An empirical investigation of different scrambling methods for Faure sequences

摘要

Quasi-Monte Carlo (QMC) methods are now widely used in scientific computation, especially in estimating integrals over multidimensional domains and in many different financial computations. The use of randomized QMC methods, where randomness can be brought to bear on quasirandom sequences through scrambling and other related randomization techniques, brings more wide applications for QMC. However, the integration variance can depend strongly on the scrambling methods. Much of the work dealing with scrambling methods has been aimed at ways of linear scrambling methods. In this paper, we take a close look at the quadratic scrambling method for Faure sequences and compare with other linear scrambling methods.We will investigate the implementation and the performance of nonlinear scrambling methods for Faure sequences. The scrambling methods will focus on quadratic and inversive scrambling. We will compare those nonlinear scrambling methods with linear scrambling methods by a set of test-integral functions. Pseudorandom number generation by nonlinear methods will be expensive when their moduli are large. In our case, the modulus is relatively small. To the best of our knowledge, such a comparison has never been done before.