Low discrepancy sequences for Monte Carlo simulations on reconfigurable platforms

Abstract

Low-discrepancy sequences, also known as ldquoquasi-randomrdquo sequences, are numbers that are better equidistributed in a given volume than pseudo-random numbers. Evaluation of high-dimensional integrals is commonly required in scientific fields as well as other areas (such as finance), and is performed by stochastic Monte Carlo simulations. Simulations which use quasi-random numbers can achieve faster convergence and better accuracy than simulations using conventional pseudo-random numbers. Such simulations are called Quasi-Monte Carlo. Conventional Monte Carlo simulations are increasingly implemented on reconfigurable devices such as FPGAs due to their inherently parallel nature. This has not been possible for Quasi-Monte Carlo simulations because, to our knowledge, no low-discrepancy sequences have been generated in hardware before. We present FPGA-optimized scalable designs to generate three different common low-discrepancy sequences: Sobol, Niederreiter and Halton. We implement these three generators on Virtex-4 FPGAs with varying degrees of fine-grained parallelization, although our ideas can be applied to a far broader class of sequences. We conclude with results from the implementation of an actual Quasi-Monte Carlo simulation for extracting partial inductances from integrated circuits.