The control variates method is a classical variance reduction technique for Monte Carlo estimators that exploits correlated auxiliary variables without introducing bias. In many applications, the quantity of interest can be expressed as a ratio of expectations. We propose a variance-reduced estimator for such ratios, which applies control variates to both the numerator and the denominator. The control variate coefficients are optimized jointly to minimize the variance of the resulting estimator. This approach theoretically guarantees variance reduction and naturally extends to approximate control variates. Simulation studies show significant variance reduction, particularly when correlations between variables and control variates are strong. The practical value of the method is illustrated with a multi-fidelity aircraft design use case.

翻译：控制变量法是一种经典的蒙特卡洛估计器方差缩减技术，它利用相关的辅助变量来降低方差而不引入偏差。在许多应用中，目标量可以表示为期望值的比值。我们针对此类比值提出了一种方差缩减估计器，该方法将控制变量同时应用于分子和分母。控制变量的系数经过联合优化，以最小化所得估计器的方差。该方法理论上保证了方差缩减，并能自然地推广到近似控制变量的情形。仿真研究表明该方法能显著降低方差，尤其在变量与控制变量之间相关性较强时效果更为明显。我们通过一个多保真度飞机设计用例展示了该方法的实用价值。