Accelerating Optimization of Parametric Linear Systems by Model Order Reduction

Abstract

Design optimization problems are often formulated as an optimization problem whose objective is a function of the output of a large-scale parametric linear system, obtained from the discretization of a PDE. To reduce the high computational cost of the objective and its gradient, model order reduction techniques can be used. This paper uses interpolatory reduced models as surrogate models in an optimization procedure. We replace the standard first-order condition by the relaxed first-order condition, which is more suitable when algebraic reduced models are used as surrogate models. The relaxed first-order condition imposes that the approximation quality of the surrogate model at the interpolation point can be measured and refined and that the surrogate model is equipped with an error bound on the entire parameter space. We propose two optimization algorithms: one uses the error bound to define a trust region and the other penalizes the objective with the error bound. We prove convergence of both methods under mild conditions in a unified framework. These methods are efficient when surrogate models are cheap to build and evaluate, since they only need these operations to achieve convergence. Numerical experiments from civil engineering show good performance of the proposed methods.