• Corpus ID: 49592937

Symplectic Model-Reduction with a Weighted Inner Product

@article{Afkham2018SymplecticMW,
  title={Symplectic Model-Reduction with a Weighted Inner Product},
  author={Babak Maboudi Afkham and Ashish Bhatt and Bernard Haasdonk and Jan S. Hesthaven},
  journal={arXiv: Numerical Analysis},
  year={2018}
}
In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more… 

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References

SHOWING 1-10 OF 41 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.
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.
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems
TLDR
The method constructs a closed reduced Hamiltonian system by coupling the full model with a canonical heat bath, which allows the reduced system to be integrated with a symplectic integrator, resulting in a correct dissipation of energy, preservation of the total energy and, ultimately, conserving the stability of the solution.
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.
Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems
TLDR
A structure-preserving model reduction approach applicable to large-scale, nonlinear port-Hamiltonian systems that ensures the retention of port- Hamiltonian structure which assures the stability and passivity of the reduced model.
Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control
Optimal control problems for partial differential equation are often hard to tackle numerically because their discretization leads to very large scale optimization problems. Therefore, different
Structure-preserving Exponential Runge-Kutta Methods
TLDR
Numerical experiments illustrate the higher-order accuracy and structure-preserving properties of various ERK methods, demonstrating clear advantages over classical conservative Runge--Kutta methods.
Structure‐preserving, stability, and accuracy properties of the energy‐conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models
The computational efficiency of a typical, projection‐based, nonlinear model reduction method hinges on the efficient approximation, for explicit computations, of the scalar projections onto a
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