Michael Mascagni

Algorithm 806: 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 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 briefly 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 different 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 htt;//sprng.sc.fsu.edu. 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 randomness to eliminate generators that show defects when used in scalable envionments.

Testing parallel random number generators

Monte Carlo computations are considered easy to parallelize. However, the results can be adversely affected by defects in the parallel pseudorandom number generator used. A parallel pseudorandom number generator must be tested for two types of correlations--(i) intra-stream correlation, as for any sequential generator, and (ii) inter-stream correlation for correlations between random number streams on different processes. Since bounds on these correlations are difficult to prove mathematically, large and thorough empirical tests are necessary. Many of the popular pseudorandom number generators in use today were tested when computational power was much lower, and hence they were evaluated with much smaller test sizes.This paper describes several tests of pseudorandom number generators, both statistical and application-based. We show defects in several popular generators. We describe the implementation of these tests in the SPRING [ACM Trans. Math. Software 26 (2000) 436; SPRNG--scalable parallel random number generators. SPRNG 1.0--http://www.ncsa.uiuc.edu/ Apps/SPRNG; SPRNG 2.0--http://sprng.cs.fsu.edu] test suite and also present results for the tests conducted on the SPRNG generators. These generators have passed some of the largest empirical random number tests.

Improving performance via computational replication on a large-scale computational grid

High performance computing on a large-scale computational grid is complicated by the heterogeneous computational capabilities of each node, node unavailability, and unreliable network connectivity. Replicating computation on multiple nodes can significantly improve performance by reducing task completion time on a grids dynamic environment. We develop an analytical model to determine the number of task replicas to meet the performance goals in different computational grid configurations. Furthermore, taking advantage of the statistical nature of grid-based Monte Carlo applications, we extend the computational replication technique to an N-out-of-M scheduling strategy for grid-based Monte Carlo applications, which can potentially form a large category of grid-computing applications. In addition, we establish a corresponding model for the N-out-of-M scheduling mechanism. Simulations are used to validate the computational replication models. Our preliminary results show that the models we use are effective in predicting the required number of replicas to achieve short task completion time with a given high probability.

The backward Euler method for numerical solution of the Hodgkin-Huxley equations of nerve conduction

A widely accepted model of nerve conduction is based on nonlinear parabolic partial differential equations (PDEs) derived for the giant axon of the squid Loligo by Hodgkin and Huxley. When consider...

On the optimal Halton sequence

Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlos pseudorandom numbers. Clearly, the generation of appropriate high-quality quasirandom sequences is crucial to the success of quasi-Monte Carlo methods. The Halton sequence is one of the standard (along with (t,s)-sequences and lattice points) low-discrepancy sequences, and one of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this phenomenon is to use a randomized (scrambled) version of the Halton sequence. An alternative approach to this is to find an optimal Halton sequence within a family of scrambled sequences. This paper presents a new algorithm for finding an optimal Halton sequence within a linear scrambling space. This optimal sequence is numerically tested and shown empirically to be far superior to the original. In addition, based on analysis and insight into the correlations between dimensions of the Halton sequence, we illustrate why our algorithm is efficient for breaking these correlations. An overview of various algorithms for constructing various optimal Halton sequences is also given.

Parameterizing parallel multiplicative lagged-Fibonacci generators

Monte Carlo computations are commonly considered to be naturally parallel. However, one needs to exercise care in parallelizing the underlying pseudorandom number generator (PRNG) to avoid correlations within, and between, random number streams. PRNGs are normally parallelized using one of the following two paradigms: (i) cycle division and (ii) parameterization. Most of the popular PRNGs based on linear recurrences, such as linear congruential generators (LCGs), generalized feedback shift-register generators (GFSRs), and additive lagged-Fibonacci generators (ALFGs) have been parallelized using both these paradigms. While the (nonlinear) multiplicative lagged-Fibonacci generators (MLFG) is considered superior to the ALFG in quality, it had so far been parallelized only via the cycle division paradigm, which has its limitations. In this paper, we describe parallelization of the MLFG using parameterization, and discuss its implementation in the Scalable Parallel Random Number Generators (SPRNG) [ACM Trans. Math. Software 26 (2000) 436] parallel pseudorandom number generation software. We also present empirical results demonstrating the quality, and quantitatively compare the parallel ALFGs and MLFGs quality.

Monte Carlo Methods for Calculating Some Physical Properties of Large Molecules

In this paper we describe Monte Carlo methods for solving some boundary-value problems for elliptic partial differential equations arising in the computation of physical properties of large molecules. The constructed algorithms are based on walk on spheres, Greens function first passage, walk in subdomains techniques, and finite-difference approximations of the boundary condition. The methods are applied to calculating the diffusion-limited reaction rate, the electrostatic energy of a molecule, and point values of an electrostatic field.

Parallel linear congruential generators with prime moduli

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.

Analysis of Large-Scale Grid-Based Monte Carlo Applications

Monte Carlo applications are widely perceived as computationally intensive but naturally parallel. Therefore, they can be effectively executed on the Grid using the dynamic bag-of-work model. In this paper we concentrate on analyzing the characteristics of large-scale Monte Carlo computation for Grid computing. Based on these analyses, we improve the efficiency of the subtask-scheduling scheme by implementing and analyzing the “N-out-of-M” strategy, and we develop a Monte-Carlo-specific lightweight checkpoint technique, which leads to a performance improvement for Monte Carlo Grid computing. Also, we enhance the trustworthiness of Monte Carlo Grid-computing applications by utilizing the statistical nature of Monte Carlo and by cryptographically validating intermediate results utilizing the random number generator already in use in the Monte Carlo application. All these techniques lead to a high-performance Grid-computing infrastructure that is capable of providing trustworthy Monte Carlo computation services.

Grid-Based Monte Carlo Application

Monte Carlo applications are widely perceived as computationally intensive but naturally parallel. Therefore, they can be effectively executed on the grid using the dynamic bag-of-work model. We improve the efficiency of the subtask-scheduling scheme by using an N-out-of-M strategy, and develop a Monte Carlo-specific lightweight checkpoint technique, which leads to a performance improvement for Monte Carlo grid computing. Also, we enhance the trustworthiness of Monte Carlo grid-computing applications by utilizing the statistical nature of Monte Carlo and by cryptographically validating intermediate results utilizing the random number generator already in use in the Monte Carlo application. All these techniques lead to a high-performance grid-computing infrastructure that is capable of providing trustworthy Monte Carlo computation services.