# Where are the logs?

@inproceedings{Owen2021WhereAT, title={Where are the logs?}, author={Art B. Owen and Zexin Pan}, year={2021} }

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

- Computer Science, MedicineNature
- 1987

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 the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

- Mathematics
- 1960

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

- Mathematics, Computer ScienceArXiv
- 2021

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

- Mathematics, Physics
- 2016

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

- Computer Science, MathematicsJ. Complex.
- 2015

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

- Computer Science, MathematicsJ. Complex.
- 2014

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

- Mathematics, Computer ScienceJ. Complex.
- 2013

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

- Computer Science
- 2009

(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

- Mathematics
- 2009

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…