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Smoothing Analysis of an All-at-Once Multigrid Approach for Optimal Control Problems Using Symbolic Computation

  • Stefan TakacsEmail author
  • Veronika Pillwein
Chapter
Part of the Texts & Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

The numerical treatment of systems of partial differential equations (PDEs) is of great interest as many problems from applications, including the optimality system of optimal control problems that is discussed here, belong to this class. These problems are not elliptic and therefore both the construction of an efficient numerical solver and its analysis are hard. In this work we will use all-at-once multigrid methods as solvers. For sake of simplicity, we will only analyze the smoothing properties of a well-known smoother. Local Fourier analysis (or local mode analysis) is a widely-used tool to analyze numerical methods for solving discretized systems of PDEs which has also been used in particular to analyze multigrid methods. The rates that can be computed with local Fourier analysis are typically the supremum of some rational function. In several publications this supremum was merely approximated numerically by interpolation. We show that it can be resolved exactly using cylindrical algebraic decomposition which is a well established method in symbolic computation.

Keywords

Multigrid Approach Local Fourier Analysis Cylindrical Algebraic Decomposition (CAD) Multigrid Method Rate Smoothing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank the anonymous referees for their remarks that have significantly improved the quality of this paper. This work is supported by the Austrian Science Fund (FWF) under grants W1214/DK12 and DK6.

References

  1. 1.
    Borzi, 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)Google Scholar
  2. 2.
    Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp. 31, 333–390 (1977)Google Scholar
  3. 3.
    Brandt, A.: Rigorous quantitative analysis of multigrid, I: Constant coefficients two-level cycle with L 2-norm. SIAM J. Numer. Anal. 31(6), 1695–1730 (1994)Google Scholar
  4. 4.
    Brown, C.W.: QEPCAD B – a program for computing with semi-algebraic sets. Sigsam Bulletin 37(4), 97–108 (2003)Google Scholar
  5. 5.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Automata theory and formal languages (Second GI Conference, Kaiserslautern, 1975), Lecture Notes in Computer Science, Vol. 33, pp. 134–183. Springer, Berlin (1975)Google Scholar
  6. 6.
    Hackbusch, W.: Multi-Grid Methods and Applications. Springer, Berlin (1985)Google Scholar
  7. 7.
    Holmes, R.B.: A formula for the spectral radius of an operator. Am. Math. Mon. 75(2), 163–166 (1968)Google Scholar
  8. 8.
    Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. J. Symbolic Comput. 24(2), 161–187 (1997); Applications of quantifier elimination (Albuquerque, NM, 1995).Google Scholar
  9. 9.
    Lass, O., Vallejos, M., Borzi, A., Douglas, C.C.: Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems. Computing 84(1–2), 27–48 (2009)Google Scholar
  10. 10.
    Schöberl, J., Simon, R., Zulehner, W.: A Robust Multigrid Method for Elliptic Optimal Control Problems. SIAM J. Numer. Anal. 49, 1482 (2011)Google Scholar
  11. 11.
    Seidl, A., Sturm, T.: A generic projection operator for partial cylindrical algebraic decomposition. In Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pp 240–247 (electronic), ACM Press, New York (2003)Google Scholar
  12. 12.
    Strzeboński, A.: Solving systems of strict polynomial inequalities. J. Symbolic Comput. 29(3), 471–480 (2000)Google Scholar
  13. 13.
    Tarski, A.: A decision method for elementary algebra and geometry. 2nd ed. University of California Press, Berkeley and Los Angeles, California (1951)Google Scholar
  14. 14.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, London (2001)Google Scholar
  15. 15.
    Vanka, S.P.: Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys. 65, 138–158 (1986)Google Scholar
  16. 16.
    Wienands, R., and Joppich, W. Practical Fourier analysis for multigrid methods. Chapman & Hall, CRC (2005)Google Scholar

Copyright information

© Springer-Verlag/Wien 2012

Authors and Affiliations

  1. 1.Doctoral Program Computational MathematicsJohannes Kepler UniversityLinzAustria
  2. 2.Research Institute for Symbolic ComputationJohannes Kepler UniversityLinzAustria

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