Saddle point matrices of a special structure arise in optimal control problems. In this paper we consider distributed optimal control for various types of scalar stationary partial differential equations and compare the efficiency of several numerical solution methods. We test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain assumptions the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and elapsed time is favourably compared with other published methods.
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Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical and computational efficiency of solvers for two-phase problems. Comput. Math. Appl. 65, 301–314 (2013)
Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)
Axelsson, O., Neytcheva, M., Bashir Ahmad, A: Comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Alg. 66, 811–841 (2014)
Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)
Bangerth, W., Hartmann, R., Kanschat, G.: deal.II-a general-purpose object-oriented finite element library. ACM T. Math. Software 33 (2007). doi:10.1145/1268776.1268779. Art. 24
Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106, 349–367 (2007)
Boyanova, P., Do-Quang, M., Neytcheva, M.: Efficient preconditioners for large scale binary Cahn-Hilliard models. Comput. Methods Appl. Math. 12, 1–22 (2012)
Choi, Y.: Simultaneous analysis and design in PDE-constrained optimization. Doctor of Philosophy Thesis, Stanford University (2012)
Greenbaum, A., Ptak, V., Strakos, Z.: Any convergence curve is possible for GMRES. SIAM Matrix Anal. Appl. 17, 465–470 (1996)
Heinkenschloss, M., Leykekhman, D.: Local error estimates for SUPG solutions of advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 47, 4607–4638 (2010)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constrains. Springer, Berlin Heidelberg New York (2009)
Kollmann, M.: Efficient iterative solvers for saddle point systems arising in PDE-constrained optimization problems with inequality constraints. Doctor of Philosophy Thesis, Johannes Kepler University Linz (2013)
Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Alg. Appl. 19, 816–829 (2012)
Pearson, J.W., Wathen, A.J.: Fast iterative solvers for convection-diffusion control problems. ETNA 40, 294–310 (2013)
Rees, T.: Preconditioning iterative methods for PDE constrained optimization. Doctor of Philosophy Thesis, University of Oxford (2010)
The Trilinos Project http://trilinos.sandia.gov/
Tröltzsch, F.: Optimal control of partial differential equations: theory, methods and applications, AMS, Graduate Studies in Mathematics (2010)
Zulehner, W.: Nonstandard norms and robust estimates for saddle-point problems. SIAM J. Matrix Anal. Appl. 32, 536–560 (2011)
Zulehner, W.: Efficient solvers for saddle point problems with applications to PDE-constrained optimization Advanced Finite Element Methods and Applications, Lecture Notes in Applied and Computational Mechanics, vol. 66, pp 197–216 (2013)
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Axelsson, O., Farouq, S. & Neytcheva, M. Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Numer Algor 73, 631–663 (2016). https://doi.org/10.1007/s11075-016-0111-1
- PDE-constrained optimization problems
- Finite elements
- Iterative solution methods