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

2026-04-21

AI Summary

This work addresses the high computational cost and poor parametric consistency of traditional pointwise approaches to computing conditional Sobol’ indices. By leveraging a pre-trained global polynomial chaos expansion (PCE) model, the authors exploit its tensor-product basis structure to analytically reconstruct the global expansion as a coefficient field conditioned on specific variables, while preserving orthogonality under the conditional probability measure. Crucially, they demonstrate for the first time that conditional Sobol’ indices are inherently embedded within the PCE basis functions. Consequently, conditional sensitivity analysis reduces to a purely algebraic post-processing step, eliminating the need for repeated surrogate modeling or additional sampling. Numerical benchmarks confirm that the proposed method substantially enhances physical consistency, numerical robustness, and computational efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

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.

Problem

• conditional Sobol' indices
• uncertainty quantification
• Polynomial Chaos Expansion
• sensitivity analysis
• parameterized systems

Innovation

• Polynomial Chaos Expansion
• Conditional Sobol' Indices
• Basis Decomposition
• Uncertainty Quantification
• Analytical Sensitivity Analysis