Fast differentiation of hyperbolic chaos
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866912759909711872 |
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| author | Ni, Angxiu |
| author_facet | Ni, Angxiu |
| contents | We derive and prove the `fast response' formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic chaotic systems. The expression is pointwisely defined so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. It has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm's cost is solving $u$, the unstable dimension, many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_00595 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Fast differentiation of hyperbolic chaos Ni, Angxiu Dynamical Systems Chaotic Dynamics We derive and prove the `fast response' formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic chaotic systems. The expression is pointwisely defined so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. It has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm's cost is solving $u$, the unstable dimension, many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods. |
| title | Fast differentiation of hyperbolic chaos |
| topic | Dynamical Systems Chaotic Dynamics |
| url | https://arxiv.org/abs/2009.00595 |