Generation of random variates using asymptotic expansions

Abstract

Monte-Carlo methods are widely used numerical tools in various fields of application, like rarefied gas dynamics, vacuum technology, stellar dynamics or nuclear physics. A central part is the generation of random variates according to a given probability law. Fundamental techniques are the inversion principle or the acceptance-rejection method—both may be quite time-consuming if the given probability law has a complicated structure. In this paper probability laws depending on a small parameter are considered and the use of asymptotic expansions to generate random variates is investigated. The results given in the paper are restricted to first order expansions. Error estimates for the discrepancy as well as for the bounded Lipschitz distance of the asymptotic expansion are derived. Furthermore the integration error for some special classes of functions is given. The efficiency of the method is proved by a numerical example from rarefied gas flows.