Asymptotic Analysis of Multilevel Best Linear Unbiased Estimators
Abstract
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.
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Published In
SIAM/ASA Journal on Uncertainty Quantification
Pages: 953 - 978
DOI: 10.1137/20M1321607
ISSN (online): 2166-2525
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© 2021, Society for Industrial and Applied Mathematics.
History
Submitted: 25 February 2020
Accepted: 12 April 2021
Published online: 1 July 2021
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Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 188264188/GRK1754
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