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Published January 2017 | Published
Journal Article Open

A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization


Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%.

Additional Information

© 2017 Elizabeth Qian, Martin Grepl, Karen Veroy & Karen Willcox. Received by the editors July 1, 2016; accepted for publication (in revised form) February 24, 2017; published electronically October 26, 2017. The work of the first author was supported by the U.S. Fulbright Student Program, the National Science Foundation Graduate Research Fellowship, and the Fannie and John Hertz Foundation. The MIT authors also acknowledge the support of the U.S. Department of Energy, Office of Advanced Scientific Computing Research (ASCR), Applied Mathematics Program, awards DE-FG02-08ER2585 and DE-SC0009297, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center. The work of the second and third authors was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through grant GSC 111.

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