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A rational deferred correction approach to parabolic optimal control problems

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https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drx046/4372128
Original languageEnglish
Pages (from-to)1861-1892
Number of pages26
JournalIMA Journal of Numerical Analysis
Volume38
Issue number4
Early online date7 Oct 2017
DOIs
Publication statusPublished - 16 Oct 2018

Abstract

The accurate and ecient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension 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 analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that 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 state-of-the-art preconditioning strategies.

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