Accelerating PDE constrained optimization by the reducedbasis method: application to batch chromatography

Abstract

In this work, we show that the reduced basis method accelerates a partial differential equation 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. To reduce the computational burden of repeatedly solving the large-scale system under parameter variations, a parametric reduced-order model with a small number of equations is derived by using the reduced basis method. As a result, the small reduced-order model, rather than the full system, is solved at each step of the optimization process. An adaptive technique for selecting the snapshots is proposed, so that the complexity and runtime for 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 and to make the construction of the reduced model more efficient. 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 acquiring the optimal solution. Copyright © 2015 John Wiley & Sons, Ltd.