逐步构造偏微分方程约束optimization的参数化降阶模型
Progressive construction of a parametric reduced-order model for PDE-constrained optimization
摘要：
提出了一种自适应降阶方法，该方法打破了传统的离线-在线降阶模型框架，将降阶模型作为偏微分方程约束优化的代理模型。定义了一系列受给定降阶模型约束的最优化问题，目标是收敛到给定偏微分方程约束的最优化问题的解。对于每个简化的最优化问题，约束ROM从在序列中的一些先前问题的解处对高维模型（HDM）进行采样来训练。简化的最优化问题配备有基于残差指示符的非线性信赖域，以将优化轨迹保持在ROM准确的参数空间的区域中。还提出了一种将灵敏度纳入降阶基（ROB）的技术，以及一种用于计算降阶模型的灵敏度的方法，该方法在合适的范数中最小化到相应HDM灵敏度的距离。将所提出的简化优化框架应用于亚音速气动外形优化，与仅使用HDM解决的最优化问题相比，将HDM的查询次数减少了4-5倍，最优解的误差远小于0.1%。
英文摘要：
An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a Reduced-Order Basis (ROB) is also presented, along with a methodology for computing sensitivities of the reduced-order model that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%.
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