Optimization of transport-reaction processes using nonlinear model reduction

Abstract

This article focuses on optimization problems arising in the context of transport-reaction processes which are governed by nonlinear elliptic partial differential equations and proposes a computationally efficient method for their solution. The central idea of the method is to discretize the infinite-dimensional optimization problem by utilizing the method of weighted residuals with empirical eigenfunctions obtained by applying Karhunen–Loéve expansion to an appropriately constructed ensemble of solutions of the PDE equality constraints for different values of the design variables. This model reduction procedure leads to low-dimensional nonlinear programs that represent accurate approximations of the original infinite-dimensional nonlinear program, and whose solution can be obtained with standard optimization algorithms. The key issues of construction of the ensemble used for the computation of the empirical eigenfunctions and validity of the optimal solutions computed from the finite-dimensional programs are addressed. The proposed method is applied to two representative transport-reaction processes and is shown to be more efficient compared to conventional optimization approaches based on spatial discretization with the finite-difference method.