\input zb-basic
\input zb-ioport
\iteman{io-port 01215244}
\itemau{Caflisch, Russel E.}
\itemti{Monte Carlo and quasi-Monte Carlo methods.}
\itemso{Acta Numerica 7, 1-49 (1998).}
\itemab
This well-written and concise survey paper brings the reader to the present state in the field of Monte Carlo quadrature. Topics discussed include the estimation of integrals via their interpretation as an expectation, the derivation of the $O(N^{-1/2})$ convergence rate law, the efficacy of Monte Carlo for high-dimensional integrals, the presentation of methods for sampling non-uniform random variables, and the derivation of several techniques for variance reduction. The survey continues with a description of the quasi-random number technique for accelerating convergence, by stating and proving the Koksma-Hlawka Theorem, by presenting specific quasi-random number generators and by examining smoothness and dimension limitation of quasi-Monte Carlo. Dimension reduction methods such as the Brownian bridge are presented. Lastly the paper treats an application of the Monte Carlo method to the problem of rarefied gas dynamics.
\itemrv{R. W. Shonkwiler (Atlanta)}
\itemcc{}
\itemut{quasi-Monte Carlo methods; quadrature; high-dimensional integrals; gas dynamics; convergence; variance reduction; quasi-random number operators; dimension reduction methods}
\itemli{}
\end