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Conditional-value-at-risk estimation via reduced-order models. (English) Zbl 1405.35263

Summary: 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 \(1-\beta\ll 1\), where \(\beta\in (0,1)\) 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.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
62H12 Estimation in multivariate analysis
91G60 Numerical methods (including Monte Carlo methods)
91G80 Financial applications of other theories
65Y20 Complexity and performance of numerical algorithms
91B30 Risk theory, insurance (MSC2010)
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