On Direct Simulation Quasi-Monte Carlo Methods

Equations derived from kinetic theory often express a desired quantity in terms of a probability density. For example, the Direct Simulation Monte Carlo (DSMC) method is a well-known powerful technique for computational rarefied gas dynamics. It uses an algorithm that begins with an initial distribution and, through random sampling, converges to a stationary distribution. Random sampling is achieved using random numbers obtained with pseudo-random number generators. Quasi-Monte Carlo methods (QMCMs) replace calls to a pseudo-random number generator by calls to a quasi-random number generator. QMCMs are known to have a better convergence rates than Monte Carlo methods for multidimensional integration, but it is not trivial to make QMCM work well in contexts outside of Monte Carlo integration, such as DSMC. In fact, naive replacement of calls to a pseudo-random number generator by calls to a quasi-random generator have been known to fail utterly. We illustrate these difficulties and discuss how to overcome them. In the context of DSMC, we conclude that little can be gained through the use of quasi-random sequences, however in the context of “direct methods” we find promising results.