Quasi-random (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 s-dimensional 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 randomly-chosen 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 quasi-random sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasi-random sequence is almost exactly the same as that of a randomly chosen sequence...

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 base-2 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.

The standard Monte Carlo approach to evaluating multi-dimensional 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 pseudo-random sequences with more uniformly distributed sequences known as quasi-random. In this paper the Halton, Sobol' and Faure quasi-random 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 Koksma-Hlawka 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...

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 leap-frog or blocking methods. We describe in detail parameterized versions of the following pseudorandom number generators: (i) linear congruential generators, (ii) shift-register generators, and (iii) lagged-Fibonacci 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.

. 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 so-called 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 d-space IE d , i.e. of all compact convex subsets of IE d with non-empty 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...

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 de-Poissonization, 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...

The need to numerically simulate stochastic processes arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudo-random sequences to simulate the randomness. This paper address the question of how to use quasi-Monte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps are required. Two such techniques, the generalized Brownian bridge and particle reordering, are described here. These methods are applied to a problem from finance, the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the effectiveness of the quasi-random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunc...

. We study the suitability of the additive lagged-Fibonacci pseudorandom number generator for parallel computation. This generator has a relatively short period with respect to the size of its seed. However, the short period is more than made up for with the huge number of full-period cycles it contains. We call these different full-period cycles equivalence classes. We show how to enumerate the equivalence classes and how to compute seeds to select a given equivalence class. The use of these equivalence classes gives an explicit parallelization suitable for a fully reproducible asynchronous MIMD implementation. To explore such an implementation we introduce an exponential sum measure of quality for the additive lagged-Fibonacci generators used in serial or parallel. We then prove the first non-trivial results we are aware of on this measure of quality. 1. Introduction. In Knuth's well known exposition on pseudorandom number generation [5], several methods of generation are considered...

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Abstract. Linear congruential generators (LCGs) remain the most popular method of pseudorandom number generation on digital computers. Ease of implementation has favored implementing LCGs with power-of-two moduli. However, prime modulus LCGs are superior in quality to power-of-two modulus LCGs, and the use of a Mersenne prime minimizes the computational cost of generation. When implemented for parallel computation, quality becomes an even more compelling issue. We use a full-period exponential sum as the measure of stream independence and present a method for producing provably independent streams of LCGs in parallel by utilizing an explicit parameterization of all of the primitive elements modulo a given prime. The minimization of this measure of independence further motivates an algorithm required in the explicit parameterization. We describe and analyze this algorithm and describe its use in a parallel LCG package. 1. Introduction. Perhaps the oldest generator still in use for the generation of uniformly distributed integers is the linear congruential generator (LCG). This generator is sometimes