Geodesic
Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems
Electronic transactions on numerical analysis, Tome 40 (2013), pp. 36-57.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 65F10, 65F50, 65N22, 35B37, 15A42, 35A15
Keywords: parabolic optimal control, reduced system, saddle point problem, inexact Krylov subspace methods, truncated Krylov subspace methods, parareal approximation, spectral bounds 
@article{ETNA_2013__40__a23,
     author = {Du, Xiuhong and Sarkis, Marcus and Schaerer, Christian E. and Szyld, Daniel B.},
     title = {Inexact and truncated parareal-in-time {Krylov} subspace methods for parabolic optimal control problems},
     journal = {Electronic transactions on numerical analysis},
     pages = {36--57},
     publisher = {mathdoc},
     volume = {40},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2013__40__a23/}
}
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Du, Xiuhong; Sarkis, Marcus; Schaerer, Christian E.; Szyld, Daniel B. Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems. Electronic transactions on numerical analysis, Tome 40 (2013), pp. 36-57. http://geodesic.mathdoc.fr/item/ETNA_2013__40__a23/