Volume 40, pp. 36-57, 2013.

Inexact and truncated Parareal-in-time Krylov subspace methods for parabolic optimal control problems

Xiuhong Du, Marcus Sarkis, Christian E. Schaerer, and Daniel B. Szyld

Abstract

We study the use of inexact and truncated Krylov subspace methods for the solution of the linear systems arising in the discretized solution of the optimal control of a parabolic partial differential equation. An all-at-once temporal discretization and a reduction approach are used to obtain a symmetric positive definite system for the control variables only, where a Conjugate Gradient (CG) method can be used at the cost of the solution of two very large linear systems in each iteration. We propose to use inexact Krylov subspace methods, in which the solution of the two large linear systems are not solved exactly, and their approximate solutions can be progressively less exact. The option we propose is the use of the parareal-in-time algorithm for approximating the solution of these two linear systems. The use of less parareal iterations makes it possible to reduce the time integration costs and to improve the time parallel scalability. We also show that truncated methods could be used without much delay in convergence but with important savings in storage. Spectral bounds are provided and numerical experiments with inexact versions of CG, the full orthogonalization method (FOM), and of truncated FOM are presented, illustrating the potential of the proposed methods.

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Key words

parabolic optimal control, reduced system, saddle point problem, inexact Krylov subspace methods, truncated Krylov subspace methods, parareal approximation, spectral bounds

AMS subject classifications

65F10, 65F50, 65N22, 35B37, 15A42, 35A15

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