Progressive Construction of a Parametric Reduced-order Model for PDE-constrained Optimization

Matthew Zahr, Stanford University

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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) also is 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 four to five, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1 percent.

Abstract Author(s): Matthew J. Zahr, Charbel Farhat