基于低秩张量近似的全局敏感性分析
Global sensitivity analysis using low-rank tensor approximations
摘要：
在全局敏感性分析的背景下，Sobol指数构成了评估模型的不确定输入参数的相对重要性的有力工具。我们在这里介绍了一种新的方法来评估这些指标在低计算成本，通过后处理多项式元模型的系数属于低秩张量近似类。这类元模型在表示高维模型的响应时特别有效，因为其一般函数形式的未知数的数量仅随输入维度线性增长。所提出的方法在示例应用中进行了验证，其中将从元模型系数导出的Sobol指数与参考指数进行比较，后者通过精确解析解或具有极大样本的蒙特卡罗模拟获得。此外，低秩张量近似面临流行的多项式混沌扩展元模型的案例研究，涉及分析秩一函数和有限元模型有关的结构力学和热传导。在检查的应用程序中，指数的基础上的新的方法往往收敛速度更快的参考解决方案的试验设计用于建立元模型的大小。
英文摘要：
In the context of global sensitivity analysis, the Sobol' indices constitute a powerful tool for assessing the relative significance of the uncertain input parameters of a model. We herein introduce a novel approach for evaluating these indices at low computational cost, by post-processing the coefficients of polynomial meta-models belonging to the class of low-rank tensor approximations. Meta-models of this class can be particularly efficient in representing responses of high-dimensional models, because the number of unknowns in their general functional form grows only linearly with the input dimension. The proposed approach is validated in example applications, where the Sobol' indices derived from the meta-model coefficients are compared to reference indices, the latter obtained by exact analytical solutions or Monte-Carlo simulation with extremely large samples. Moreover, low-rank tensor approximations are confronted to the popular polynomial chaos expansion meta-models in case studies that involve analytical rank-one functions and finite-element models pertinent to structural mechanics and heat conduction. In the examined applications, indices based on the novel approach tend to converge faster to the reference solution with increasing size of the experimental design used to build the meta-model.
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