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An adaptive approach to using reduced-order models as surrogates inPDE-constrained optimization is introduced that breaks the traditionaloffline-online framework of model order reduction. A sequence of optimizationproblems constrained by a given Reduced-Order Model (ROM) is defined with thegoal of converging to the solution of a given PDE-constrained optimizationproblem. For each reduced optimization problem, the constraining ROM is trainedfrom sampling the High-Dimensional Model (HDM) at the solution of some of theprevious problems in the sequence. The reduced optimization problems areequipped with a nonlinear trust-region based on a residual error indicator tokeep the optimization trajectory in a region of the parameter space where theROM is accurate. A technique for incorporating sensitivities into aReduced-Order Basis (ROB) is also presented, along with a methodology forcomputing sensitivities of the reduced-order model that minimizes the distanceto the corresponding HDM sensitivity, in a suitable norm. The proposed reducedoptimization framework is applied to subsonic aerodynamic shape optimizationand 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 inthe optimal solution far less than 0.1%.

Progressive construction of a parametric reduced-order model for PDE-constrained optimization