论文标题
多级最佳线性无偏估计器的渐近分析
Asymptotic Analysis of Multilevel Best Linear Unbiased Estimators
论文作者
论文摘要
我们研究了[D. Schaden和E. Ullmann，Siam/Asa J. Uncert。 Quantif。，（2020）]。我们将这项工作的结果专门用于基于PDE的模型，这些模型通过离散数量（例如有限元网格大小）进行了参数化。特别是，我们研究了所谓样品分配最佳最佳线性无偏估计器（SAOBS）的渐近复杂性。鉴于固定的计算预算，这些估计器的方差最小。但是，SAOB是通过解决优化问题而被隐式定义的，并且难以分析。另外，我们根据参数模型家族的理查森外推研究了一类辅助估计器。这使我们能够为SAOBS的复杂性提供上限，这表明它们的复杂性在特定类别的线性无偏估计器中是最佳的。此外，蓝绿色的复杂性并不大于多级蒙特卡洛的复杂性。通过用椭圆PDE进行数值实验来说明理论结果。
We study the computational complexity and variance of multilevel best linear unbiased estimators introduced in [D. Schaden and E. Ullmann, SIAM/ASA J. Uncert. Quantif., (2020)]. 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.
