Approximating maps into manifolds with lower curvature bounds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915753675980800 |
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| author | Jacobsson, Simon Vandebril, Raf van der Veken, Joeri Vannieuwenhoven, Nick |
| author_facet | Jacobsson, Simon Vandebril, Raf van der Veken, Joeri Vannieuwenhoven, Nick |
| contents | Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for functions into linear spaces in such a way that we can upper bound the forward error in terms of a lower bound on the manifold's sectional curvature. Furthermore, when the sectional curvature is nonnegative, such as for compact Lie groups, the error is guaranteed to not be worse than in the linear case.
We implement the algorithm in a Julia package ManiFactor.jl and apply it to two example problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16785 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximating maps into manifolds with lower curvature bounds Jacobsson, Simon Vandebril, Raf van der Veken, Joeri Vannieuwenhoven, Nick Numerical Analysis Differential Geometry 15A69, 53Z99, 65D15 Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for functions into linear spaces in such a way that we can upper bound the forward error in terms of a lower bound on the manifold's sectional curvature. Furthermore, when the sectional curvature is nonnegative, such as for compact Lie groups, the error is guaranteed to not be worse than in the linear case. We implement the algorithm in a Julia package ManiFactor.jl and apply it to two example problems. |
| title | Approximating maps into manifolds with lower curvature bounds |
| topic | Numerical Analysis Differential Geometry 15A69, 53Z99, 65D15 |
| url | https://arxiv.org/abs/2403.16785 |