltonian structure in model order reduction by analyzing randomized symplectic basis generation. It develops two error bounds for rcSVD that depend on hyperparameters and proves quasi-optimality within the set of ortho-symplectic bases, showing rcSVD’s projection error is at most a constant factor worse than the optimal cSVD under suitable choices. A refined real-arithmetic formulation and extensive numerical experiments on a large 2D wave problem demonstrate that rcSVD, especially with nonzero power iterations, achieves near-equivalent accuracy to cSVD while offering significant offline-speedups. The work provides practical guidance for hyperparameter selection and lays groundwork for adaptive error estimation and broader randomized, structure-preserving MOR approaches for Hamiltonian systems.

Abstract

Solving high-dimensional dynamical systems in multi-query or real-time applications requires efficient surrogate modelling techniques, as e.g., achieved via model order reduction (MOR). If these systems are Hamiltonian systems their physical structure should be preserved during the reduction, which can be ensured by applying symplectic basis generation techniques such as the complex SVD (cSVD). Recently, randomized symplectic methods such as the randomized complex singular value decomposition (rcSVD) have been developed for a more efficient computation of symplectic bases that preserve the Hamiltonian structure during MOR. In the current paper, we present two error bounds for the rcSVD basis depending on the choice of hyperparameters and show that with a proper choice of hyperparameters, the projection error of rcSVD is at most a constant factor worse than the projection error of cSVD. We provide numerical experiments that demonstrate the efficiency of randomized symplectic basis generation and compare the bounds numerically.
Paper Structure (10 sections, 11 theorems, 100 equations, 4 figures, 2 algorithms)

This paper contains 10 sections, 11 theorems, 100 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Structure-Preserving MOR
  3. Hamiltonian Systems and Symplectic MOR
  4. Symplectic basis generation using the complex SVD (cSVD)
  5. Symplectic basis generation using the randomized complex SVD
  6. Quasi-optimality for the rcSVD in the set of ortho-symplectic matrices
  7. Influence of Power Iterations on the Error (Bound)
  8. Formulation of rcSVD based on real numbers
  9. Numerical Experiments
  10. Conclusion and Outlook

Key Result

Lemma 1

Consider a matrix ${\bm{W}}\in {\mathbb{C}}^{n \times k}$ with orthonormal columns, and define the quantity $M := n \max\limits_{j=1,...,n}\vert\vert{\bm{e}}_j^{\textsf{T}} {\bm{W}}\vert\vert_2^2$ where ${\bm{e}}_j$ denotes the $j$-th unit vector. For a positive parameter $\alpha$, select the sampl with failure probability at most i.e., singbound holds with probability at least $1 - \mathcal{P}_

Figures (4)

  • Figure 1: Projection error for different values for $p_\mathrm{ovs}$ and $q_\mathrm{pow}$
  • Figure 2: Runtimes for different values for $p_\mathrm{ovs}$ and $q_\mathrm{pow}$
  • Figure 3: Effectivity of deterministic error bound for different values of $p_\mathrm{ovs}$ and $q_\mathrm{pow}$
  • Figure 4: Effectivity of probabilistic error bound for different values of $p_\mathrm{ovs}$ and $q_\mathrm{pow}$

Theorems & Definitions (23)

  • Definition 1
  • Lemma 1: Row sampling Tropp2011Ana
  • Lemma 2: Row norms Tropp2011Ana
  • proof
  • Proposition 1: The SRFT preserves geometry Tropp2011Ana
  • proof
  • Proposition 2: Deterministic error bound Halko2011
  • Theorem 1
  • proof
  • Proposition 3: Gu2015
  • ...and 13 more