Optimization of Quasi Random Number Generators

Monte Carlo simulations are based on Random Number Generators (RNG). Drawing random numbers often constitutes a considerable part of simulation time. Unrolling random numbers is a way to accelerate Monte Carlo simulations appreciably by speeding up any RNG. This method consists in a pre-generation and storage of random numbers in an array of values. This array is either directly included in the binary code during the compilation process or is written in binary files, which are mapped in memory at run time. The convoluted RNG algorithm is then bypassed and the numbers are directly picked in sequence in the array. This method has been proved to be quite efficient with small series of pseudo random numbers [1]. Another type of generator is very interesting for stochastic simulations, the Quasi Random Number Generator (QRNG) [2]. QRNGs gain time by improving the convergence in numerous simulations. Due to their fast convergence and the lack of replication, quasi Monte Carlo simulations are naturally inclined to be unrolled in arrays stored in random access memory. In this paper we present a benchmark on different recent computer architectures of the ”unrolling” technique using random access memory. We then discuss results obtained with classical RNGs (Quick and dirty, ran2 from numerical recipes [5], and Mersenne Twister [3]) and with quasi-random numbers using Sobol [6] and Nidereiter [4] (using the Gnu Scientific Library). By reducing by 30 the generation time of the Sobol algorithm in one dimension, the ”unrolling” optimization technique turned out to be really efficient on QRNGs. However multiple dimensions QRNGs are harder to optimize using only the random access memory. Actually, it creates huge arrays in the source files that are impossible to compile. More than that, the computation time taken to generate a 40 dimensions low discrepancy sequence, is only twice as important as that taken to generate a single dimension sequence. A solution to address those issues is explored, the memory mapping. This optimization technique cannot be as efficient as the ”unrolling” one, but has the advantage of being able to support very long number series. The gain variations are presented and solutions are discussed to adapt optimization techniques to multidimensional quasi random number generation.