Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems
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.
KeywordsPDE-constrained optimization problems Finite elements Iterative solution methods Preconditioning
Unable to display preview. Download preview PDF.
- 5.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
- 8.Choi, Y.: Simultaneous analysis and design in PDE-constrained optimization. Doctor of Philosophy Thesis, Stanford University (2012)Google Scholar
- 12.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)Google Scholar
- 16.Rees, T.: Preconditioning iterative methods for PDE constrained optimization. Doctor of Philosophy Thesis, University of Oxford (2010)Google Scholar
- 17.The Trilinos Project http://trilinos.sandia.gov/
- 18.Tröltzsch, F.: Optimal control of partial differential equations: theory, methods and applications, AMS, Graduate Studies in Mathematics (2010)Google Scholar
- 20.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)Google Scholar