Analytical Extraction of Conditional Aleatory Sensitivities Across Epistemic Space via a Single PCE Model

2026-07-06

AI Summary

This work addresses the computational inefficiency of traditional methods for evaluating conditional Sobol’ indices under mixed aleatory and epistemic uncertainties, which typically require costly double-loop sampling and struggle to capture how stochastic sensitivities vary with epistemic variables. The authors propose a unified Bayesian framework that leverages the tensor-product structure of a single global polynomial chaos expansion (PCE) model to analytically decompose coefficient fields dependent on epistemic variables, enabling closed-form algebraic computation of conditional Sobol’ indices. For the first time, continuous conditional sensitivity fields are extracted from a single PCE without repeated surrogate modeling. By integrating reversible-jump Markov chain Monte Carlo (RJMCMC), the approach adaptively selects basis functions and provides Bayesian credible intervals for sensitivity measures. Demonstrated on a high-dimensional groundwater flow model, the method substantially reduces computational cost while yielding smooth, statistically consistent sensitivity fields and robust uncertainty quantification.

Abstract

In hybrid uncertainty quantification, evaluating how aleatory sensitivities vary under epistemic uncertainty, referred to as conditional Sobol' indices, is typically hindered by the computationally expensive double-loop procedure. Classical Polynomial Chaos Expansion (PCE) provides efficient access to global sensitivity measures but cannot directly resolve sensitivity variation across the epistemic space without repeated surrogate reconstruction. This study proposes a unified Bayesian framework that extracts continuous conditional Sobol' fields from a single global PCE representation. By exploiting the tensor-product structure of orthogonal polynomial bases in an augmented stochastic space, the global expansion is analytically decomposed into epistemic-dependent coefficient fields, enabling a closed-form variance decomposition. As a result, conditional Sobol' indices can be computed through a purely algebraic post-processing step without additional model evaluations or retraining. In addition, a Reversible Jump Markov Chain Monte Carlo (RJMCMC) scheme is incorporated to perform adaptive basis selection and trans-dimensional inference, while simultaneously providing Bayesian credible intervals for the conditional sensitivity measures. Numerical experiments on a high-dimensional groundwater flow model demonstrate that the proposed method significantly reduces computational cost while maintaining smooth sensitivity fields and statistically consistent uncertainty quantification across the epistemic domain.

Problem

• conditional Sobol' indices
• hybrid uncertainty quantification
• aleatory sensitivity
• epistemic uncertainty
• computational cost

Innovation

• Polynomial Chaos Expansion
• Conditional Sobol' Indices
• Epistemic Uncertainty
• Bayesian Inference
• RJMCMC