Risk-Averse PDE-Constrained Optimization Using the Conditional Value-At-Risk
针对偏微分方程系数和输入存在不确定性的优化问题，提出了两种近似方法来最小化条件风险值（CVaR），并证明了光滑近似的可微性和一致性，最后通过数值实验验证了方法的有效性。
Uncertainty is inevitable when solving science and engineering application problems. In the face of uncertainty, it is essential to determine robust and risk-averse solutions. In this work, we consider a class of PDE-constrained optimization problems in which the PDE coefficients and inputs may be uncertain. We introduce two approximations for minimizing the conditional value-at-risk (CVaR) for such PDE-constrained optimization problems. These approximations are based on the primal and dual formulations of CVaR. For the primal problem, we introduce a smooth approximation of CVaR in order to utilize derivative-based optimization algorithms and to take advantage of the convergence properties of quadrature-based discretizations. For this smoothed CVaR, we prove differentiability as well as consistency of our approximation. For the dual problem, we regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes advantage of the structure of the regularized optimality conditions and provides a means of calculating worst-case probability distributions based on the given probability level. We conclude with numerical results.