Summary of Quasi-Monte Carlo ---- Discrepancy between Theory
and Practice
Yu Zhang
yz08@fsu.edu
Quasi-Monte Carlo is widely used in many theory and practice areas. In this paper,
the author surveyed the current status of Quasi-Monte Carlo. In order to tell the
difference between Monte Carlo and Quasi-Monte Carlo, the author introduced the
definition of discrepancy and L∞ , L2 discrepancy. Sequences have low discrepancy if
the points distribute uniformly. Quasi-Monte Carlo methods use low-discrepancy
sequences while Monte Carlo methods use random numbers. The Koksma-Hlawka
theorem gave the relationship between discrepancy and the integration error.
After that, the author defined (t, k)-sequences in base b, and pointed that, if t and b
are constants or only depend on k, the (t, k)-sequence is a low-discrepancy sequence.
Moreover, the smaller value of t is the lower discrepancy the sequence will be. So, (0,
k)-sequence is the best. Such sequences also called generalized Faure sequences.
Then, the author described how to construct such sequences.
The author also talked about derandomization of RQMC. The process focuses on
finding good performance sequences with good convergence rate. From the analysis
on Halton sequences, lattice rules and generalized Faure sequences,
digit-permutation is proved to be useful to improve the performance.
The following of the paper mentioned the small effective dimension and isotropic
problem. The author showed that Quasi-Monte Carlo or RQMC methods are more
suitable to solve these problems than the ordinary Monte Carlo methods. Then, the
author discussed the application of low-discrepancy sequences in financial area. The
experiment results show that GFaure has good performance on solving such
problems.