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

Markhasin, Lev

doi:10.4064/aa168-2-4

Cited by 16 publications

(12 citation statements)

References 30 publications

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“…Recently, Dick [5] proved that digital nets over Z 2 with large minimum Dick weight achieve the best possible order of the L p discrepancy for 1 < p < ∞ and for any number of dimensions. His result was generalized more recently by Markhasin [14] to digital nets over Z b for arbitrary b ≥ 2. We specialize the results of Dick [5, Corollary 2.2] and Markhasin [14, Theorem 1.8] on the L p discrepancy of digital nets for the two-dimensional case.…”

Section: Dick Weightmentioning

confidence: 93%

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On the L p discrepancy of two-dimensional folded Hammersley point sets

Goda

2014

Arch. Math.

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We give an explicit construction of two-dimensional point sets whose Lp discrepancy is of best possible order for all 1 ≤ p ≤ ∞. It is provided by folding Hammersley point sets in base b by means of the b-adic baker's transformation which has been introduced by Hickernell (2002) for b = 2 and Goda, Suzuki and Yoshiki (2013) for arbitrary b ∈ N, b ≥ 2. We prove that both the minimum Niederreiter-Rosenbloom-Tsfasman weight and the minimum Dick weight of folded Hammersley point sets are large enough to achieve the best possible order of Lp discrepancy for all 1 ≤

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Section: Dick Weightmentioning

confidence: 93%

“…As with Lemma 1, the following lemma shows how the minimum Dick weight connects with a structure of generating matrices of a digital net P , see [5,Chapter 15] and [14,Lemma 4.3].…”

Section: Dick Weightmentioning

confidence: 99%

On the L p discrepancy of two-dimensional folded Hammersley point sets

Goda

2014

Arch. Math.

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“…We will use Haar coefficients of the discrepancy function which are given by [33,Proposition 5.7] . Proposition 4.3.…”

Section: The Discrepancy Function and Its Haar Coefficientsmentioning

confidence: 99%

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

et al. 2015

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We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces S r p,q B(T d ) with dominating mixed smoothness 1/ p < r < 2. We show that order 2 digital nets achieve the optimal rate of convergence N −r (log N ) (d−1)(1−1/q) . The logarithmic term does not depend on r and hence improves the known bound of Dick (SIAM J Numer Anal 45: 2007) for the special case of Sobolev spaces H r mix (T d ). Secondly, the rate of convergence is independent of the integrability p of the Besov space, which allows for sacrificing integrability in order to gain Besov regularity. Our method combines characterizations of periodic Besov spaces with dominating mixed smoothness via Faber bases with sharp estimates of Haar coefficients for the discrepancy function of order 2 digital nets. Moreover, we provide numerical computations which indicate that this bound also holds for the case r = 2.Mathematics Subject Classification 11K06 · 11J71 · 42C10 · 46E35 · 65C05 · 65D30 · 65D32 · 91G60

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“…Explicit constructions are known and comprise certain digital nets introduced by Chen and Skriganov [2] and Skriganov [29] and higher order digital nets [4,9,10,22]. So for p = 2 the minimal extreme L p discrepancy for N-element point sets in dimension d is of exact order of magnitude (1 + log N) d−1 2 .…”

Section: Introductionmentioning

confidence: 99%

Point sets with optimal order of extreme and periodic discrepancy

Kritzinger¹,

Pillichshammer²

2021

Preprint

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We study the extreme and the periodic L p discrepancy of point sets in the d-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on periodic intervals modulo one. This is in contrast to the classical star discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. In a recent paper the authors together with Aicke Hinrichs studied relations between the L 2 versions of these notions of discrepancy and presented exact formulas for typical twodimensional quasi-Monte Carlo point sets. In this paper we study the general L p case and deduce the exact order of magnitude of the respective minimal discrepancy in the number N of elements of the considered point sets, for arbitrary but fixed dimension d, which is (log N ) (d−1)/2 .

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

Cited by 16 publications

References 30 publications

On the L p discrepancy of two-dimensional folded Hammersley point sets

On the L p discrepancy of two-dimensional folded Hammersley point sets

Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions

Point sets with optimal order of extreme and periodic discrepancy

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