Discrepancy, Integration and Tractability

Hinrichs, Aicke

doi:10.1007/978-3-642-41095-6_6

Cited by 10 publications

(16 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We are thus interested in the inverse star discrepancy problem with d = 8 and ε = 0.125. By similar arguments (see [17] for the details), we can solve the 15-dimensional problem also by finding a 4-dimensional point configuration of star discrepancy at most 0.0136 or a 5-dimensional point configuration of discrepancy at most 0.0575 with at most 764 points each. As we can see from Table 5: Approximated discrepancy results for the sequences presented in [24,25].…”

Section: Inverse Star Discrepancymentioning

confidence: 96%

See 1 more Smart Citation

Constructing low star discrepancy point sets with genetic algorithms

Doerr

Rainville

2013

Proceedings of the 15th Annual Conference on Genetic and Evolutionary Computation

View full text Add to dashboard Cite

Geometric discrepancies are standard measures to quantify the irregularity of distributions. They are an important notion in numerical integration. One of the most important discrepancy notions is the so-called star discrepancy. Roughly speaking, a point set of low star discrepancy value allows for a small approximation error in quasi-Monte Carlo integration. It is thus the most studied discrepancy notion.In this work we present a new algorithm to compute point sets of low star discrepancy. The two components of the algorithm (for the optimization and the evaluation, respectively) are based on evolutionary principles. Our algorithm clearly outperforms existing approaches. To the best of our knowledge, it is also the first algorithm which can be adapted easily to optimize inverse star discrepancies.

show abstract

Section: Inverse Star Discrepancymentioning

confidence: 96%

“…crepancy 0.1248 and size n = 104. This is much less than the requested 764 size bound asked for in [17,22]. Also the 5-dimensional version of the problem is solved easily; the point configuration has only 172 instead of the requested upper bound of 764 points.…”

Section: Inverse Star Discrepancymentioning

confidence: 96%

Constructing low star discrepancy point sets with genetic algorithms

Doerr

Rainville

2013

Proceedings of the 15th Annual Conference on Genetic and Evolutionary Computation

View full text Add to dashboard Cite

show abstract

“…As general references for studies of the discrepancy function we refer to the monographs [2], [16], [25], [31], [33] and surveys [4], [22], [29].…”

Section: Resultsmentioning

confidence: 99%

BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

Bilyk

Markhasin

2018

JAMA

View full text Add to dashboard Cite

In the current paper we obtain discrepancy estimates in exponential Orlicz and BMO spaces in arbitrary dimension d ≥ 3. In particular, we use dyadic harmonic analysis to prove that the dyadic product BMO and exp L 2/(d−1) norms of the discrepancy function of so-called digital nets of order two are bounded above by (log N ) d−1 2 . The latter bound has been recently conjectured in several papers and is consistent with the best known low-discrepancy constructions. Such estimates play an important role as an intermediate step between the well-understood L p bounds and the notorious open problem of finding the precise L ∞ asymptotics of the discrepancy function in higher dimensions, which is still elusive.

show abstract

“…As general references for studies of the discrepancy function we refer to the monographs [DP10], [NW10], [M99], [KN74] and surveys [B11], [Hi14], [M13c]. The problem of point disribution is closely related to numerical integration, we refer to [KN74,Chapter 2] and [DP10,Section 2.4] for more on this subject.…”

Section: Introduction and Resultsmentioning

confidence: 99%

L_p- and S_p,q^rB-discrepancy of (order 2) digital nets

Markhasin¹

2015

Acta Arith.

View full text Add to dashboard Cite

Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the L p -discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range and enlarge that range for order 2 digitals nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness is considered as well.

show abstract

Discrepancy, Integration and Tractability

Cited by 10 publications

References 49 publications

Constructing low star discrepancy point sets with genetic algorithms

Constructing low star discrepancy point sets with genetic algorithms

BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

L_p- and S_p,q^rB-discrepancy of (order 2) digital nets

Contact Info

Product

Resources

About

Discrepancy, Integration and Tractability

Cited by 10 publications

References 49 publications

Constructing low star discrepancy point sets with genetic algorithms

Constructing low star discrepancy point sets with genetic algorithms

BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension

Lp- and Sp,qrB-discrepancy of (order 2) digital nets

Contact Info

Product

Resources

About

L_p- and S_p,q^rB-discrepancy of (order 2) digital nets