Computing and Visualization in Science

, Volume 13, Issue 4, pp 153–160 | Cite as

Multigrid and sparse-grid schemes for elliptic control problems with random coefficients

Regular Article

Abstract

A multigrid and sparse-grid computational approach to solving nonlinear elliptic optimal control problems with random coefficients is presented. The proposed scheme combines multigrid methods with sparse-grids collocation techniques. Within this framework the influence of randomness of problem’s coefficients on the control provided by the optimal control theory is investigated. Numerical results of computation of stochastic optimal control solutions and formulation of mean control functions are presented.

Keywords

Multigrid method Sparse grids Reaction-diffusion problems Random fields Optimal control theory 

Mathematics Subject Classification (2000)

35J55 60H25 49K20 65N55 65C20 

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References

  1. 1.
    Babuška I., Nobile F., Tempone R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bebernes J., Eberly D.: Mathematical Problems from Combustion Theory. Springer, New York (1988)Google Scholar
  3. 3.
    Borzì A.: High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems. J. Comput. Appl. Math. 200, 67–85 (2007)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Borzì A.: Smoothers for control- and state-constrained optimal control problems. Comput. Vis. Sci. 11, 59–66 (2008)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Borzì A., Borzì G.: An efficient algebraic multigrid method for solving optimality systems. Comput. Vis. Sci. 7(3/4), 183–188 (2004)MATHMathSciNetGoogle Scholar
  6. 6.
    Borzì A., Kunisch K.: The numerical solution of the steady state solid fuel ignition model and its optimal control. SIAM J. Sci. Comput. 22(1), 263–284 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Borzì A., Schulz V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361–395 (2009)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Borzì A., von Winckel G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comput. 31, 2172–2192 (2009)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Brandt A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31, 333–390 (1977)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bungartz H.-J., Griebel M.: Sparse grids. Acta Numerica 13, 147–269 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Elman H., Furnival D.: Solving the stochastic steady-state diffusion problem using multigrid. IMA J. Numer. Anal. 27, 675–688 (2007)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ganapathysubramanian B., Zabaras N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gasca M., Sauer T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12, 377–410 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ghanem R.G., Spanos P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)MATHGoogle Scholar
  15. 15.
    Hackbusch W.: Multi-grid Methods and Applications. Springer, New York (1985)MATHGoogle Scholar
  16. 16.
    Huang S.P., Mahadevan S., Rebba R.: Collocation-based stochastic finite element analysis for random field problems. Probabilistic Eng. Mech. 22, 194–205 (2007)CrossRefGoogle Scholar
  17. 17.
    Ito K., Kunisch K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)MATHGoogle Scholar
  18. 18.
    Klimke A., Wohlmuth B.: Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Trans. Math. Softw. 31, 561–579 (2005)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)MATHGoogle Scholar
  20. 20.
    Lions J.-L.: Control of Distributed Singular Systems. Gauthier-Villars, Paris (1985)Google Scholar
  21. 21.
    Loeve M.: Probability Theory. Vols. I & II, IV edn. Springer, New York (1978)Google Scholar
  22. 22.
    Marzouk Y.M., Najm H.N., Rahn L.A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224, 560–586 (2007)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Nobile F., Tempone R., Webster C.: A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 46, 2309–2345 (2008)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Putko M.M., Newman P.A., Taylor A.C. III, Green L.L.: Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives. J. Fluids Eng. 124, 60–69 (2002)CrossRefGoogle Scholar
  25. 25.
    Schulz V., Schillings C.: On the nature and treatment of uncertainties in aerodynamic design. AIAA J. 47, 646–654 (2009)CrossRefGoogle Scholar
  26. 26.
    Schwab Ch, Todor R.A.: Sparse finite elements for stochastic elliptic problems higher order moments. Computing 71, 43–63 (2003)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Seynaeve B., Rosseel E., Nicolaï B., Vandewalle S.: Fourier mode analysis of multigrid methods for partial differential equations with random coefficients. J. Comput. Phys. 224, 132–149 (2007)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Trottenberg U., Oosterlee C., Schüller A.: Multigrid. Academic Press, London (2001)MATHGoogle Scholar
  29. 29.
    Xiu D., Hesthaven J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Xiu D., Karniadakis G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Zabaras N., Ganapathysubramanian B.: A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach. J. Comput. Phys. 227, 4697–4735 (2008)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Università degli Studi del Sannio, Dipartimento e Facoltà di IngegneriaPalazzo Dell’Aquila Bosco LucarelliBeneventoItaly
  2. 2.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-Universität GrazGrazAustria

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