Abstract
In optimal control formulations of partial differential equations the aim is to find a control function that steers the solution to a desired form. A Lagrange multiplier, i.e. an adjoint variable is introduced to handle the PDE constraint. One can reduce the problem to a two-by-two block matrix form with square blocks for which a very efficient preconditioner, PRESB can be applied. This method gives sharp and tight eigenvalue bounds, which hold uniformly with respect to regularization, mesh size and problem parameters, and enable use of the second order inner product free Chebyshev iteration method, which latter enables implementation on parallel computers without any need to use global data communications. Furthermore this method is insensitive to round-off errors. It outperforms other earlier published methods. Implementational and spectral properties of the method, and a short survey of applications, are given.
The work is supported by The Ministry of Education, Youth and Sports of the Czech Republic from the National Programme of Sustainability (NPU II), project “IT4Innovations excellence in science - LQ1602”.
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References
Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems: stokes control. Numer. Algorithms 74, 19–37 (2017)
Axelsson, O., Lukáš, D.: Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems. J. Numer. Math. 27(1), 1–21 (2019). https://doi.org/10.1515/jnma-2017-0064
Liang, Z.-Z., Axelsson, O., Neytcheva, M.: A robust structured preconditioner for time-harmonic parabolic optimal control problems. Numer. Algorithms 79, 575–596 (2018)
Axelsson, O., Liang, Z.-Z.: A note on preconditioning methods for time-periodic eddy current optimal control problems. J. Comput. Appl. Math. 352, 262–277 (2019)
Axelsson, O., Neytcheva, M., Ström, A.: An efficient preconditioning method for state box-constrained optimal control problems. J. Numer. Math. 26(4), 185–207 (2018)
Axelsson, O., Blaheta, R., Kohut, R.: Preconditioning methods for high-order strongly stable time integration methods with an application for a DAE problem. Numer. Linear Algebra Appl. 22(6), 930–949 (2015)
Axelsson, O., Liang, Z.-Z.: An optimal control framework for the iterative solution of a missing boundary data problem. Work in Progress
Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 19(5), 816–829 (2012)
Axelsson, O., Salkuyeh, D.K.: A new version of a preconditioning method for certain two-by-two block matrices with square blocks. BIT Numer. Math. 59(2), 321–342 (2019)
Bai, Z.Z., Benzi, M., Chen, F., Wang, Z.Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J. Numer. Anal. 33(1), 343–369 (2012)
Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods, II. SIAM J. Numer. Anal. 27, 1569–1590 (1990)
Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)
Varga, R.S.: Matrix Iterative Analysis. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-642-05156-2. Originally published by Prentice-Hall 1962 edition
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Axelsson, O. (2020). An Introduction and Summary of Use of Optimal Control Methods for PDE’s. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_31
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