SIAM/ASA Journal on Uncertainty Quantification, Volume 8, Issue 2, Page 668-692, January 2020.
This paper shows how to systematically and efficiently improve a reduced-order model (ROM) to obtain a better ROM-based estimate of the Conditional Value-at-Risk (CVaR) of a computationally expensive quantity of interest (QoI). Efficiency is gained by exploiting the structure of CVaR, which implies that a ROM used for CVaR estimation only needs to be accurate in a small region of the parameter space, called the $\epsilon$-risk region. Hence, any full-order model (FOM) queries needed to improve the ROM can be restricted to this small region of the parameter space, thereby substantially reducing the computational cost of ROM construction. However, an example is presented which shows that simply constructing a new ROM that has a smaller error with the FOM is in general not sufficient to yield a better CVaR estimate. Instead a combination of previous ROMs is proposed that achieves a guaranteed improvement, as well as $\epsilon$-risk regions that converge monotonically to the FOM risk region with decreasing ROM error. Error estimates for the ROM-based CVaR estimates are presented. The gains in efficiency obtained by improving a ROM only in the small $\epsilon$-risk region over a traditional greedy procedure on the entire parameter space are illustrated numerically.

中文翻译：

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条件风险值的自适应降阶模型构造

SIAM / ASA不确定性量化期刊，第8卷，第2期，第668-692页，2020年1月。
本文展示了如何系统地，有效地改进降阶模型（ROM），以获取基于ROM的更好的基于ROM的条件估计值（CVaR），以计算感兴趣的计算量（QoI）。通过利用CVaR的结构可以获得效率，这意味着用于CVaR估计的ROM仅在参数空间的一小部分区域（称为\ε风险区域）中才需要是准确的。因此，改善ROM所需的任何全阶模型（FOM）查询都可以限制在参数空间的这一小区域内，从而大大降低了ROM构建的计算成本。但是，给出了一个示例，该示例表明仅构造一个具有FOM误差较小的新ROM通常不足以产生更好的CVaR估计。取而代之的是，提出了可实现有保证的改进的先前ROM的组合，以及随着ROM误差减小，ε风险区域单调收敛到FOM风险区域。给出了基于ROM的CVaR估计的误差估计。用数值表示了在整个参数空间上仅通过传统的贪婪过程仅在较小的$ε风险区域中改进ROM所获得的效率增益。