Model reduction on manifolds: A differential geometric framework
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911822710308864 |
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| author | Buchfink, Patrick Glas, Silke Haasdonk, Bernard Unger, Benjamin |
| author_facet | Buchfink, Patrick Glas, Silke Haasdonk, Bernard Unger, Benjamin |
| contents | Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01963 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Model reduction on manifolds: A differential geometric framework Buchfink, Patrick Glas, Silke Haasdonk, Bernard Unger, Benjamin Numerical Analysis 34A26, 34C20, 37C05, 37N30, 65P10 Using nonlinear projections and preserving structure in model order reduction (MOR) are currently active research fields. In this paper, we provide a novel differential geometric framework for model reduction on smooth manifolds, which emphasizes the geometric nature of the objects involved. The crucial ingredient is the construction of an embedding for the low-dimensional submanifold and a compatible reduction map, for which we discuss several options. Our general framework allows capturing and generalizing several existing MOR techniques, such as structure preservation for Lagrangian- or Hamiltonian dynamics, and using nonlinear projections that are, for instance, relevant in transport-dominated problems. The joint abstraction can be used to derive shared theoretical properties for different methods, such as an exact reproduction result. To connect our framework to existing work in the field, we demonstrate that various techniques for data-driven construction of nonlinear projections can be included in our framework. |
| title | Model reduction on manifolds: A differential geometric framework |
| topic | Numerical Analysis 34A26, 34C20, 37C05, 37N30, 65P10 |
| url | https://arxiv.org/abs/2312.01963 |