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259
QuasiRandom Sequences and Their Discrepancies
 SIAM J. Sci. Comput
, 1994
"... Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is meas ..."
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Cited by 73 (6 self)
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Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is measured by its discrepancy, which is of size (log N) s N \Gamma1 for large N , as opposed to discrepancy of size (log log N) 1=2 N \Gamma1=2 for a random sequence (i.e. for almost any randomlychosen sequence). Several types of discrepancy, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancy are presented for a wide choice of dimension s, number of points N and different quasirandom sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence...
On the Random Character of Fundamental Constant Expansions
 EXPERIMENTAL MATHEMATICS
, 2001
"... We propose a theory to explain random behavior for the digits
in the expansions of fundamental mathematical constants. At
the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base2 no ..."
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Cited by 51 (14 self)
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We propose a theory to explain random behavior for the digits
in the expansions of fundamental mathematical constants. At
the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base2 normality—namely bit randomness in a specific technical sense— for a collection of celebrated constants, including , log 2, (3), and others. Also on the hypothesis, the number (5) is either rational or normal to base 2. We indicate a research connection between our dynamical model and the theory of pseudorandom number
generators.
Quasirandom methods for estimating integrals using relatively small samples
 SIAM Review
, 1994
"... Abstract. Much of the recent work dealing with quasirandom methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finitedimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to ..."
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Cited by 43 (1 self)
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Abstract. Much of the recent work dealing with quasirandom methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finitedimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to concentrate on asymptotic convergence rates, this paper emphasizes quasirandom methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finitedimensional integrals is used to illustrate the major ideas, although much of what is done applies equally to the problem of solving certain Fredholm integral equations. Some new techniques, based on errorreducing transformations of the integrand, are described that have been shown to be useful both in estimating highdimensional integrals and in solving integral equations. These techniques illustrate the utility of carrying over to the quasiMonte Carlo method certain devices that have proven to be very valuable in statistical (pseudorandom) Monte Carlo applications. Key words, quasiMonte Carlo, asymptotic rate of convergence, numerical integration
QuasiMonte Carlo Integration
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1995
"... The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved con ..."
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Cited by 42 (6 self)
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The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved convergence may be obtained by replacing the pseudorandom sequences with more uniformly distributed sequences known as quasirandom. In this paper the Halton, Sobol' and Faure quasirandom sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand, including variance, variation, smoothness and dimension. The results show that variation, which plays an important role in the theoretical upper bound given by the KoksmaHlawka inequality, does not affect convergence; while variance, the determining factor in random Monte Carlo, is shown to provide a rough upper bound, but does not accurately predict performance. In ge...
Monte Carlo and QuasiMonte Carlo methods
 Acta Numerica
, 1998
"... Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N ~ 1 ^ 2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including conve ..."
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Cited by 35 (1 self)
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Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N ~ 1 ^ 2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasirandom (also called lowdiscrepancy) sequences, which are a deterministic alternative to random or pseudorandom sequences. The points in a quasirandom sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasiMonte Carlo, has a convergence rate of approximately O((log N^N ' 1). For quasiMonte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less
Asymptotic estimates for best and stepwise approximation of convex bodies III
, 1997
"... . We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the difference of the mean width of the convex body and the approximating polytopes. The following results ..."
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Cited by 32 (3 self)
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. We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the difference of the mean width of the convex body and the approximating polytopes. The following results are obtained. (i) An asymptotic formula for best approximation. (ii) Upper and lower estimates for stepby step approximation in terms of the socalled dispersion. (iii) For a sequence of best approximating inscribed polytopes the sequence of vertex sets is uniformly distributed in the boundary of the convex body where the density is specified explicitly. 1. Introduction and statement of results 1.1 Let C denote the space of convex bodies in Euclidean dspace IE d , i.e. of all compact convex subsets of IE d with nonempty interior. For notions not explained below we refer to [20]. Given C 2 C and k = 0; : : : ; d, let W k (C) be the kth quermassintegral of C. W 0 = V is the volume, dW 1...
SPRNG: A Scalable Library for Pseudorandom Number Generation
"... In this article we present background, rationale, and a description of the Scalable Parallel Random
Number Generators (SPRNG) library. We begin by presenting some methods for parallel pseudorandom number generation. We will focus on methods based on parameterization, meaning that we will not conside ..."
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Cited by 29 (6 self)
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In this article we present background, rationale, and a description of the Scalable Parallel Random
Number Generators (SPRNG) library. We begin by presenting some methods for parallel pseudorandom number generation. We will focus on methods based on parameterization, meaning that we will not consider splitting methods such as the leapfrog or blocking methods. We describe in detail
parameterized versions of the following pseudorandom number generators: (i) linear congruential
generators, (ii) shiftregister generators, and (iii) laggedFibonacci generators. We briey describe
the methods, detail some advantages and disadvantages of each method, and recount results from
number theory that impact our understanding of their quality in parallel applications.
SPRNG was designed around the uniform implementation of dierent families of parameterized random number
generators. We then present a short description of
SPRNG. The description contained within this
document is meant only to outline the rationale behind and the capabilities of SPRNG. Much more
information, including examples and detailed documentation aimed at helping users with putting
and using SPRNG on scalable systems is available at the URL:
http://sprng.cs.fsu.edu/RNG. In this description of SPRNG we discuss the random number generator library as well as the suite of
tests of randomness that is an integral part of SPRNG. Random number tools for parallel Monte
Carlo applications must be subjected to classical as well as new types of empirical tests of ran
domness to eliminate generators that show defects when used in scalable environments.
On the Distribution for the Duration of a Randomized Leader Election Algorithm
 Ann. Appl. Probab
, 1996
"... We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressio ..."
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Cited by 27 (10 self)
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We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic fluctuations. In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization, and dePoissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis metho...
Analysis of an Asymmetric Leader Election Algorithm
 Electronic J. Combin
, 1996
"... We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who rec ..."
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Cited by 26 (9 self)
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We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who received, say a tail, survive for the next round. The process continues until only one objects remains. Our interest is in evaluating the limiting distribution and the first two moments of the number of rounds needed to select a leader. We establish precise asymptotics for the first two moments, and show that the asymptotic expression for the duration of the algorithm exhibits some periodic fluctuations and consequently no limiting distribution exists. These results are proved by analytical techniques of the precise analysis of algorithms such as: analytical poissonization and depoissonization, Mellin transform, and complex analysis.
Diffractive point sets with entropy
"... Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of ..."
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Cited by 24 (16 self)
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Dedicated to HansUde Nissen on the occasion of his 65th birthday After a brief historical survey, the paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in welldefined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e., site dependent) random variables. Several examples are shown.