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

• URL: http://arxiv.org/abs/2604.19165v1
• Date: Tue, 21 Apr 2026 07:20:47 GMT
• Title: Analytical Extraction of Conditional Sobol' Indices via Basis Decomposition of Polynomial Chaos Expansions
• Abstract summary: In uncertainty quantification, evaluating sensitivity measures under specific conditions is essential for systems with parameterized responses.<n>Traditional approaches often rely on point-wise modeling, which is computationally expensive and may lack consistency across the parameter space.<n>This paper demonstrates that for a pre-trained global Polynomial Chaos Expansion (PCE) model, the analytical conditional Sobol' indices are embedded within its basis functions.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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