Hammersley Point Sets Research Articles

Dispersion of digital (0, m, 2)-nets

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.

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

AbstractThe notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasi- Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over ℤb. 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 ℤbin 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 Lpdiscrepancy for all 1 ≤ p < ∞.

L2 discrepancy of symmetrized generalized Hammersley point sets in base b

Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of Lp 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 Lp discrepancy is of optimal order O(log⁡N/N) for p∈[1,∞) in contrast to the classical Hammersley point set. We prove an exact formula for the L2 discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest L2 discrepancy for every base b∈{2,…,27} by employing computer search algorithms.

An exact formula for the [formula omitted] discrepancy of the symmetrized Hammersley point set

The process of symmetrization is often used to construct point sets with low Lp discrepancy. In the current work we apply this method to the shifted Hammersley point set. It is known that for every shift this symmetrized point set achieves an Lp discrepancy of order O(logN/N) for p∈[1,∞), which is best possible in the sense of results by Roth, Schmidt and Halász. In this paper we present an exact formula for the L2 discrepancy of the symmetrized Hammersley point set, which shows in particular that it is independent of the choice for the shift.

Formula omitted]- and [formula omitted]-discrepancy of the symmetrized van der Corput sequence and modified Hammersley point sets in arbitrary bases

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 Sp,qrB([0,1)s), which will also give us bounds on the Lp-discrepancy. Our sequence and point sets will achieve the known optimal order for the Lp- and Sp,qrB-discrepancy. The results in this paper generalize several previous results on Lp- and Sp,qrB-discrepancy estimates and provide a sharp upper bound on the Sp,qrB-discrepancy of one-dimensional sequences for r>0. We will use the b-adic Haar function system in the proofs.

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

We give an explicit construction of two-dimensional point sets whose $L_p$ discrepancy is of best possible order for all $1\le p\le \infty$. 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 $b\in \mathbb{N}$, $b\ge 2$. 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 $L_p$ discrepancy for all $1\le p\le \infty$.

On an example of finite hybrid quasi-Monte Carlo point sets

In this paper, we consider finite hybrid point sets in the unit cube. The components of these stem from two well known types of low discrepancy point sets, namely Hammersley point sets on the one hand, and lattice point sets in the sense of Korobov and Hlawka on the other hand. As a quality measure, we consider the star discrepancy, which gives information about the quality of distribution of finite or infinite sequences. We present existence results for finite hybrid point sets with low discrepancy. Thereby, we make analogous results for infinite sequences more explicit in the sense that, theoretically, it is now possible to find such finite hybrid low discrepancy point sets.

L_2 discrepancy of generalized Zaremba point sets

We give an exact formula for the $L_2$ discrepancy of a class of generalized two-dimensional Hammersley point sets in base $b$, namely generalized Zaremba point sets. These point sets are digitally shifted Hammersley point sets with an arbitrary number of different digital shifts in base $b$. The Zaremba point set introduced by White in 1975 is the special case where the $b$ shifts are taken repeatedly in sequential order, hence needing at least $b^b$ points to obtain the optimal order of $L_2$ discrepancy. On the contrary, our study shows that only one non-zero shift is enough for the same purpose, whatever the base $b$ is.

Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness

AbstractWe study the discrepancy function of two‐dimensional Hammersley type point sets in the unit square. It is well‐known that the symmetrized Hammersley point set achieves the asymptotically best possible rate for the L2‐norm of the discrepancy function. In this paper we consider the norm of the discrepancy function of Ham¬mersley type point sets in Besov spaces of dominating mixed smoothness and show that it achieves the optimal rate under appropriate assumptions on the set and the smoothness parameter of the Besov space. Our proof relies on a characterization of the Besov spaces of dominating mixed smoothness via coefficients in the Haar expansion and the computation of the Haar expansion of the discrepancy function of Hammersley type point sets (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

L2discrepancy of generalized two-dimensional Hammersley point sets scrambled with arbitrary permutations

The L2 discrepancy is a quantitative measure for the irregularity of distribution of a finite point set. In this paper we consider the L2 discrepancy of so-called generalized Hammersley point sets which can be obtained from the classical Hammersley point sets by introducing some permutations on the base b digits. While for the classical Hammersley point set it is not possible to achieve the optimal order of L2 discrepancy with respect to a general lower bound due to Roth this disadvantage can be overcome with the generalized version thereof. For special permutations we obtain an exact formula for the L2 discrepancy from which we obtain two-dimensional finite point sets with the lowest value of L2 discrepancy known so far. AMS subject classification: 11K06, 11K38.

L p discrepancy of generalized two-dimensional Hammersley point sets

We determine the Lp discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the Lp discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on Lp discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the Lp discrepancy of the generalized Hammersley point set is of best possible order. For the L2 discrepancy such permutations are given explicitly.

Point sets with low L p-discrepancy

Abstract In this paper we study the L p-discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L p-discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L p-discrepancy (p an even integer) of order $$\sqrt {\log N} /N$$ which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L 2-discrepancy of digitally shifted Hammersley point sets which show that the value of the L 2-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these.

On some remarkable properties of the two-dimensional Hammersley point set in base 2

We study a special class of (0,m,2)-nets in base 2. In particular, we are concerned with the two-dimensional Hammersley net that plays a special role among these since we prove that it is the worst distributed with respect to the star discrepancy. By showing this, we also improve an existing upper bound for the star discrepancy of digital (0,m,2)-nets over ℤ 2 . Moreover, we show that nets with very low star discrepancy can be obtained by transforming the Hammersley point set in a suitable way.

On the -Discrepancy of the Hammersley Point Set

We give a formula for the \(\)-discrepancy of the 2-dimensional Hammersley point set in base 2 for all integers p, \(\).

Error bounds for quasi-Monte Carlo integration with nets

We analyze the error introduced by approximately calculating the s s -dimensional Lebesgue measure of a Jordan-measurable subset of I s = [ 0 , 1 ) s I^s=[0,1)^s . We give an upper bound for the error of a method using a ( t , m , s ) (t,m,s) -net, which is a set with a very regular distribution behavior. When the subset of I s I^s is defined by some function of bounded variation on I ¯ s − 1 {\bar I}^{s-1} , the error is estimated by means of the variation of the function and the discrepancy of the point set which is used. A sharper error bound is established when a ( t , m , s ) (t,m,s) -net is used. Finally a lower bound of the error is given, for a method using a ( 0 , m , s ) (0,m,s) -net. The special case of the 2-dimensional Hammersley point set is discussed.