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Conditional-Value-at-Risk Estimation via Reduced-Order Models

机译：Conditional-Value-at-Risk评估通过降维模型

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摘要

This paper proposes and analyzes two reduced-order model (ROM) based approaches for the efficient and accurate evaluation of the Conditional-Value-at-Risk (CVaR) of quantities of interest (QoI) in engineering systems with uncertain parameters. CVaR is used to model objective or constraint functions in risk-averse engineering design and optimization applications under uncertainty. Evaluating the CVaR of the QoI requires sampling in the tail of the QoI distribution and typically requires many solutions of an expensive full-order model of the engineering system. Our ROM approaches substantially reduce this computational expense. Both ROM-based approaches use Monte Carlo (MC) sampling. The first approach replaces the computationally expensive full-order model by an inexpensive ROM. The resulting CVaR estimation error is proportional to the ROM error in the so-called risk region, a small region in the space of uncertain system inputs. The second approach uses a combination of full-order model and ROM evaluations via importance sampling and is effective even if the ROM has large errors. In the importance sampling approach, ROM samples are used to estimate the risk region and to construct a biasing distribution. A few full-order model samples are then drawn from this biasing distribution. Asymptotically, as the ROM error goes to zero, the importance sampling estimator reduces the variance by a factor of 1-β<<1, where β∈2 (0, 1) is the quantile level at which CVaR is computed. Numerical experiments on a system of semilinear convection-diffusion-reaction equations illustrate the performance of the approaches.
机译：本文提出并分析了两种降维基于模型(ROM)的有效方法和准确的评估Conditional-Value-at-Risk (CVaR)的数量兴趣(QoI)工程系统不确定的参数。在规避风险的目标或约束功能工程设计和优化应用程序下的不确定性。需要抽样QoI的尾巴分布和通常需要很多一个昂贵的全阶模型的解决方案工程系统。大大减少计算费用。rom的方法都使用蒙特卡罗(MC)抽样。计算昂贵的全阶模型廉价的罗CVaR估计错误罗误差成正比所谓的风险区域,一个小区域空间不确定系统的输入。方法使用全阶模型的结合和罗评估通过采样和重要性是有效的,即使罗有大量错误。重要性抽样方法,罗样本用于估计风险区域和构造偏置分布。样品然后来自偏压分布。趋于零,重要性抽样估计量减少方差因子的1 -β< < 1,β2∈(0,1)是CVaR的分位数水平计算。半线性convection-diffusion-reaction方程说明的性能方法。

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