发表机构
Technical University of Munich（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对空间随机场输出，提出基于有限元求积的高效敏感性指数估计方法，避免蒙特卡洛采样，并通过氢气燃烧实验验证了观测窗口对输入重要性排序的影响。
AI 中文摘要
在这项工作中，我们考虑具有随机输入参数的数值模型，其中模型输出是一个空间分布的随机场。我们对空间集合进行统计敏感性分析，这些集合例如在模型输出超过临界阈值时出现。我们考虑两种方法：（i）基于核的敏感性分析，使用希尔伯特-施密特独立性准则；（ii）函数值敏感性分析，使用广义Sobol'指数，其中模型输出是感兴趣空间集合的指示函数。我们为敏感性指数开发了高效估计器，特别是对于基于有限元的数值模型，其中我们通过有限元求积近似敏感性度量中的体积积分，从而避免在空间域上进行蒙特卡洛采样。这改进了基于核的指数的现有技术水平，并使这些方法对于昂贵的数值模型在实践中可访问。对于广义Sobol'指数，我们利用相同的有限元求积进行高效计算。我们针对氢气燃烧过程进行了数值实验，该过程由二维空间域中的对流-扩散-反应方程组耦合系统建模，并询问在燃烧域的选定区域中温度是否保持在临界值以下。我们的结果表明，在此示例中，基于核的指数和广义Sobol'指数给出了定性相似的输入重要性排序。此外，排序取决于所选的观测窗口，并且可以在不同窗口之间翻转。用于实现所提出方法的算法在补充材料和开源代码库中提供。
英文摘要
In this work we consider numerical models with random input parameters, where the model output is a spatially distributed random field. We carry out a statistical sensitivity analysis of spatial sets which arise for instance when the model output exceeds a critical threshold. We consider two approaches: (i) a kernel-based sensitivity analysis working with the Hilbert--Schmidt Independence Criterion, and (ii) a function-valued sensitivity analysis working with generalized Sobol' indices, where the model output is the indicator function of the spatial set of interest. We develop efficient estimators for the sensitivity indices, especially for finite-element-based numerical models, where we approximate volume integrals in the sensitivity measures by finite element quadrature, thereby avoiding Monte Carlo sampling over the spatial domain. This improves the state-of-the-art for the kernel-based indices and makes these methods practically accessible for expensive numerical models. For the generalized Sobol' indices we utilize the same finite element quadrature for efficient computation. We present numerical experiments for a hydrogen combustion process, modeled by a coupled system of convection-diffusion-reaction equations in a two-dimensional spatial domain, and ask whether the temperature remains below a critical value in selected regions of the combustion domain. Our results show that in this example the kernel-based and the generalized Sobol' indices give qualitatively similar input importance rankings. Moreover, the ranking depends on the chosen observation window and can flip between different windows. The algorithms for implementing the proposed methods are provided in the supplementary materials and in an open source code repository.
