A rational deferred correction approach to parabolic optimal control problems

Abstract

The accurate and efficient solution of time-dependent partial differential equation (PDE)-constrained optimization problems is a challenging task, in large part due to the very high dimensions of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time-stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyse their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to those achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems as well as the state-of-the-art preconditioning strategies.