Minimal Subsampled Rank-1 Lattices for Multivariate Approximation with Optimal Convergence Rate
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914320348086272 |
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| author | Bartel, Felix Gilbert, Alexander D. Kuo, Frances Y. Sloan, Ian H. |
| author_facet | Bartel, Felix Gilbert, Alexander D. Kuo, Frances Y. Sloan, Ian H. |
| contents | In this paper we show error bounds for randomly subsampled rank-1 lattices. We pay particular attention to the ratio of the size of the subset to the size of the initial lattice, which is decisive for the computational complexity. In the special case of Korobov spaces, we achieve the optimal polynomial sampling complexity whilst having the smallest initial lattice possible. We further characterize the frequency index set for which a given lattice is reconstructing by using the reciprocal of the worst-case error achieved using the lattice in question. This connects existing approaches used in proving error bounds for lattices. We make detailed comments on the implementation and test different algorithms using the subsampled lattice in numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal Subsampled Rank-1 Lattices for Multivariate Approximation with Optimal Convergence Rate Bartel, Felix Gilbert, Alexander D. Kuo, Frances Y. Sloan, Ian H. Numerical Analysis 41A25, 94A20 In this paper we show error bounds for randomly subsampled rank-1 lattices. We pay particular attention to the ratio of the size of the subset to the size of the initial lattice, which is decisive for the computational complexity. In the special case of Korobov spaces, we achieve the optimal polynomial sampling complexity whilst having the smallest initial lattice possible. We further characterize the frequency index set for which a given lattice is reconstructing by using the reciprocal of the worst-case error achieved using the lattice in question. This connects existing approaches used in proving error bounds for lattices. We make detailed comments on the implementation and test different algorithms using the subsampled lattice in numerical experiments. |
| title | Minimal Subsampled Rank-1 Lattices for Multivariate Approximation with Optimal Convergence Rate |
| topic | Numerical Analysis 41A25, 94A20 |
| url | https://arxiv.org/abs/2506.07729 |