Metamodel-based Variance Estimation and Global Sensitivity Analysis

We propose a multilevel Monte Carlo metamodeling approach for variance function estimation.  we show that, under some mild conditions, the proposed MLMC metamodeling approach for variance function estimation can achieve a computational complexity superior to the standard Monte Carlo while achieving a target accuracy level. Additionally, we establish the asymptotic normality of the MLMC metamodeling estimator under a set of sufficient conditions, providing valuable utility for uncertainty quantification. An application in global sensitivity analysis shows that the approach can achieve competitive performance in supporting Sobol' indices estimation.

Sobol' indices are widely used in global sensitivity analysis for assessing the input parameters' impact on the model output,  with successful applications in epidemiological modeling, defect detection in manufacturing, pollutant transport modeling, etc. We propose joint metamodel-based Sobol' index estimators  which rely on estimation of both the mean and variance functions implied by a stochastic simulation experiment. We prove the estimators' asymptotic normality, based on which asymptotic confidence intervals can be constructed for uncertainty quantification. 

We provide a comprehensive examination of conventional methods for computing Sobol's indices and introduce a nested simulation estimator. We investigate the asymptotically optimal allocation of computational resources for this nested estimator to optimize the convergence rate of its mean square error. Additionally, we propose two jackknife estimators: one that achieves unbiasedness and another that surpasses the nested simulation estimator in terms of the convergence rate. Through extensive numerical studies, we highlight the strengths and weaknesses of each estimator, providing a detailed comparison of their effectiveness.