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Dispersion of digital (0,m,2)-nets [article]

Ralph Kritzinger
2020 arXiv   pre-print
We study the dispersion of digital (0,m,2)-nets; i.e. the size of the largest axes-parallel box within such point sets. Digital nets are an important class of low-discrepancy point sets. We prove tight lower and upper bounds for certain subclasses of digital nets where the generating matrices are of triangular form and compute the dispersion of special nets such as the Hammersley point set exactly.
arXiv:2004.14760v1 fatcat:xni4g7vzwbbs3n4hc5jveykrky

L_2 discrepancy of symmetrized generalized Hammersley point sets in base b [article]

Ralph Kritzinger, Lisa M. Kritzinger
2016 arXiv   pre-print
Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of L_p discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize b-adic Hammersley point sets scrambled with arbitrary permutations. It is already known that these modifications indeed assure that the L_p discrepancy is of optimal order O(√(N)/N) for p∈ [1,∞) in contrast to the classical Hammersley point set. We prove an exact formula for the
more » ... _2 discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest L_2 discrepancy for every base b∈{2,...,27} by employing computer search algorithms.
arXiv:1511.04937v2 fatcat:52tlppko7reg3dog4moupfjtdy

Extremal Distributions of Discrepancy functions [article]

Ralph Kritzinger, Markus Passenbrunner
2019 arXiv   pre-print
The irregularities of a distribution of N points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form [0,t)⊂ [0,1) or arbitrary subintervals of the unit interval. In the former case, it is a well known fact in discrepancy theory that the N-element point set in the with the lowest L_2 or L_∞ norm of the discrepancy function is the centered regular grid Γ_N:={2n+1/2N: n=0,1,...,N-1}. We show a
more » ... onger result on the distribution of discrepancy functions of point sets in [0,1], which basically says that the distribution of the discrepancy function of Γ_N is in some sense minimal among all N-element point sets. As a consequence, we can extend the above result to rearrangement-invariant norms, including L_p, Orlicz and Lorentz norms. We study the same problem for the discrepancy notions with respect to arbitrary subintervals. In this case, we will observe that we have to deal with integrals of convolutions of functions. To this end, we prove a general upper bound on such expressions, which might be of independent interest as well.
arXiv:1902.09877v2 fatcat:fdpyjs3vs5dtjhjg4yshe3ooxu

L_p-discrepancy of the symmetrized van der Corput sequence [article]

Ralph Kritzinger, Friedrich Pillichshammer
2015 arXiv   pre-print
It is well known that the L_p-discrepancy for p ∈ [1,∞] of the van der Corput sequence is of exact order of magnitude O(( N)/N). This however is for p ∈ (1,∞) not best possible with respect to the lower bounds according to Roth and Proinov. For the case p=2 it is well known that the symmetrization trick due to Davenport leads to the optimal L_2-discrepancy rate O(√( N)/N) for the symmetrized van der Corput sequence. In this note we show that this result holds for all p ∈ (1,∞). The proof is
more » ... d on an estimate of the Haar coefficients of the corresponding local discrepancy and on the use of the Littlewood-Paley inequality.
arXiv:1501.02552v1 fatcat:ubs2iyih2jadfgr275obaaw54y

Lower bounds on the L_p discrepancy of digital NUT sequences [article]

Ralph Kritzinger, Friedrich Pillichshammer
2019 arXiv   pre-print
We study the L_p discrepancy of digital NUT sequences which are an important sub-class of digital (0,1)-sequences in the sense of Niederreiter. The main result is a lower bound for certain sub-classes of digital NUT sequences.
arXiv:1904.01433v1 fatcat:2whxbkgfs5aqvjflcq2jhny22q

Point sets with optimal order of extreme and periodic discrepancy [article]

Ralph Kritzinger, Friedrich Pillichshammer
2021 arXiv   pre-print
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
more » ... hese notions of discrepancy and presented exact formulas for typical two-dimensional 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.
arXiv:2109.05781v1 fatcat:mkxt557m5zdnvcs36vdqg5ppnu

Optimal discrepancy rate of point sets in Besov spaces with negative smoothness [article]

Ralph Kritzinger
2017 arXiv   pre-print
We consider the local discrepancy of a symmetrized version of Hammersley type point sets in the unit square. As a measure for the irregularity of distribution we study the norm of the local discrepancy in Besov spaces with dominating mixed smoothness. It is known that for Hammersley type points this norm has the best possible rate provided that the smoothness parameter of the Besov space is nonnegative. While these point sets fail to achieve the same for negative smoothness, we will prove in
more » ... s note that the symmetrized versions overcome this defect. We conclude with some consequences on discrepancy in further function spaces with dominating mixed smoothness and on numerical integration based on quasi-Monte Carlo rules.
arXiv:1701.01970v1 fatcat:5uvfxaurqvdaxkovssdy42dvei

Exact order of extreme L_p discrepancy of infinite sequences in arbitrary dimension [article]

Ralph Kritzinger, Friedrich Pillichshammer
2021 arXiv   pre-print
We study the extreme L_p discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star L_p discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any p>1 the extreme L_p discrepancy of every infinite sequence in [0,1)^d is at least of order of magnitude (log N)^d/2, where N is the number of considered initial
more » ... ms of the sequence. For p ∈ (1,∞) this order of magnitude is best possible.
arXiv:2109.06461v1 fatcat:rj7aexw6angbxb3fngam3mzfnu

Digital nets in dimension two with the optimal order of L_p discrepancy [article]

Ralph Kritzinger, Friedrich Pillichshammer
2018 arXiv   pre-print
. ✷ Authors' Address: Ralph Kritzinger and Friedrich Pillichshammer, Institut für Finanzmathematik und angewandte Zahlentheorie, Johannes Kepler Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria  ... 
arXiv:1804.04891v1 fatcat:qljasfnoxrg4veoi3sxyshswz4

Extreme and periodic $L_2$ discrepancy of plane point sets

Aicke Hinrichs, Ralph Kritzinger, Friedrich Pillichshammer
2021 Acta Arithmetica  
The authors are supported by the Austrian Science Fund (FWF), Projects F5513-N26 (Hinrichs) and F5509-N26 (Kritzinger and Pillichshammer), which are parts of the Special Research Program "Quasi-Monte Carlo  ... 
doi:10.4064/aa200520-22-12 fatcat:bhffrsub5jcwjemd7mv3qugi5m

Optimal order of L_p-discrepancy of digit shifted Hammersley point sets in dimension 2 [article]

Aicke Hinrichs, Ralph Kritzinger, Friedrich Pillichshammer
2014 arXiv   pre-print
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 L_p-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 √( N)/N of L_2-discrepancy, where N is the number of points. Among these are for example digit shifts or
more » ... he symmetrization. In this paper we show that these modified Hammersley point sets also achieve optimal order of L_p-discrepancy for all p ∈ (1,∞).
arXiv:1410.4315v1 fatcat:raq6cqhhi5btlncfhaw72ycf3e

Finding exact formulas for the L_2 discrepancy of digital (0,n,2)-nets via Haar functions [article]

Ralph Kritzinger
2017 arXiv   pre-print
We use the Haar function system in order to study the L_2 discrepancy of a class of digital (0,n,2)-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the unit interval. We will obtain such formulas not only for the classical digital nets, but also for shifted and symmetrized versions thereof. The basic idea of our proofs is to calculate all Haar coefficents of the discrepancy function exactly and insert them into
more » ... arseval's identity. We will also discuss reasons why certain (symmetrized) digital nets fail to achieve the optimal order of L_2 discrepancy and use the Littlewood-Paley inequality in order to obtain results on the L_p discrepancy for all p∈ (1,∞).
arXiv:1711.06058v1 fatcat:6hjapaoccrgxjhzrj3v3unt32y

A reduced fast construction of polynomial lattice point sets with low weighted star discrepancy [article]

Ralph Kritzinger, Helene Laimer, Mario Neumüller
2017 arXiv   pre-print
The weighted star discrepancy is a quantitative measure for the performance of point sets in quasi-Monte Carlo algorithms for numerical integration. We consider polynomial lattice point sets, whose generating vectors can be obtained by a component-by-component construction to ensure a small weighted star discre-pancy. Our aim is to significantly reduce the construction cost of such generating vectors by restricting the size of the set of polynomials from which we select the components of the
more » ... tors. To gain this reduction we exploit the fact that the weights of the spaces we consider decay very fast.
arXiv:1701.02525v1 fatcat:qoeyz6vksvaf5o7osgcnixo6be

Extreme and periodic L_2 discrepancy of plane point sets [article]

Aicke Hinrichs, Ralph Kritzinger, Friedrich Pillichshammer
2020 arXiv   pre-print
In this paper we study the extreme and the periodic L_2 discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen as intervals on the torus. The periodic L_2 discrepancy is, up to a multiplicative factor, also known as diaphony. The main results are exact formulas for these kinds of discrepancies for the Hammersley point set and for rational lattices. In order to value the
more » ... btained results we also prove a general lower bound on the extreme L_2 discrepancy for arbitrary point sets in dimension d, which is of order of magnitude (log N)^(d-1)/2, like the standard and periodic L_2 discrepancies, respectively. Our results confirm that the extreme and periodic L_2 discrepancies of the Hammersley point set are of best possible asymptotic order of magnitude. This is in contrast to the standard L_2 discrepancy of the Hammersley point set. Furthermore our exact formulas show that also the L_2 discrepancies of the Fibonacci lattice are of the optimal order. We also prove that the extreme L_2 discrepancy is always dominated by the standard L_2 discrepancy, a result that was already conjectured by Morokoff and Caflisch when they introduced the notion of extreme L_2 discrepancy in the year 1994.
arXiv:2005.09933v2 fatcat:b63s4mcmdjaz3c2gexovopnpze

A reduced fast component-by-component construction of lattice point sets with small weighted star discrepancy [article]

Ralph Kritzinger, Helene Laimer
2015 arXiv   pre-print
The weighted star discrepancy of point sets appears in the weighted Koksma-Hlawka inequality and thus is a measure for the quality of point sets with respect to their performance in quasi-Monte Carlo algorithms. A special choice of point sets are lattice point sets whose generating vector can be obtained one component at a time such that the resulting lattice point set has a small weighted star discrepancy. In this paper we consider a reduced fast component-by-component algorithm which
more » ... ntly reduces the construction cost for such generating vectors provided that the weights decrease fast enough.
arXiv:1501.07073v1 fatcat:pb364zbgkrgcpoco3dtlpndtsm
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