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

Goda, Takashi

doi:10.1007/s00013-014-0698-1

Cited by 4 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further examples of two-dimensional finite point sets with optimal order of L 2 -discrepancy which are based on scrambled digital nets can be found in [17,18,19,20,22,23,26]. One prominent instance in this class are digit shifted Hammersley point sets.…”

Section: A Brief Survey Of Known Resultsmentioning

confidence: 99%

“…Finally it should be remarked that recently Goda [22] presented another modification of two-dimensional Hammersley point sets (in arbitrary base b) with optimal order of L p -discrepancy. He considered so-called two-dimensional folded Hammersley point sets which result from the application of the so-called tent (or bakers) transformation to the elements of the two-dimensional Hammersley point set.…”

Section: Final Remarksmentioning

confidence: 99%

See 1 more Smart Citation

Optimal order of $L_p$-discrepancy of digit shifted Hammersley point sets in dimension 2

Hinrichs,

Kritzinger,

Pillichshammer

2014

Preprint

View full text Add to dashboard Cite

It is well known that the two-dimensional Hammersley point set consisting of N = 2 n elements (also known as Roth net) does not have optimal order of Lp-discrepancy for p ∈ (1, ∞) in the sense of the lower bounds according to Roth (for p ∈ [2, ∞)) and Schmidt (for p ∈ (1, 2)). On the other hand, it is also known that slight modifications of the Hammersley point set can lead to the optimal order √ log N /N of L2-discrepancy, where N is the number of points. Among these are for example digit shifts or the symmetrization. In this paper we show that these modified Hammersley point sets also achieve optimal order of Lp-discrepancy for all p ∈ (1, ∞).

show abstract

Section: A Brief Survey Of Known Resultsmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Optimal order of $L_p$-discrepancy of digit shifted Hammersley point sets in dimension 2

Hinrichs,

Kritzinger,

Pillichshammer

2014

Preprint

View full text Add to dashboard Cite

show abstract

“…As P sym,n H consists of b m+2 points and the minimum Dick weight of P sym,n H is larger than 2m + 1 = 2(m + 2) − 3, the t-value in Proposition 27 of P sym,n H always equals 3 for any m ∈ N. This is why we simply write C p instead of C p,t in the above proposition. This t-value is same as that for two-dimensional folded Hammersley point sets as given in [7,Lemma 3.3]. Moreover, if one wants to get an explicit value of C p , it might be better to use the Littlewood-Paley inequality in conjunction with the Haar coefficients of the local discrepancy function for P sym,n H as done in [11], instead of our proof based on the linear independence properties.…”

mentioning

confidence: 84%

“…There are several ways to modify P H such that the modified point set has optimal order of L p discrepancy for p ∈ [1, ∞), see for instance [1,7,11].…”

Section: Introductionmentioning

confidence: 99%

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

Goda

2017

Uniform Distribution Theory

Self Cite

View full text Add to dashboard Cite

ABSTRACT. The notion of symmetrization, also known as Davenport's reflection principle, is well known in the area of the discrepancy theory and quasiMonte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over Z b . Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base b ∈ N, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over Z b in a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base b, and prove that the minimum Dick weight is large enough to achieve the best possible order of L p discrepancy for all 1 ≤ p < ∞.

show abstract

“…This fact has already been utilized to study the L p discrepancy of two-dimensional folded Hammersley point sets [9].…”

Section: Folded Digital Netsmentioning

confidence: 99%

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

Goda

Suzuki

Yoshiki

2016

Journal of Complexity

Self Cite

View full text Add to dashboard Cite

In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over Z b by means of the b-adic tent transformation, which has recently been introduced by the authors, and employ such folded digital nets as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with "infinite digit expansions", which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good folded higher order polynomial lattice rules that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.

show abstract

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

Cited by 4 publications

References 23 publications

Optimal order of $L_p$-discrepancy of digit shifted Hammersley point sets in dimension 2

Optimal order of $L_p$-discrepancy of digit shifted Hammersley point sets in dimension 2

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

Contact Info

Product

Resources

About