On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems

Numerische Mathematik

Volumn 111, Issue 3, 2009, Pages 445-468

  Simon, René ^a   Zulehner, Walter ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

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[No Author keywords available]

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EID: 58149396680     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-008-0187-1     Document Type: Article

References (23)

1. Arian, E., Ta'asan, S.: Multigri one-shot methods for optimal control problems: Infinite dimensional control. ICASE-Report 94-52, NASA Langley Research Center, Hampton (1994)
2. I. Babuška 1973 The finite element method with Lagrangian multipliers Numer. Math. 20 179 192
3. Battermann A., Heinkenschloss M. et al.: Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems. In: Desch, W. (eds) Control and Estimation of Distributed Parameter Systems. Int. Ser. Numer. Math. 126., pp. 15-32. Birkhäuser, Basel (1998)
4. Battermann A., Sachs E.W. et al.: Block preconditioners for KKT systems in PDE-governed optimal control problems. In: Hoffmann, Karl-Heinz. (eds) Fast solution of discretized optimization problems. Int. Ser. Numer. Math. 138, pp. 1-18. Birkhäuser, Basel (2001)
5. G. Biros O. Ghattas 2005 Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. Part I: The Krylov-Schur solver SIAM J. Sci. Comput. 27 2 687 713
6. A. Borzì K. Kunisch D.Y. Kwak 2003 Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system SIAM J. Control Optim. 41 5 1477 1497
7. D. Braess R. Sarazin 1997 An efficient smoother for the Stokes problem Appl. Numer. Math. 23 1 3 19
8. S.C. Brenner 1996 Multigrid methods for parameter dependent problems RAIRO, Modelisation Math. Anal. Numer. 30 265 297
9. Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
10. F. Brezzi 1974 On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers R.A.I.R.O. 8 129 151
11. T. Dreyer B. Maar V. Schulz 2000 Multigrid optimization in applications J. Comput. Appl. Math. 120 1-2 67 84
12. W. Hackbusch 1980 Fast solution of elliptic control problems J. Optim. Theory Appl. 31 565 581
13. Hackbusch W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)
14. S.B. Hazra V. Schulz 2004 Simultaneous pseudo-timestepping for PDE-model based optimization problems BIT 44 3 457 472
15. Lions J.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
16. J. Schöberl W. Zulehner 2003 On Schwarz-type smoothers for saddle point problems Numer. Math. 95 377 399
17. J. Schöberl W. Zulehner 2007 Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems SIAM J. Matrix Anal. Appl. 29 3 752 773
18. Ta'asan, S.: One-shot methods for optimal control of distributed parameter systems I: The finite dimensional case. ICASE-Report 91-2, NASA Langley Research Center, Hampton (1991)
19. Tröltzsch F.: Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen. Vieweg, Wiesbaden (2005)
20. S. Vanka 1986 Block-implicit multigrid solution of Navier-Stokes equations in primitive variables J. Comput. Phys. 65 138 158
21. G. Wittum 1988 Multi-grid methods for Stokes and Navier-Stokes equations. Transforming smoothers: Algorithms and numerical results Numer. Math. 54 5 543 563
22. G. Wittum 1990 On the convergence of multi-grid methods with transforming smoothers Numer. Math. 57 1 15 38
23. W. Zulehner 2000 A class of smoothers for saddle point problems Computing 65 227 246