Accelerating PDE constrained optimization by the reduced basis method : application to batch chromatography / Yongjin Zhang, Lihong Feng, Suzhou Li and Peter Benner

Abstract

Abstract: In this work, we show that the reduced basis method accelerates a PDE constrained optimization problem, where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization. Such an optimization problem arises, for example, from batch chromatography. Instead of solving the full system of equations, a reduced model with a small number of equations is derived by the reduced basis method, such that only the small reduced system is solved at each step of the optimization process. An adaptive technique of selecting the snapshots is proposed, so that the complexity and runtime of generating the reduced basis are largely reduced. An output-oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically. An early-stop criterion is proposed to circumvent the stagnation of the error bound and make the construction of the reduced model more efficiently. Numerical examples show that the adaptive technique is very efficient in reducing the offline time. The optimization based on the reduced model is successful in terms of the accuracy and the runtime for getting the optimal solution.