Introduction to finite fields and their applications
Introduction to finite fields and their applications
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Quasi-random sequences and their discrepancies
SIAM Journal on Scientific Computing
Computational investigations of low-discrepancy sequences
ACM Transactions on Mathematical Software (TOMS)
On the L2-discrepancy for anchored boxes
Journal of Complexity
Algorithm 647: Implementation and Relative Efficiency of Quasirandom Sequence Generators
ACM Transactions on Mathematical Software (TOMS)
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Mathematical and Computer Modelling: An International Journal
Proceedings of the 38th conference on Winter simulation
Efficient Generation of Parallel Quasirandom Faure Sequences Via Scrambling
ICCS '07 Proceedings of the 7th international conference on Computational Science, Part I: ICCS 2007
Generalized Halton sequences in 2008: A comparative study
ACM Transactions on Modeling and Computer Simulation (TOMACS)
Computational investigations of scrambled Faure sequences
Mathematics and Computers in Simulation
Applied Numerical Mathematics
On the scrambled soboĺ sequence
ICCS'05 Proceedings of the 5th international conference on Computational Science - Volume Part III
Evolutionary optimization of low-discrepancy sequences
ACM Transactions on Modeling and Computer Simulation (TOMACS)
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Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ highly uniform quasirandom numbers in place of Monte Carlo's pseudorandom numbers. Clearly, the generation of appropriate high-quality quasirandom sequences is crucial to the success of quasi-Monte Carlo methods. The Halton sequence is one of the standard (along with (t,s)-sequences and lattice points) low-discrepancy sequences, and one of its important advantages is that the Halton sequence is easy to implement due to its definition via the radical inverse function. However, the original Halton sequence suffers from correlations between radical inverse functions with different bases used for different dimensions. These correlations result in poorly distributed two-dimensional projections. A standard solution to this phenomenon is to use a randomized (scrambled) version of the Halton sequence. An alternative approach to this is to find an optimal Halton sequence within a family of scrambled sequences. This paper presents a new algorithm for finding an optimal Halton sequence within a linear scrambling space. This optimal sequence is numerically tested and shown empirically to be far superior to the original. In addition, based on analysis and insight into the correlations between dimensions of the Halton sequence, we illustrate why our algorithm is efficient for breaking these correlations. An overview of various algorithms for constructing various optimal Halton sequences is also given.