Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66, 811–841 (2014)
Babuska, I., Aziz, A. K.: Survey lectures on the mathematical foundations of the finite element method, pp 1–359. Academic Press, New York (1972)
Cahouet, J., Chabard, J. P.: Some fast 3d finite element solvers for the generalized Stokes problem. Int. J. Numer. Meth. Fl. 8, 869–895 (1988)
Elman, H. C., Golub, G. H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer-Verlag (1991)
Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006)
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
Ahrens, J., Geveci, B., Law, C.: Paraview. Elsevier (2005)
Lions, J. -L.: Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin (1971)
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
Pearson, J.W: On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems. ETNA 44, 53–72 (2015)
Wathen, A. J., Rees, T.: Chebyshev semi-iteration in preconditioning for problems including the mass matrix. ETNA 34(09), 125–135 (2008)
Olshanskii, M. A., Peters, J., Reusken, A.: Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations. Numer. Math. 105, 159–191 (2006)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)
Paige, C., Saunders, M.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)
Borzí, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations, volume 8 of Computational Science and Engineering. Philadelphia (2012)
Drǎgǎnescu, A., Soane, A. M.: Multigrid solution of a distributed optimal control problem constrained by the Stokes equations. Appl. Math. Comput. 219, 5622–5634 (2013)
Hinze, M., Pinnau, R., Ulbrich, M, Ulbrich, S.: Optimization with PDE Constraints. Springer-Verlag (2009)
Bramble, J. H., Pasciak, J. E., Vassilev, A. T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34, 1072–1092 (1997)
Elman, H. C., Silvester, D. J., Wathen, A. J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, New York (2005)
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)
Rees, T., Wathen, A. J.: Preconditioning iterative methods for the optimal control of the Stokes equations. SIAM J. Sci. Comput. 33, 2903–2926 (2011)
Heroux, M. A., Willenbring, J. M.: Trilinos users guide. Technical Report SAND2003-2952, Sandia National Laboratories (2003)
Mardal, K. -A., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98, 305–327 (2004)
Mardal, K. -A., Winther, R.: Erratum. Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 103, 171–172 (2006)
Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical computational efficiency of solvers for two-phase problems. Comput. Math Appl. 65, 301–314 (2013)
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, pp 197–216. Springer (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Axelsson, O., Farouq, S. & Neytcheva, M. Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems. Numer Algor 74, 19–37 (2017). https://doi.org/10.1007/s11075-016-0136-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0136-5