Analytical Extraction of Conditional Sobol' Indices via Basis Decomposition of Polynomial Chaos Expansions
TLDR
This paper introduces an analytical method to extract conditional Sobol' indices directly from Polynomial Chaos Expansions, offering a computationally efficient approach.
Key contributions
- Derives analytical conditional Sobol' indices directly from pre-trained Polynomial Chaos Expansions (PCE).
- Leverages PCE basis decomposition to reformulate global expansion into analytical coefficient fields.
- Provides closed-form expressions for conditional variances and Sobol' indices, preserving orthogonality.
- Transforms conditional sensitivity analysis into an efficient, purely algebraic post-processing step.
Why it matters
Traditional conditional sensitivity analysis is computationally expensive and inconsistent. This paper provides a robust, efficient, and analytical method that transforms it into a simple algebraic post-processing step for pre-trained PCE models. This significantly improves uncertainty quantification for parameterized systems.
Original 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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