Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems
The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and 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 the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, 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 execution time is favorably compared with other published methods.
KeywordsPDE-constrained optimization problems Finite elements Iterative solution methods Preconditioning
Unable to display preview. Download preview PDF.
- 5.Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer-Verlag (1991)Google Scholar
- 7.Bangerth, W., Hartmann, R., Kanschat, G.: deal.ii-a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (2007). doi: 10.1145/1268776.1268779
- 8.Ahrens, J., Geveci, B., Law, C.: Paraview. Elsevier (2005)Google Scholar
- 10.Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Poisson and convection-diffusion control. Numerical Algorithms. In press. doi: 10.1007/s11075-016-0111-1
- 16.Borzí, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations, volume 8 of Computational Science and Engineering. Philadelphia (2012)Google Scholar
- 18.Hinze, M., Pinnau, R., Ulbrich, M, Ulbrich, S.: Optimization with PDE Constraints. Springer-Verlag (2009)Google Scholar
- 23.Heroux, M. A., Willenbring, J. M.: Trilinos users guide. Technical Report SAND2003-2952, Sandia National Laboratories (2003)Google Scholar
- 28.Zulehner, W.: Efficient solvers for saddle point problems with applications to PDE-constrained optimization, pp 197–216. Springer (2013)Google Scholar