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

Elizabeth Qian, Martin Grepl, Karen Veroy, Karen Willcox

Research output: Contribution to journalArticleAcademicpeer-review


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%.
Original languageEnglish
Pages (from-to)434-460
Number of pages27
JournalSIAM Journal on Scientific Computing
Issue number5
Publication statusPublished - 26 Oct 2017
Externally publishedYes


  • model reduction
  • optimization
  • trust region methods
  • partial differential equations
  • reduced basis methods
  • error bounds
  • parametrized systems


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