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

• Published in: International journal for numerical methods in engineering Vol. 102; no. 5; pp. 1111 - 1135
• Main Authors: Zahr, Matthew J., Farhat, Charbel
• Format: Journal Article
• Language: English
• Published: Bognor Regis Blackwell Publishing Ltd 04.05.2015; Wiley Subscription Services, Inc
• Subjects: Constraining; Errors; Mathematical models; model order reduction; Nonlinearity; Numerical analysis; Optimization; Partial differential equations; PDE-constrained optimization; proper orthogonal decomposition (POD); reduced-order model (ROM); Sampling; Shape optimization; singular value decomposition (SVD); Trajectories
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.4770

SummaryAn adaptive approach to using reduced‐order models (ROMs) as surrogates in partial differential equations (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 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 is also presented, along with a methodology for computing sensitivities of the ROM 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 with the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%. Copyright © 2014 John Wiley & Sons, Ltd.