  This paper analyzes the approximate control variate (ACV) approach to
multifidelity uncertainty quantification in the case where weighted estimators
are combined to form the components of the ACV. The weighted estimators enable
one to precisely group models that share input samples to achieve improved
variance reduction. We demonstrate that this viewpoint yields a generalized
linear estimator that can assign any weight to any sample. This generalization
shows that other linear estimators in the literature, particularly the
multilevel best linear unbiased estimator (ML-BLUE) of Schaden and Ullman in
2020, becomes a specific version of the ACV estimator of Gorodetsky, Geraci,
Jakeman, and Eldred, 2020. Moreover, this connection enables numerous
extensions and insights. For example, we empirically show that having
non-independent groups can yield better variance reduction compared to the
independent groups used by ML-BLUE. Furthermore, we show that such grouped
estimators can use arbitrary weighted estimators, not just the simple Monte
Carlo estimators used in ML-BLUE. Furthermore, the analysis enables the
derivation of ML-BLUE directly from a variance reduction perspective, rather
than a regression perspective.
