The authors present the results of numerical experiments on variance reduction techniques based on “multipliers”. As usual this helps in estimation of multidimensional integrals. Instead of looking for a function \(v(x)\) that has known expectation and variance and that is related to the target function \(f(x)\) the authors investigate a family of functions: \[ v(x)=\lambda + (1-\lambda) a(x). \] Here \(a(x)\) is an “easy function” or a “reference function”. The optimum value \(\lambda_0\) that minimizes the variation is calculated. As usual good results are obtained in case of close correlations between \(f(x)\) and \(a(x)\). Considering the quasi Monte Carlo method the authors show that multipliers can reduce errors but rates of convergence remain unchanged.
The paper is a new approach to variance reduction multipliers described in I. M. Sobol’s book: [Numerical Monte Carlo methods (Nauka, Moskow, 1973; Zbl 0289.65001)] and a complement to P. H. Eberhard and O. P. Schneider [Reference functions to decrease errors in Monte Carlo integrals, Computer Phys. Communs. 67, 378-388 (1992)].