L p -discrepancy of the symmetrized van der Corput sequence

Kritzinger, Ralph; Pillichshammer, Friedrich

doi:10.1007/s00013-015-0760-7

Cited by 8 publications

(14 citation statements)

References 20 publications

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“…Proof. The proof follows exactly the same lines as the proof of [18,Lemma 2]. ✷ In the following, let µ N,sym j,m,ℓ , µ N,ϕ j,m,ℓ and µ N,1−ϕ j,m,ℓ be the Haar coefficients of the local discrepancy of the first N elements of the sequences V sym b , (ϕ(n)) n≥0 and (1 − ϕ(n)) n≥0 , respectively.…”

Section: Lemmamentioning

confidence: 82%

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Lp- and Sp,qr
Kritzinger
¹

2016
Journal of Complexity
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We study the local discrepancy of a symmetrized version of the well-known van der Corput sequence and of modified two-dimensional Hammersley point sets in arbitrary base b. We give upper bounds on the norm of the local discrepancy in Besov spaces of dominating mixed smoothness S r p,q B([0, 1) s ), which will also give us bounds on the L p -discrepancy. Our sequence and point sets will achieve the known optimal order for the L p -and S r p,q B-discrepancy. The results in this paper generalize several previous results on L p -and S r p,q B-discrepancy estimates and provide a sharp upper bound on the S r p,q B-discrepancy of one-dimensional sequences for r > 0. We will use the b-adic Haar function system in the proofs.

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Section: Lemmamentioning

confidence: 82%

“…✷ In the following, let µ N,sym j,m,ℓ , µ N,ϕ j,m,ℓ and µ N,1−ϕ j,m,ℓ be the Haar coefficients of the local discrepancy of the first N elements of the sequences V sym b , (ϕ(n)) n≥0 and (1 − ϕ(n)) n≥0 , respectively. The next lemma may be proved in complete analogy to [18,Corollary 1].…”

Section: Lemmamentioning

confidence: 99%

Lp- and Sp,qr
Kritzinger
¹

2016
Journal of Complexity
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“…Considering the result in Theorem 13, one might hope that there is a sequence Σ such that the order of magnitude in N of L 2,N (Y Σ 2 ) is lower than that of L 2,N (Y 2 ). Unfortunately, these hopes are crushed by the following result that was also shown in [82].…”

Section: Remarkmentioning

confidence: 96%

“…We now summarize the results from [82], where the L 2 -discrepancy of Y Σ 2 is considered. From Theorem 4, we know that…”

Section: Remarkmentioning

confidence: 99%

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

Faure

Kritzer

Pillichshammer

2015

Indagationes Mathematicae

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In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results.On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput's ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter's (t, s)-sequences.

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“…Furthermore, it is interesting to note that the construction corresponding to the gray line also describes the symmetrized binary van der Corput sequence. Symmetrization has been shown to improve the asymptotic behavior of the L p -discrepancy [16]. We also computed the star discrepancy for two dimensional sequences constructed with the Stirling matrices over F 2 (see [13]) in Figure 4.…”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

An Extension of the Digital Method Based on b-Adic Integers

Hofer

Piršić

2018

Uniform Distribution Theory

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ABSTRACT. We introduce a hybridization of digital sequences with uniformly distributed sequences in the domain of b-adic integers, Z b , b ∈ N \ {1}, by using such sequences as input for generating matrices. The generating matrices are then naturally required to have finite row-lengths. We exhibit some relations of the 'classical' digital method to our extended version, and also give several examples of new constructions with their respective quality assessments in terms of t, T and discrepancy.

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L p -discrepancy of the symmetrized van der Corput sequence

Cited by 8 publications

References 20 publications

Lp- and Sp,qr
Kritzinger
¹

2016
Journal of Complexity
Self Cite
6
1
6
0
View full text Add to dashboard Cite

Lp- and Sp,qr
Kritzinger
¹

2016
Journal of Complexity
Self Cite
6
1
6
0
View full text Add to dashboard Cite

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

An Extension of the Digital Method Based on b-Adic Integers

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L p -discrepancy of the symmetrized van der Corput sequence

Cited by 8 publications

References 20 publications

Lp- and Sp,qrKritzinger1 2016Journal of ComplexitySelf Cite6160View full textAdd to dashboardCite

Lp- and Sp,qrKritzinger1 2016Journal of ComplexitySelf Cite6160View full textAdd to dashboardCite

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

An Extension of the Digital Method Based on b-Adic Integers

Contact Info

Product

Resources

About

Lp- and Sp,qr
Kritzinger
¹

2016
Journal of Complexity
Self Cite
6
1
6
0
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Lp- and Sp,qr
Kritzinger
¹

2016
Journal of Complexity
Self Cite
6
1
6
0
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