A generalized approximate control variate framework for multifidelity uncertainty quantification

Abstract

We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models -which are typically lower fidelity with unknown statistics -are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multi-model variance reduction schemes as special cases. We demonstrate that these existing strategies use recursive sampling strategies, and as a result, their maximum possible variance reduction is limited to that of a control variate algorithm that uses only a single low-fidelity model with known mean. This theoretical result holds regardless of the number of low-fidelity models and/or samples used to build the estimator. We then derive new sampling strategies within our framework that circumvent this limitation to make efficient use of all available information sources. In particular, we demonstrate that a significant gap can exist, of orders of magnitude in some cases, between the variance reduction achievable by using a single low-fidelity model and our non-recursive approach. We also present initial sample allocation approaches for exploiting this gap. They yield the greatest benefit when augmenting the high-fidelity model evaluations is impractical because, for instance, they arise from a legacy database. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

“…Such a hierarchy can be formed from a sequence of finite element discretizations to the solutions of a partial differential equation (PDE), for example, which converges to the exact solution of the governing equations as the finite element mesh is refined. Some attention has been given to building multi‐fidelity approximations using models that do not admit a strict ordering of fidelity; however, literature in this area is limited.…”

“…Multi‐level/multi‐index surrogate methods are closely related to many multilevel/multifidelity sampling algorithms . These sampling algorithms leverage correlation between the outputs of multiple models to reduce the variance in statistical estimators of quantities such as expectation.…”

Accelerated Multilevel Monte Carlo With Kernel‐Based Smoothing and Latinized Stratification

“…Multilevel methods belong to the wider class of “multifidelity” approaches which involve a combination of models with varying degrees of fidelity (Müller et al., 2014; O'Malley et al., 2018; Peherstorfer et al., 2016). The maximum variance reduction (and, hence, speedup) MLMC can achieve depends on the degree of correlation between the levels (Gorodetsky et al., 2020). For complex physics, where refining the grid actually resolves more features, this correlation will be lower, and so will be the variance reduction achieved by going from coarser to finer grids.…”

On Multilevel Best Linear Unbiased Estimators

“…Recall that an important motivation for the introduction of the ACV estimators is their increased variance reduction capacity compared to MLMC and MFMC. In fact, the ACV estimators in [9] reach the exact same lower variance bound in the infinite low fidelity data limit as the multilevel BLUEs (cf. [19,Sec.…”

“…Typically, multilevel estimators couple an expensive, high resolution PDE model with cheap, low resolution PDE models. Examples for multilevel estimators are multilevel Monte Carlo (MLMC) [7,8], multifidelity Monte Carlo (MFMC) [14,15] and approximate control variates (ACVs) [9]. In this work we revisit the multilevel best linear unbiased estimator (BLUE) introduced in [19].…”