Grouped approximate control variate estimators

TL;DR

The paper presents a generalized grouped approximate control variate (GACV) framework that unifies multifidelity variance reduction methods, showing that ML-BLUE is a specific instantiation of ACV while enabling broader groupings and non-independent samples. It derives optimal weightings to minimize variance under unbiasedness constraints and demonstrates that non-independent, nested groupings can outperform traditional independent-group BLUE estimators. By converting ML-BLUE estimators into nested GACV estimators, the work provides a practical pathway to harness sample reuse for variance reduction. Theoretical results are complemented by numerical experiments comparing ACV-IS, ACV-MF, and nested GACV constructions, with findings that strategic grouping and nesting can yield substantial variance reductions in multifidelity UQ settings.

Abstract

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.

Figures (2)

• Figure 1: Ratio of estimator variance $\mathbb{V}\left[\hat{Q}^{ACV-IS}\right] / \mathbb{V}\left[\hat{Q}^{ACV-MF}\right]$ for fixed cost using a three model setting for two choices of correlation structures as a function of number of evaluations of each model. The $y$ axis is the number of additional samples used for $Q_2$ over $Q_1$. Red lines indicate a variance ratio of one, values higher than one indicate the ACV-MF estimator obtains greater variance reduction. These results indicate that ACV-MF estimator can out-perform the ACV-IS estimator (which is BLUE), depending on correlation and computational cost of models.
• Figure 2: Histogram of the ratio of the ML-BLUE SAOB-M estimator variance to the GACV estimator variance for several $(L,M)$ pairs. The black line denotes equal performance between estimator where experiments to the right of the line indicate that the variance reduction of GACV is better. The GACV often exhibits better variance reduction, and this improvement is typically larger when $M$ is lower compared to $L$. The fully recursive case $(4,5)$ results in a variance ratio of 1.

Theorems & Definitions (14)

• Proposition 1
• proof
• Proposition 2: ML-BLUE Unbiasedness
• Theorem 3: ML-BLUE is an ACV
• proof
• Definition 4: Generalized Linear Grouped ACV estimator
• Proposition 5: Variance of the GACV
• proof
• Theorem 6: Optimal weights of the GACV
• proof