A CERTIFIED TRUST REGION REDUCED BASIS APPROACH TO PDE-CONSTRAINED OPTIMIZATION

作者：Qian Elizabeth^*; Grepl Martin; Veroy Karen; Willcox Karen

来源：SIAM Journal on Scientific Computing, 2017, 39(5): S434-S460.

DOI：10.1137/16M1081981

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

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%.

• 出版日期2017
• 单位MIT