Abstract
We present a method for solving optimal control problems constrained by a partial differential equation, where we simultaneously impose sparsity-promoting L 1-regularization on the control as well as box constraints on both the control and the state. We focus on numerical implementation aspects and on preconditioners used when solving the arising linear systems.
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References
Dravins I., Neytcheva M.: PDE-Constrained Optimization: Matrix Structures and Preconditioners. Lirkov I., Margenov S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Comput. Sci. 11958, 315–323 (2020)
Stadler G.: Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44, 159–181 (2009)
Herzog R., Sachs E. W.: Preconditioned conjugate gradient method for optimal control problems with control and state constraints. SIAM J. Matrix. Anal. Appl. 31, 2291–2317 (2010)
Pearson J.W., Stoll M., Wathen A.J.: Preconditioners for state–constrained optimal control problems with Moreau–Yosida penalty function. Numer. Lin. Alg. Appl. 21, 81–97 (2014)
Hintermüller M., Hinze M.: Moreau-Yosida regularization in state constrained elliptic control problems: error estimates and parameter adjustment. SIAM J. Numer. Anal. 47, 1666–1683 (2009)
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)
Ito K., Kunisch K.: Semi-smooth Newton methods for state-constrained optimal control problems. Systems & Control Letters 5, 221–228 (2003)
Axelsson O., Kaporin I.: On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces. SIAM J. Sci. Comput. 16(1), 228–249 (1995)
Axelsson O., On mesh independence and Newton-type methods. Appl. Math. 39, 249–265 (1993)
Bezanson J., Edelman A., Karpinski S., Shah V.: Julia: A Fresh Approach to Numerical Computing. SIAM Review 50, 65–98 (2017)
Alnaes M. S., Blechta J., Hake J., Johansson A., Kehlet B., Logg A., Richardson C., Ring J., Rognes M. E., Wells G. N.: The FEniCS Project Version 1.5. Arch. Num. Soft. (2015)
Pearson J.W., Porcelli M., Stoll M.: Interior Point Methods and Preconditioning for PDE-Constrained Optimization Problems Involving Sparsity Terms. arXiv: 1806.05896 (2019)
Acknowledgements
This work was supported by VR Grant 2017-03749 Mathematics and numerics in PDE-constrained optimization problems with state and control constraints, 2018–2022.
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Dravins, I., Neytcheva, M. (2021). PDE-Constrained Optimization: Optimal control with L 1-Regularization, State and Control Box Constraints. 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_31
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DOI: https://doi.org/10.1007/978-3-030-55874-1_31
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