Where are the logs?

@inproceedings{Owen2021WhereAT,
  title={Where are the logs?},
  author={Art B. Owen and Zexin Pan},
  year={2021}
}
  • A. Owen, Zexin Pan
  • Published 13 October 2021
  • Mathematics, Computer Science
The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence isO (n−1 log(n) ) with r = d for extensible sequences and r = d − 1 otherwise. Such rates hold uniformly over all d dimensional integrands of Hardy-Krause variation one when using n evaluation points. Implicit in those bounds is that for any sequence of QMC points, the integrand can be chosen to depend on n. In this paper we show that rates with any r < (d − 1)/2 can hold when f is held fixed as n… 

References

SHOWING 1-10 OF 30 REFERENCES
Information-based complexity
Information-based complexity seeks to develop general results about the intrinsic difficulty of solving problems where available information is partial or approximate and to apply these results to
On irregularities of distribution.
§
On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals
by formulae of the form ~'iwi/(x,1 . . . . . xi~ ). He points out tha t the efficiency of such an integration formula m a y be gauged by considering how it fares when [(x 1 . . . . . xk) is the
The nonzero gain coefficients of Sobol's sequences are always powers of two
TLDR
This paper studies nets in base 2 and shows that Γ 6 2 for nets is a simple, but apparently unnoticed, consequence of a microstructure analysis in Niederreiter and Pirsic (2001).
Randomized quasi-Monte Carlo: An introduction for practitioners
We survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard
Proof techniques in quasi-Monte Carlo theory
TLDR
This survey paper discusses some tools and methods which are of use in quasi-Monte Carlo (QMC) theory, including reproducing and covariance kernels, Littlewood-Paley theory, Riesz products, Minkowski's fundamental theorem, exponential sums, diophantine approximation, Hoeffding's inequality and empirical processes.
A variant of Atanassov's method for (t, s)-sequences and (t, e, s)-sequences
TLDR
By using a combinatorial argument coupled with a careful worst-case analysis, the discrepancy bounds from Faure and Lemieux (2012) for ( t, s ) -sequences are improved and the asymptotic behavior of the bounds from Tezuka (2013) in the case of even bases is improved.
On the Koksma-Hlawka inequality
TLDR
A Koksma Hlawka type inequality is state which applies to piecewise smooth functions f\chi_{\Omega}$, with f smooth and $\Omega $ a Borel subset of $[0,1]^{d}$.
MINT – New Features and New Results
(t,m,s)-nets are among the best methods for the construction of low-discrepancy point sets in the s-dimensional unit cube. Various types of constructions and bounds are known today. Additionally
Some of Roth's ideas in Discrepancy Theory
We give a brief survey on some of the main ideas that Klaus Roth introduced into the study of irregularities of point distribution and, through a small selection of results, indicate how some of
...
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3
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