SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 2, Page 601-635, January 2020.
We present a general variance reduction technique for the estimation of the expectation of a scalar-valued quantity of interest associated with a family of model evaluations. The key idea is to reformulate the estimation as a linear regression problem. We then show that the associated estimators are variance minimal within the class of linear unbiased estimators. By solving a sample allocation problem we further construct a variance minimal, linear, and unbiased estimator for a given computational budget. We compare our proposed estimator to other multilevel estimators, such as multilevel Monte Carlo, multifidelity Monte Carlo, and approximate control variates. In addition, we provide a sharp lower bound for the variance of any linear unbiased multilevel estimator and show that our estimator approaches this bound in the infinite low fidelity data limit. The results are illustrated by numerical experiments where the underlying output quantity of interest is generated by an elliptic partial differential equation.

中文翻译：

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关于多级最佳线性无偏估计量

SIAM / ASA不确定性量化期刊，第8卷，第2期，第601-635页，2020年1月。
我们提出了一种用于估计与模型评估族相关的标量值数量的期望值的一般方差减少技术。关键思想是将估计重新表述为线性回归问题。然后，我们证明在线性无偏估计量类别中，相关估计量是方差最小的。通过解决样本分配问题，我们进一步为给定的计算预算构造了方差最小，线性和无偏估计量。我们将我们提出的估计量与其他多层估计量进行比较，例如多层蒙特卡洛，多保真度蒙特卡洛和近似控制变量。此外，我们为任何线性无偏多级估计量的方差提供了一个尖锐的下限，并表明我们的估计量在无限的低保真度数据限制内逼近该范围。结果通过数值实验说明，其中感兴趣的基础输出量由椭圆偏微分方程生成。