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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 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 http://sprng.cs.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 environments.The SPRNG software was developed with funding from DARPA Contract Number DABT63-95-C-0123 for ITO: Scalable Systems and Software, entitled A Scalable Pseudorandom Number

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) intrastream 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 SPRNG [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.

Abstract. 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, Green's 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.Key words. Monte Carlo method, random walk, diffusion, reaction rate, electrostatic energy, molecule AMS subject classifications. 65C05, 65N99, 78M25, 92C451. Introduction. Elliptic partial differential equations such as the Laplace, Poisson, Poisson-Boltzmann, etc. are effectively used as mathematical models in different branches of computational biophysics and chemistry. Calculation of the diffusionlimited reaction rate, the electrostatic potential and field, the internal energy -are all problems that can be reduced to the solution of a diffusion equation with some conditions on the boundary and at the infinity. The intrinsic analogy between diffusion and electrostatics makes it possible to apply the same computational techniques to solve problems coming from these different fields. It is worth noting that the solution domain in this class of problems is usually infinite. So, it is natural that since the days of Maxwell, the boundary-element method has been used as an effective tool for solving such problems. In particular, one can calculate the capacitance and the diffusionlimited reaction rate as a surface integral [18]. The boundary-element method, as well as finite-difference and finite-element methods, are still commonly used for solving electrostatics and diffusion problems arising in biophysics. Review of these and other techniques used in the computation of molecular electrostatic properties is given in [2].Another possible way of computationally treating these problems comes from the probabilistic representation of solutions to elliptic and parabolic partial differential equations as functionals of diffusion process trajectories [15,7,8]. Direct computational simulation of physical diffusion in this case coincides with the approximation of Brownian motion as the solution to a stochastic differential equation via a firstorder Euler scheme [19,16]. This approach was applied [4] to simulation studies of diffusion-limited reactions. Though computationally far from being optimal, it allows one to include different physical phenomena (hydrodynamic, electrostatic, etc.) into the computational scheme, and, what is essential, it is efficient enough to be competitive with deterministic methods. Later, this algorithm was modified [30] by incorporating different boundary conditions [28,26]. In addition, other variants of the algorithm were suggested. In [31], in particular, the Brownian dynamics simulation method was compared to the algorithms based on survival probab...

Recent research shows that Monte Carlo diffusion methods are often the most efficient algorithms for solving certain elliptic boundary value problems. In this paper, we extend this research by providing two efficient algorithms based on the concept of "last-passage diffusion." These algorithms are qualitatively compared with each other (and with the best first-passage diffusion algorithm) in solving the classical problem of computing the charge distribution on a conducting disk held at unit voltage. All three algorithms show detailed agreement with the known analytic solution to this problem.