Randomized Halton sequences

Abstract

The Halton sequence is a well-known multi-dimensional low-discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence. This randomization makes use of the description of Halton sequence using the von Neumann-Kakutani transformation. We randomize the starting point of the sequence. This method combines the potential accuracy advantage of Halton sequence in multi-dimensional integration with the practical error estimation advantage of Monte Carlo methods. Theoretically, using multiple randomized Halton sequences as a variance reduction technique we can obtain an efficiency improvement over standard Monte Carlo. Numerical results show that randomized Halton sequences have better performance not only than Monte Carlo, but also than randomly shifted Halton sequences. They have similar performance with the randomly digit-scrambled Halton sequences but require much less generating time.