## Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

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Josef Dick, Takashi Goda and Kosuke Suzuki
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## Abstract:

We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters, $\gamma _j$, satisfy the summability condition $\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }<\infty$, where $\alpha$ is a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces.

In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition $\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }<\infty$ holds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.

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## Additional Information

**Josef Dick**- Affiliation: School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW, Australia
- MR Author ID: 735535
- ORCID: 0000-0003-0142-6022
- Email: josef.dick@unsw.edu.au
**Takashi Goda**- Affiliation: School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
- MR Author ID: 936012
- ORCID: 0000-0001-6055-8055
- Email: goda@frcer.t.u-tokyo.ac.jp
**Kosuke Suzuki**- Affiliation: Graduate School of Advanced Science and Engineering, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima City, Hiroshima 739-8526, Japan
- MR Author ID: 1057946
- Email: kosuke-suzuki@hiroshima-u.ac.jp
- Received by editor(s): September 23, 2021
- Received by editor(s) in revised form: March 21, 2022, and May 12, 2022
- Published electronically: August 5, 2022
- Additional Notes: The first author was partly supported by the Australian Research Council Discovery Project DP190101197. The second author was supported by JSPS KAKENHI Grant Number 20K03744. The third author was supported by JSPS KAKENHI Grant Number 20K14326.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2771-2801 - MSC (2020): Primary 11K45, 65C05, 65D30, 65D32
- DOI: https://doi.org/10.1090/mcom/3769
- MathSciNet review: 4473103