Advances in Sobol' Index Estimation: Metamodeling, Multilevel Monte Carlo Metamodeling, and Nested Simulation Techniques

Abstract

Sobol' indices are widely used in global sensitivity analysis to quantify input variable contributions to output variance in complex computational models. Traditional methods become impractical due to prohibitive computational costs, difficulties managing model stochasticity and multivariate outputs, and experimental design constraints. This dissertation addresses these limitations through advanced techniques based on metamodeling, multilevel Monte Carlo (MLMC) metamodeling, and nested simulation. We develop two joint metamodel-based estimators for Sobol' indices with established asymptotic normality, enabling reliable uncertainty quantification. Our proposed MLMC metamodeling approach for variance function estimation substantially reduces computational complexity, yielding competitive estimators with superior performance. Additionally, we leverage nested simulation frameworks with robust jackknife-based estimators and introduce a novel method combining nested simulation with Latin hypercube sampling for enhanced efficiency. For scenarios where experimental design is infeasible, we propose a partition-based approach enabling Sobol' index estimation from existing datasets, eliminating new experiment requirements. This extends into a comprehensive metamodeling framework supporting state-of-the-art estimators using available data. Finally, we introduce generalized Sobol' indices for quantifying global sensitivity in stochastic models with multivariate outputs.