Abstract
Optimal control problems constrained by partial differential equations arise in a multitude of important applications. They lead mostly to the solution of very large scale algebraic systems to be solved, which must be done by iterative methods. The problems should then be formulated so that they can be solved fast and robust, which requires the construction of an efficient preconditioner. After reduction of a variable, a two-by-two block matrix system with square blocks arises for which such a preconditioner, PRESB is presented, involving the solution of two algebraic systems which are a linear combination of the matrix blocks. These systems can be solved by inner iterations, involving some available classical solvers to some relative, not very demanding tolerance.
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References
Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)
Axelsson, O., Liang, Z.-Z., Kruzik, J., Horak, D.: Inner product free iterative solution and elimination methods for linear systems of a three-by-three block matrix form. J. Comput. Appl. Math. 383, 113–117 (2021)
Axelsson, O., Blaheta R., Béreš, M.: A boundary optimal control identification problem. Uppsala University, Department of Technology, TR 2020-002, May 2020.
Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Poisson and convection-diffusion control. Numer. Alg. 73, 631–663 (2016)
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., Lukáš, D.: Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems. J. Numer. Math. 27, 1–21 (2019)
Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numerical Algorithms 66, 811–841 (2014)
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, 321–342 (2019)
Axelsson, O., Vassilevski, P.S.: Algebraic multilevel preconditioning methods. I. Numerische Mathematik 56, 157–177 (1989)
Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (2006)
Liang, Z.-Z., Axelsson, O., Neytcheva, M.: A robust structured preconditioner for time-harmonic parabolic optimal control problems. Numer. Algor. 79, 575–596 (2018)
Martin, T.J., Dulikravich, G.S.: Inverse determination of temperatures and heat fluxes on inaccessible surfaces. WIT Transactions on Modelling and Simulation 8, (1994)
Notay, Y.: AGMG software and documentation. See http://homepages.ulb.ac.be/textasciitildeynotay/ (2015)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)
Vassilevski, P.S.: Multilevel Block Factorization Preconditioners. Springer-Verlag, New York (2008)
Acknowledgements
The research of O. Axelsson was supported by the Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4 Innovations excellence in science LQ1602”
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Axelsson, O. (2021). Applications of the PRESB Preconditioning Method for OPT-PDE Problems. In: Vermolen, F.J., Vuik, C. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2019. Lecture Notes in Computational Science and Engineering, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-030-55874-1_6
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DOI: https://doi.org/10.1007/978-3-030-55874-1_6
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