Error Analysis for Quasi-Monte Carlo Methods

Fred J. Hickernell

Published 2017-02-06Version 1

Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity comprised of the variation of the integrand, the discrepancy of the sampling measure, and the confounding. The trio identity has different versions, depending on whether the integrand is deterministic or Bayesian and whether the sampling measure is deterministic or random. Whereas the variation and the discrepancy are common in the literature, the confounding is a factor deserving more attention. Theory and examples are used to show how the cubature error may be reduced by employing the low discrepancy sampling that defines quasi-Monte Carlo methods. The error may also be reduced by rewriting the integral in terms of a different integrand. Finally, the confounding explains why the cubature error might not decay at a rate that one would expect from looking at the discrepancy.