• Corpus ID: 245353850

Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

@article{Buchfink2021SymplecticMR,
  title={Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds},
  author={Patrick Buchfink and Silke Glas and Bernard Haasdonk},
  journal={ArXiv},
  year={2021},
  volume={abs/2112.10815}
}
Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-nwidths such as certain transport-dominated problems, however, classical linear-subspace reducedorder models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like Model Order Reduction (MOR) on manifolds. Moreover, as we… 

Figures and Tables from this paper

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale mechanical systems
TLDR
This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian mechanical systems that exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangia ROM which can be learned from projected snapshot data.

References

SHOWING 1-10 OF 61 REFERENCES
Symplectic Model Reduction of Hamiltonian Systems
TLDR
This paper introduces three algorithms for PSD, which are based upon the cotangent lift, complex singular value decomposition, and nonlinear programming, and shows how PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems.
Dynamical reduced basis methods for Hamiltonian systems
TLDR
A nonlinear structure-preserving model reduction where the reduced phase space evolves in time and the resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluated flow velocity has a comparable cost.
Rank-adaptive structure-preserving reduced basis methods for Hamiltonian systems
TLDR
This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena that demonstrates the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches.
Structure Preserving Model Reduction of Parametric Hamiltonian Systems
TLDR
It is demonstrated that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure of Hamiltonian systems to ensure stability of the reduced model.
Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Hamiltonian Systems
TLDR
This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Hamiltonian systems that exploits knowledge of the Hamiltonian functional to define and parametrize a Hamiltonian ROM form which can be learned from data projected via symplectic projectors.
Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics
TLDR
This work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence and retains key properties such as energy conservation and symplectic time-evolution maps.
Symplectic Model Order Reduction with Non-Orthonormal Bases
Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time
PSD-GREEDY BASIS GENERATION FOR STRUCTURE-PRESERVING MODEL ORDER REDUCTION OF HAMILTONIAN SYSTEMS∗
TLDR
It is proved that this algorithm computes a symplectic basis when symplectic techniques are used for compression, and improvements of up to one order of magnitude in the relative reduction error are achievable with the new basis generation technique, the PSD-greedy.
Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
...
...