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%. Key words: model reduction, optimization, trust region methods, partial differential equations, reduced basis methods, error bounds, parametrized systemsFulbright U.S. Student ProgramNational Science Foundation (U.S.). Graduate Research Fellowship ProgramHertz FoundationUnited States. Department of Energy. Office of Advanced Scientific Computing Research (Award DEFG02-08ER2585)United States. Department of Energy. Office of Advanced Scientific Computing Research (Award DE-SC0009297