Articlegreen2021

Asymptotic Analysis of Multilevel Best Linear Unbiased Estimators

17 citations0/0 files storedSIAM/ASA Journal on Uncertainty Quantification

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Probabilistic and Robust Engineering Design Markov Chains and Monte Carlo Methods Mathematical Approximation and Integration

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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References

1999Interpolation of Spatial Data 2008Multilevel Monte Carlo Path Simulation 2018Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization 2011Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients 2016Optimal Model Management for Multifidelity Monte Carlo Estimation 2020A generalized approximate control variate framework for multifidelity uncertainty quantification 2017Multilevel Richardson–Romberg extrapolation 2020On Multilevel Best Linear Unbiased Estimators

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