Analytical Extraction of Conditional Sobol' Indices via Basis Decomposition of Polynomial Chaos Expansions

TL;DR

This paper introduces an analytical method to compute conditional Sobol' indices from Polynomial Chaos Expansions, enabling efficient and consistent sensitivity analysis under various conditions without additional sampling.

Contribution

It derives closed-form expressions for conditional Sobol' indices directly from PCE basis functions, improving computational efficiency and robustness over traditional methods.

Findings

Method ensures physical coherence of sensitivity measures.

Offers superior numerical robustness and efficiency.

Eliminates need for additional sampling in conditional analysis.

Abstract

In uncertainty quantification, evaluating sensitivity measures under specific conditions (i.e., conditional Sobol' indices) is essential for systems with parameterized responses, such as spatial fields or varying operating conditions. Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space. This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are inherently embedded within its basis functions. By leveraging the tensor-product property of PCE bases, we reformulate the global expansion into a set of analytical coefficient fields that depend on the conditioning variables. Based on the preservation of orthogonality under conditional probability measures, we derive closed-form expressions for conditional variances and Sobol' indices. This framework bypasses the need for repetitive modeling or additional sampling, transforming conditional sensitivity analysis into a purely algebraic post-processing step. Numerical benchmarks indicate that the proposed method ensures physical coherence and offers superior numerical robustness and computational efficiency compared to conventional point-wise approaches.