iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://unpaywall.org/10.1007/S00211-008-0187-1
On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems | Numerische Mathematik Skip to main content
Springer Nature Link
Log in

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

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper we consider a (one-shot) multigrid strategy for solving the discretized optimality system (KKT system) of a PDE-constrained optimization problem. In particular, we discuss the construction of an additive Schwarz-type smoother for a certain class of optimal control problems. A rigorous multigrid convergence analysis is presented. Numerical experiments are shown which confirm the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. Babuška I.: The finite element method with Lagrangian multipliers. Numer. Math. 20, 179–192 (1973)

    Article  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  5. Biros G., Ghattas O.: Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: The Krylov–Schur solver. SIAM J. Sci. Comput. 27(2), 687–713 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borzì A., Kunisch K., Kwak D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Optim. 41(5), 1477–1497 (2003)

    Article  MATH  Google Scholar 

  7. Braess D., Sarazin R.: An efficient smoother for the Stokes problem. Appl. Numer. Math. 23(1), 3–19 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brenner S.C.: Multigrid methods for parameter dependent problems. RAIRO, Modelisation Math. Anal. Numer. 30, 265–297 (1996)

    MATH  MathSciNet  Google Scholar 

  9. Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    MATH  Google Scholar 

  10. Brezzi F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. R.A.I.R.O. 8, 129–151 (1974)

    MathSciNet  Google Scholar 

  11. Dreyer T., Maar B., Schulz V.: Multigrid optimization in applications. J. Comput. Appl. Math. 120(1–2), 67–84 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hackbusch W.: Fast solution of elliptic control problems. J. Optim. Theory Appl. 31, 565–581 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hackbusch W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)

    MATH  Google Scholar 

  14. Hazra S.B., Schulz V.: Simultaneous pseudo-timestepping for PDE-model based optimization problems. BIT 44(3), 457–472 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lions J.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  16. Schöberl J., Zulehner W.: On Schwarz-type smoothers for saddle point problems. Numer. Math. 95, 377–399 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schöberl J., Zulehner W.: Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29(3), 752–773 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  20. Vanka S.: Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. J. Comput. Phys. 65, 138–158 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wittum G.: Multi-grid methods for Stokes and Navier–Stokes equations. Transforming smoothers: Algorithms and numerical results. Numer. Math. 54(5), 543–563 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wittum G.: On the convergence of multi-grid methods with transforming smoothers. Numer. Math. 57(1), 15–38 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zulehner W.: A class of smoothers for saddle point problems. Computing 65, 227–246 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SFB 013, Johannes Kepler University Linz, Altenbergerstraße 69, 4040, Linz, Austria

    René Simon

  2. Institute of Computational Mathematics, Johannes Kepler University, Altenbergerstraße 69, 4040, Linz, Austria

    Walter Zulehner

Authors
  1. René Simon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Zulehner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to René Simon.

Additional information

The work was supported by the Austrian Science Fund (FWF) under grant SFB 013/F1309.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Simon, R., Zulehner, W. On Schwarz-type smoothers for saddle point problems with applications to PDE-constrained optimization problems. Numer. Math. 111, 445–468 (2009). https://doi.org/10.1007/s00211-008-0187-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-008-0187-1

Mathematics Subject Classification (2000)

Search

Navigation