Uncertainty quantification (UQ) and inference involving a large number of parameters are valuable tools for problems associated with heterogeneous and non-stationary behaviors. The difficulty with these problems is exacerbated when these parameters are statistically dependent requiring statistical characterization over joint measures. Probabilistic modeling methodologies stand as effective tools in the realms of UQ and inference. Among these, polynomial chaos expansions (PCE), when adapted to low-dimensional quantities of interest (QoI), provide effective yet accurate approximations for these QoI in terms of an adapted orthogonal basis. These adaptation techniques have been cast as projection pursuits in Gaussian Hilbert space in what has been referred to as a projection pursuit adaptation (PPA) by Xiaoshu Zeng and Roger Ghanem (2023). The PPA method efficiently identifies an optimal low-dimensional space for representing the QoI and simultaneously evaluates an optimal PCE within that space. The quality of this approximation clearly depends on the size of the training dataset, which is typically a function of the adapted reduced dimension. The complexity of the problem is thus mediated by the complexity of the low-dimensional quantity of interest and not the complexity of the high-dimensional parameter space.

中文翻译：

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数据驱动的投影追踪对相关高维参数的多项式混沌展开的适应


涉及大量参数的不确定性量化 （UQ） 和推理是解决与异构和非平稳行为相关的问题的宝贵工具。当这些参数在统计上依赖于需要统计表征而不是联合测量时，这些问题的困难就会加剧。概率建模方法是 UQ 和推理领域的有效工具。其中，多项式混沌展开 （PCE） 在适应低维目标量 （QoI） 时，根据适应的正交基为这些 QoI 提供了有效而准确的近似值。这些适应技术已被 Xiaoshu Zeng 和 Roger Ghanem （2023） 称为高斯希尔伯特空间中的投影追求 （PPA）。PPA 方法有效地确定了表示 QoI 的最佳低维空间，并同时评估了该空间内的最佳 PCE。这种近似的质量显然取决于训练数据集的大小，这通常是适应后的缩减维度的函数。因此，问题的复杂性是由感兴趣的低维量的复杂性而不是高维参数空间的复杂性来中介的。