Fast expansion into harmonics on the ball
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915268544954368 |
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| author | Kileel, Joe Marshall, Nicholas F. Mickelin, Oscar Singer, Amit |
| author_facet | Kileel, Joe Marshall, Nicholas F. Mickelin, Oscar Singer, Amit |
| contents | We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in $\mathbb{R}^3$. Given $\varepsilon > 0$, our algorithms achieve relative $\ell^1$ - $\ell^\infty$ accuracy $\varepsilon$ in time $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$, while the naïve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast expansion into harmonics on the ball Kileel, Joe Marshall, Nicholas F. Mickelin, Oscar Singer, Amit Numerical Analysis Classical Analysis and ODEs 65R10, 65D18, 42-04, 33C10, 33C55 We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in $\mathbb{R}^3$. Given $\varepsilon > 0$, our algorithms achieve relative $\ell^1$ - $\ell^\infty$ accuracy $\varepsilon$ in time $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$, while the naïve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples. |
| title | Fast expansion into harmonics on the ball |
| topic | Numerical Analysis Classical Analysis and ODEs 65R10, 65D18, 42-04, 33C10, 33C55 |
| url | https://arxiv.org/abs/2406.05922 |