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The nonzero gain coefficients of Sobol's sequences are always powers of two.
- Source :
-
Journal of Complexity . Apr2023, Vol. 75, pN.PAG-N.PAG. 1p. - Publication Year :
- 2023
-
Abstract
- When a plain Monte Carlo estimate on n samples has variance σ 2 / n , then scrambled digital nets attain a variance that is o (1 / n) as n → ∞. For finite n and an adversarially selected integrand, the variance of a scrambled (t , m , s) -net can be at most Γ σ 2 / n for a maximal gain coefficient Γ < ∞. The most widely used digital nets and sequences are those of Sobol'. It was previously known that Γ ⩽ 2 t 3 s for any nets in base 2. For digital nets, Dick and Pillichshammer (2010) obtained the bound 2 t + s. In this paper we study digital nets in base 2 and show that Γ ⩽ 2 t + s − 1 for such nets. This bound is a simple, but apparently unnoticed, consequence of a microstructure analysis by Niederreiter and Pirsic in 2001. We obtain a sharper bound that is smaller than this for some digital nets. Our main finding is that all nonzero gain coefficients must be powers of two. A consequence of this latter fact is a simplified algorithm for computing gain coefficients of digital nets in base 2. [ABSTRACT FROM AUTHOR]
- Subjects :
- *PLAINS
*FINITE, The
*MICROSTRUCTURE
*ALGORITHMS
Subjects
Details
- Language :
- English
- ISSN :
- 0885064X
- Volume :
- 75
- Database :
- Academic Search Index
- Journal :
- Journal of Complexity
- Publication Type :
- Academic Journal
- Accession number :
- 161401051
- Full Text :
- https://doi.org/10.1016/j.jco.2022.101700