SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 3, Page 953-978, January 2021.
We study the computational complexity and variance of multilevel best linear unbiased estimators introduced in [D. Schaden and E. Ullmann, SIAM/ASA J. Uncertain. Quantif., 8 (2020), pp. 601--635]. We specialize the results in this work to PDE-based models that are parameterized by a discretization quantity, e.g., the finite element mesh size. In particular, we investigate the asymptotic complexity of the so-called sample allocation optimal best linear unbiased estimators (SAOBs). These estimators have the smallest variance given a fixed computational budget. However, SAOBs are defined implicitly by solving an optimization problem and are difficult to analyze. Alternatively, we study a class of auxiliary estimators based on the Richardson extrapolation of the parametric model family. This allows us to provide an upper bound for the complexity of the SAOBs, showing that their complexity is optimal within a certain class of linear unbiased estimators. Moreover, the complexity of the SAOBs is not larger than the complexity of multilevel Monte Carlo. The theoretical results are illustrated by numerical experiments with an elliptic PDE.

中文翻译：

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多级最佳线性无偏估计量的渐近分析

SIAM/ASA 不确定性量化杂志，第 9 卷，第 3 期，第 953-978 页，2021 年 1 月。
我们研究了 [D. Schaden 和 E. Ullmann，SIAM/ASA J. 不确定。Quantif., 8 (2020), pp. 601--635]。我们将这项工作的结果专门用于基于离散化量参数化的基于 PDE 的模型，例如有限元网格尺寸。特别是，我们研究了所谓的样本分配最优最佳线性无偏估计量 (SAOB) 的渐近复杂性。给定固定的计算预算，这些估计量具有最小的方差。然而，SAOB 是通过解决优化问题隐式定义的，并且难以分析。或者，我们研究了一类基于参数模型族的理查森外推的辅助估计量。这使我们能够为 SAOB 的复杂性提供一个上限，表明它们的复杂性在某一类线性无偏估计器中是最优的。而且，SAOB 的复杂度并不比多级蒙特卡罗的复杂度大。通过椭圆偏微分方程的数值实验说明了理论结果。