Niederreiter Sequences Research Articles

Pair correlations of Halton and Niederreiter Sequences are not Poissonian

Niederreiter and Halton sequences are two prominent classes of higher-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper we show that these sequences—even though they are uniformly distributed—fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.

Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules and polynomial lattice rules, which are special classes of QMC methods, have been intensively studied and the so-called component-by-component (CBC) algorithm has been well-established to construct rules which achieve the almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are used for function classes with different smoothness and/or weights.In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights which satisfy some summability conditions. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.

О множествах ограниченных остатков для (t, s)-последовательностей I

Пусть $$(x_n)_{n \geq 0} $$ - s-мерная последовательность типа Холтона, полученная из глобального функционального поля, $$b \geq 2$$, $$\gamma =(\gamma_1,..., \gamma_s)$$, $$\gamma_i \in [0, 1)$$ с b-адическим разложением $$\gamma_i= \gamma_{i,1}b^{-1}+ \gamma_{i,2}b^{-2}+...$$, $$i=1,...,s$$. В этой статье мы докажем, что $$[0,\gamma_1) \times ...\times [0,\gamma_s)$$ - множество ограниченного остатка относительно последовательности $$(x_n)_{n \geq 0}$$ тогда и только тогда, когда \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; \gamma_{i,j} \neq 0 \} < \infty. \end{equation}Мы также получим аналогичные результаты для обобщенных последовательностей Нидеррайтера, последовательностей Хинга-Нидеррайтера и последовательностей Нидеррайтера-Хинга.

Implementation of irreducible Sobol’ sequences in prime power bases

We present different implementations for the irreducible Sobol’ (IS) sequences introduced in Faure and Lemieux (2016). For this purpose we retain two strategies: first we use the connection between IS and Niederreiter sequences to provide a very simple implementation requiring no computer search; then we use criteria measuring the equidistribution to search for good parameters. Numerical results comparing these IS sequences to known implementations show promise for the proposed approaches.

Comparison of Sobol’ sequences in financial applications

Abstract Sobol’ sequences are widely used for quasi-Monte Carlo methods that arise in financial applications. Sobol’ sequences have parameter values called direction numbers, which are freely chosen by the user, so there are several implementations of Sobol’ sequence generators. The aim of this paper is to provide a comparative study of (non-commercial) high-dimensional Sobol’ sequences by calculating financial models. Additionally, we implement the Niederreiter sequence (in base 2) with a slight modification, that is, we reorder the rows of the generating matrices, and analyze and compare it with the Sobol’ sequences.

Tractability properties of the weighted star discrepancy of the Halton sequence

We study the weighted star discrepancy of the Halton sequence. In particular, we show that the Halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights $(\gamma_j)_{j \ge 1}$ under the mildest condition on the weight sequence known so far for explicitly constructive sequences. The condition requires $\sup_{d \ge 1} \max_{\emptyset \not= \mathfrak{u} \subseteq [d]} \prod_{j \in \mathfrak{u}} (j \gamma_j) < \infty$. The same result holds for Niederreiter sequences and for other types of digital sequences. Our results are true also for the weighted unanchored discrepancy.

Improved Markov Chain Monte Carlo method for cryptanalysis substitution-transposition cipher

In this paper we first investigate the use of Markov Chain Monte Carlo (MCMC) methods to attack classical ciphers. MCMC has previously been used to break simple substitution, transposition and substitution-transposition ciphers. Here, we improve the accuracy of obtained results by these algorithms and we show the performance of the algorithms using quasi random numbers such as Faure, Sobol and Niederreiter sequences.

Tractability of Multivariate Integration Using Low-Discrepancy Sequences

AbstractWe propose a notion of (t,e,s)-sequences in multiple bases, which unifies the Halton sequence and (t, s)-sequences under one roof, and obtain an upper bound of their discrepancy consisting only of the leading term. By using this upper bound, we improve the tractability results currently known for the Halton sequence, the Niederreiter sequence, the Sobol’ sequence, and the generalized Faure sequence, and also give tractability results for the Xing-Niederreiter sequence and the Hofer-Niederreiter sequence, for which no results have been known so far.

Improved Markov Chain Monte Carlo method for cryptanalysis substitution-transposition cipher

Abstract In this paper we first investigate the use of Markov Chain Monte Carlo (MCMC) methods to attack classical ciphers. MCMC has previously been used to break simple substitution, transposition and substitution-transposition ciphers. Here, we improve the accuracy of obtained results by these algorithms and we show the performance of the algorithms using quasi random numbers such as Faure, Sobol and Niederreiter sequences.

Halton-type sequences in rational bases in the ring of rational integers and in the ring of polynomials over a finite field

The aim of this paper is to generalize the well-known Halton sequences from integer bases to rational number bases and to translate this concept of Halton-type sequences in rational bases from the ring of integers to the ring of polynomials over a finite field. These two new classes of Halton-type sequences are low-discrepancy sequences. More exactly, the first class, based on the ring of integers, satisfies the discrepancy bounds that were recently obtained by Atanassov for the ordinary Halton sequence, and the second class, based on the ring of polynomials over a finite field, satisfies the discrepancy bounds that were recently introduced by Tezuka and by Faure & Lemieux for the generalized Niederreiter sequences.

On the lower bound of the discrepancy of (t,s) sequences: I

We find the exact lower bound of the discrepancy of shifted Niederreiter's sequences.

Irreducible Sobol’ sequences in prime power bases

Sobol’ sequences are a popular family of low-discrepancy sequences, in spite of requiring primitive polynomials instead of irreducible ones in later constructions by Niederreiter and Tezuka. We introduce a generalization of Sobol’ sequences that removes this shortcoming and that we believe has the potential of becoming useful for practical applications. Indeed, these sequences preserve two important properties of the original construction proposed by Sobol’: their generating matrices are non-singular upper triangular matrices, and they have an easy-to-implement column-by-column construction. We prove they form a subfamily of the wide family of generalized Niederreiter sequences, hence satisfying all known discrepancy bounds for this family. Further, their connections with Niederreiter sequences show these two families only have a small intersection (after reordering the rows of generating matrices of Niederreiter sequences in that intersection).

Generalized Hofer–Niederreiter sequences and their discrepancy from an [formula omitted]-point of view

Recently Tezuka introduced the concept of (u,e,s)-sequences which generalizes (t,s)-sequences by Niederreiter. This generalization can be used to point out a deeper regularity of certain digital sequences, which can be exploited to obtain improvements on their discrepancy bounds. Earlier Larcher and Niederreiter introduced so-called (T,s)-sequences with similar consequences. In this paper we generalize both concepts by introducing (U,e,s)-sequences, we work out relations between the concept of (U,e,s)-sequences and the concepts of (t,s)-, (u,e,s)-, and (T,s)-sequences, we introduce an explicit construction of (U,e,s)-sequences by generalizing Hofer–Niederreiter sequences, we give a discrepancy bound for (U,e,s)-sequences which is based on bounds for(u,m,e,s)-nets, and we relate our results to earlier ones.

On the discrepancy of generalized Niederreiter sequences

First, we propose a notion of (t,e,s)-sequences in base b, where e is an integer vector (e1,…,es) with ei≥1 for i=1,…,s, which are identical to (t,s)-sequences in base b when e=(1,…,1), and show that a generalized Niederreiter sequence in base b is a (0,e,s)-sequence in base b, where ei is equal to the degree of the base polynomial for the i-th coordinate. Then, by using the signed splitting technique invented by Atanassov, we obtain a discrepancy bound for a (t,e,s)-sequence in base b. It follows that a (unanchored) discrepancy bound for the first N>1 points of a generalized Niederreiter sequence in base b is given as NDN≤(1s!∏i=1s2⌊bei/2⌋eilogb)(logN)s+O((logN)s−1), where the constant in the leading term is asymptotically much smaller than the one currently known.

Ultra-fast Carrier Transport Simulation on the Grid. Quasi-Random Approach

The problem for simulation ultra-fast carrier transport in nano-electronics devices is a large scale computational problem and requires HPC and/or Grid computing resources. The most widely used techniques for modeling this carrier transport are Monte Carlo methods. In this work we consider a set of stochastic algorithms for solving quantum kinetic equations describing quantum effects during the femtosecond relaxation process due to electron-phonon interaction in one-band semiconductors or quantum wires. The algorithms are integrated in a Grid-enabled package named S tochas-tic A Lgorithms for U ltra-fast T ransport in s E miconductors (SALUTE). There are two main reasons for running this package on the computational Grid: (i) quantum problems are very computationally intensive; (ii) the inherently parallel nature of Monte Carlo applications makes efficient use of Grid resources. Grid (distributed) Monte Carlo applications require that the underlying random number streams in each subtask are independent in a statistical sense. The efficient application of quasi-Monte Carlo algorithms entails additional difficulties due to the possibility of job failures and the inhomogeneous nature of the Grid resource. In this paper we study the quasi-random approach in SALUTE and the performance of the corresponding algorithms on the grid, using the scrambled Halton, Sobol and Niederreiter sequences. A large number of tests have been performed on the EGEE and SEEGRID grid infrastructures using specially developed grid implementation scheme. Novel results for energy and density distribution, obtained in the inhomogeneous case with applied electric field are presented.

On the exact [formula omitted]-value of Niederreiter and Sobol’ sequences

This paper studies several well-known families of ( t , s ) -sequences. First we determine the exact t -value of Niederreiter sequences. Then we analyze the exact t -value of generalized Niederreiter sequences and we show that, for a range of dimensions of practical interest, Niederreiter–Xing sequences are demonstrably better than Sobol’ sequences in terms of the exact t -value. Previously, such a conclusion was not possible since only upper bounds on the exact t -value of these sequences and a general lower bound for all ( t , s ) -sequences were available.

Error trends in Quasi-Monte Carlo integration

Several test functions, whose variation could be calculated, were integrated with up to 10 10 trials using different low-discrepancy sequences in dimensions 3, 6, 12, and 24. The integration errors divided by the variation of the functions were compared with exact and asymptotic discrepancies. These errors follow an approximate power law, whose constant is essentially given by the variance of the integrand, and whose power depends on its effective dimension. Included were also some calculations with scrambled Niederreiter sequences, and with Niederreiter–Xing sequences. A notable result is that the pre-factors of the asymptotic discrepancy function D ∗(N) , which are so often used as a quality measure of the sequences, have little relevance in practical ranges of N.

Discrepancy behaviour in the non-asymptotic regime

We have computed true star-discrepancies D ∗(N) for Halton and Niederreiter point sequences in dimensions s=2, 3, 4 , and 6 with N up to 250 000 , 10 000 , 2000 and 300, respectively. For comparison, we also calculated some L 2 discrepancies T ∗ . The mean behaviour of D ∗(N) can well be approximated by power laws with an exponent between about −0.7 and −0.9, which slowly decreases with N. This behaviour is far from the generally assumed asymptotic one ∼ C( s)(log N) s / N. The factor between the true and the asymptotic behaviour increases strongly with s and reaches many orders of magnitude for large s. The ratios of D ∗(N) for different low discrepancy sequences are not proportional to the presumed asymptotic pre-factors C( s). Especially for the range of bases investigated, Niederreiter sequences have about the same D ∗ as Halton ones in a range of N, which we conjecture to be at least of the order of 10 10.

Finite-order weights imply tractability of multivariate integration

Multivariate integration of high dimension s occurs in many applications. In many such applications, for example in finance, integrands can be well approximated by sums of functions of just a few variables. In this situation the superposition (or effective) dimension is small, and we can model the problem with finite-order weights, where the weights describe the relative importance of each distinct group of variables up to a given order (where the order is the number of variables in a group), and ignore all groups of variables of higher order. In this paper we consider multivariate integration for the anchored and unanchored (non-periodic) Sobolev spaces equipped with finite-order weights. Our main interest is tractability and strong tractability of QMC algorithms in the worst-case setting. That is, we want to find how the minimal number of function values needed to reduce the initial error by a factor ε depends on s and ε −1. If there is no dependence on s, and only polynomial dependence on ε −1, we have strong tractability, whereas with polynomial dependence on both s and ε −1 we have tractability. We show that for the anchored Sobolev space we have strong tractability for arbitrary finite-order weights, whereas for the unanchored Sobolev space we have tractability for all bounded finite-order weights. In both cases, the dependence on ε −1 is quadratic. We can improve the dependence on ε −1 at the expense of polynomial dependence on s. For finite-order weights, we may achieve almost linear dependence on ε −1 with a polynomial dependence on s whose degree is proportional to the order of the weights. We show that these tractability bounds can be achieved by shifted lattice rules with generators computed by the component-by-component (CBC) algorithm. The computed lattice rules depend on the weights. Similar bounds can also be achieved by well-known low discrepancy sequences such as Halton, Sobol and Niederreiter sequences which do not depend on the weights. We prove that these classical low discrepancy sequences lead to error bounds with almost linear dependence on n −1 and polynomial dependence on d. We present explicit worst-case error bounds for shifted lattice rules and for the Niederreiter sequence. Better tractability and error bounds are possible for finite-order weights, and even for general weights if they satisfy certain conditions. We present conditions on general weights that guarantee tractability and strong tractability of multivariate integration.

Numerical integration of singular integrands using low-discrepancy sequences

We investigate quasi-Monte Carlo integration for functions on thes-dimensional unit cube having point singularities. Error bounds are proved and the theoretical results are verified by computations using Halton, Sobol’ and Niederreiter sequences.