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

Author(s)
Grepl, Martin; Veroy, Karen; Qian, Elizabeth Y.; Willcox, Karen E
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Abstract
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%. Key words: model reduction, optimization, trust region methods, partial differential equations, reduced basis methods, error bounds, parametrized systems
Date issued
2017-10
URI
http://hdl.handle.net/1721.1/116912
Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Journal
SIAM Journal on Scientific Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Qian, Elizabeth, et al. “A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization.” SIAM Journal on Scientific Computing, vol. 39, no. 5, Jan. 2017, pp. S434–60.
Version: Final published version
ISSN
1064-8275
1095-7197

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