无需重复采样，直接从多项式混沌展开中解析计算条件敏感性指标。
Analytical Extraction of Conditional Sobol' Indices via Basis Decomposition of Polynomial Chaos Expansions
- 利用PCE基函数的张量积特性，将全局展开重构为依赖条件变量的系数场。
- 推导出条件方差与Sobol'指数的闭式表达式，实现代数化后处理。
- 计算高效且物理一致性好，适合参数化系统敏感性分析场景。
在不确定性量化中，评估特定条件下的敏感性度量（即条件Sobol'指数）对具有参数化响应的系统（如空间场或变化工况）至关重要。传统方法常依赖逐点建模，计算成本高且参数空间内不一致。本文证明，对于预训练的全局多项式混沌展开（PCE）模型，其条件Sobol'指数天然嵌入于基函数之中。通过利用PCE基的张量积性质，我们将全局展开重构为依赖于条件变量的解析系数场。基于条件概率测度下正交性的保持，我们推导出条件方差和Sobol'指数的闭式表达式。该框架避免了重复建模或额外采样，使条件敏感性分析转变为纯粹的代数后处理步骤。数值基准测试表明，该方法保证了物理一致性，相比传统逐点方法具有更优的数值鲁棒性和计算效率。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。