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 context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MO...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-04-01
|
| Series: | Mathematical and Computational Applications |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2297-8747/24/2/43 |
| _version_ | 1818982306047590400 |
|---|---|
| author | Patrick Buchfink Ashish Bhatt Bernard Haasdonk |
| author_facet | Patrick Buchfink Ashish Bhatt Bernard Haasdonk |
| author_sort | Patrick Buchfink |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-20T17:45:07Z |
| format | Article |
| id | doaj.art-de790f1112164df58d5652893a4bb718 |
| institution | Directory Open Access Journal |
| issn | 2297-8747 |
| language | English |
| last_indexed | 2024-12-20T17:45:07Z |
| publishDate | 2019-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-de790f1112164df58d5652893a4bb7182022-12-21T19:31:01ZengMDPI AGMathematical and Computational Applications2297-87472019-04-012424310.3390/mca24020043mca24020043Symplectic Model Order Reduction with Non-Orthonormal BasesPatrick Buchfink0Ashish Bhatt1Bernard Haasdonk2Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569 Stuttgart, GermanyIndian Institute of Technology (ISM), Dhanbad, Jharkhand 826004, IndiaInstitute of Applied Analysis and Numerical Simulation, University of Stuttgart, 70569 Stuttgart, GermanyParametric 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.https://www.mdpi.com/2297-8747/24/2/43symplectic model order reductionproper symplectic decomposition (PSD)structure preservation of symplecticityHamiltonian system |
| spellingShingle | Patrick Buchfink Ashish Bhatt Bernard Haasdonk Symplectic Model Order Reduction with Non-Orthonormal Bases Mathematical and Computational Applications symplectic model order reduction proper symplectic decomposition (PSD) structure preservation of symplecticity Hamiltonian system |
| title | Symplectic Model Order Reduction with Non-Orthonormal Bases |
| title_full | Symplectic Model Order Reduction with Non-Orthonormal Bases |
| title_fullStr | Symplectic Model Order Reduction with Non-Orthonormal Bases |
| title_full_unstemmed | Symplectic Model Order Reduction with Non-Orthonormal Bases |
| title_short | Symplectic Model Order Reduction with Non-Orthonormal Bases |
| title_sort | symplectic model order reduction with non orthonormal bases |
| topic | symplectic model order reduction proper symplectic decomposition (PSD) structure preservation of symplecticity Hamiltonian system |
| url | https://www.mdpi.com/2297-8747/24/2/43 |
| work_keys_str_mv | AT patrickbuchfink symplecticmodelorderreductionwithnonorthonormalbases AT ashishbhatt symplecticmodelorderreductionwithnonorthonormalbases AT bernardhaasdonk symplecticmodelorderreductionwithnonorthonormalbases |