Symplectic Model Order Reduction with Non-Orthonormal Bases
1
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569 Stuttgart, Germany
2
Indian Institute of Technology (ISM), Dhanbad, Jharkhand 826004, India
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(2), 43; https://doi.org/10.3390/mca24020043
Received: 26 February 2019
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Revised: 15 April 2019
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Accepted: 17 April 2019
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Published: 21 April 2019
(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)
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 context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.
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Keywords:
symplectic model order reduction; proper symplectic decomposition (PSD); structure preservation of symplecticity; Hamiltonian system
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MDPI and ACS Style
Buchfink, P.; Bhatt, A.; Haasdonk, B. Symplectic Model Order Reduction with Non-Orthonormal Bases. Math. Comput. Appl. 2019, 24, 43. https://doi.org/10.3390/mca24020043
AMA Style
Buchfink P, Bhatt A, Haasdonk B. Symplectic Model Order Reduction with Non-Orthonormal Bases. Mathematical and Computational Applications. 2019; 24(2):43. https://doi.org/10.3390/mca24020043
Chicago/Turabian StyleBuchfink, Patrick, Ashish Bhatt, and Bernard Haasdonk. 2019. "Symplectic Model Order Reduction with Non-Orthonormal Bases" Mathematical and Computational Applications 24, no. 2: 43. https://doi.org/10.3390/mca24020043
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