453 | RWTH Publication No: 659956 2016   IGPM453.pdf |
TITLE | A certified trust region reduced basis approach to PDE-constrained optimization |
AUTHORS | Elizabeth Qian, Martin Grepl, Karen Veroy, Karen Willcox |
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-dimension 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%. |
KEYWORDS | model reduction, optimization, trust region methods, partial differential equations, reduced basis methods, error bounds, parametrized systems |
DOI | 10.1137/16M1081981 |
PUBLICATION | SIAM Journal of Scientific Computing 39(5), pp 434– 460, 2017 |