Fixed length low discrepancy sequences

Low discrepancy sequences (sometimes called “quasi-Monte Carlo” or “sub-random” sequences) are often used to try to improve the convergence properties of multi-dimensional Monte Carlo simulation exercises. These sequences aim to sample the relevant multi-dimensional domain space of the simulation exercise more uniformly than the basic Monte Carlo approach of choosing sequence elements purely at random. However, the convergence improvements provided by traditional low discrepancy sequence methodologies such as Halton or Sobol’ sequences can seem puzzlingly intermittent. We explore why this is so. We also propose an approach that addresses many of the relevant issues. It involves defining in advance the targeted length of the sequence and then optimising how sequence elements are selected bearing in mind this target length. A particularly appealing example involves a sequence length that is an integer power of two and choosing sequence elements using a method similar to that used to generate Sobol’ sequences but with scrambling and length-specific overlays.