A randomized lattice rule without component-by-component construction
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909392990896128 |
|---|---|
| author | Goda, Takashi |
| author_facet | Goda, Takashi |
| contents | We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors $r$ times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing $r$ logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A randomized lattice rule without component-by-component construction Goda, Takashi Numerical Analysis We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors $r$ times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing $r$ logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings. |
| title | A randomized lattice rule without component-by-component construction |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2403.02660 |