Reducing the number of MC runs with antithetic and common random fields

In this paper we adapt two variance reduction techniques, namely antithetic variates and common random numbers, to a sequential simulation scheme which uses copulas as spatial dependence functions to simulate Gaussian and non-Gaussian random fields. The resulting antithetic random fields (ARF) are highly negatively correlated, while common random fields (CRF) exhibit strong positive correlation. We further extend the method in such a way that ARF can be constructed not only as pairs of fields, but also as antithetic triplets, quadruples and any n-tuple of higher dimension. If such ARF or CRF are used as input in Monte Carlo frameworks, this negative or positive correlation of the input random fields is propagated through the physical model to a negative or positive correlation of the output. Ultimately, this enables a significant reduction of simulation runs required for convergence of an estimator. The performances of the proposed methods are examined with two typical applications of stochastic hydrogeology.