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Computing and Visualization in Science

, Volume 14, Issue 3, pp 91–103 | Cite as

A POD framework to determine robust controls in PDE optimization

  • A. BorzìEmail author
  • G. von Winckel
Article

Abstract

A strategy for the fast computation of robust controls of PDE models with random-field coefficients is presented. A robust control is defined as the control function that minimizes the expectation value of the objective over all coefficient configurations. A straightforward application of the adjoint method on this problem results in a very large optimality system. In contrast, a fast method is presented where the expectation value of the objective is minimized with respect to a reduced POD basis of the space of controls. Comparison of the POD scheme with the full optimization procedure in the case of elliptic control problems with random reaction terms and with random diffusivity demonstrates the superior computational performance of the POD method.

Keywords

PDE-constrained optimization Proper orthogonal decomposition Random fields 

Mathematics Subject Classification (2000)

15A18 49K20 60H35 93B11 

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References

  1. 1.
    Atwell J.A., Borggaard J.T., King B.B.: Reduced order controllers for Burgers’ equation with a nonlinear observer. Int. J. Appl. Math. Comput. 11, 1311–1330 (2001)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Babuška I., Nobile F., Tempone R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Num. Anal. 45(3), 1005–1034 (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Babuska I.M., Tempone R., Zouraris G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42, 800–825 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Borzì A.: Multigrid and sparse-grid schemes for elliptic control problems with random coefficients. Comput. Vis. Sci. 13, 153–160 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Borzì A., Schulz V.: Multigrid methods for PDE optimization. SIAM Rev. 51, 361–395 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Borzì A., von Winckel G.: Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comp. 31, 2172–2192 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Borzì A., Schulz V., Schillings C., von Winckel G.: On the treatment of distributed uncertainties in PDE-constrained optimization. GAMM Mitteilungen 33, 230–246 (2010)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bungartz H.-J., Griebel M.: Sparse grids. Acta Numer. 13, 147–269 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Demmel, J.W.: Applied Numerical Linear Algebra. SIAM (1997)Google Scholar
  10. 10.
    Ganapathysubramanian B., Zabaras N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652–685 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ghanem R.G., Spanos P.D.: Stochastic finite elements: a spectral approach. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  12. 12.
    Graham, I.G., Kuo, F.Y., Nuyens, D., Scheichl, R., Sloan, I.H.: Quasi-monte carlo methods for computing flow in random porous media. Bath Institute For Complex Systems. Preprint 04/10 (2010)Google Scholar
  13. 13.
    Haasdonk B., Ohlberger M.: Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math. Comp. Model. Dyn. Syst. 17, 145–161 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ito K., Kunisch K.: Lagrange Multiplier Approach to Variational Problems and Applications. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kerschen G., Golival J.-C., Vakakis A.F., Bergman L.A.: The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn. 41, 147–169 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Klimke A., Wohlmuth B.: Algorithm 847: Spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB. ACM Trans. Math. Softw. 31, 561–579 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kunisch K., Volkwein S.: Proper orthogonal decomposition for optimality systems. ESAIM Math. Model. Numer. Anal. 42, 1–23 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Lions J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)zbMATHGoogle Scholar
  19. 19.
    Loeve M.: Probability Theory, vols. I & II, 4th edn. Springer, New York (1978)Google Scholar
  20. 20.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Nguyen, N.C., Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation. In: Biegler, L. Biros, G., Ghattas, O., Heinkenschloss, M., Keyes, D., Mallick, B., Marzouk, Y. Tenorio, L., van Bloemen Waanders, B., Willcox, K. (eds.) Large Scale Inverse Problems and Quantification of Uncertainty, Chap. 8, pp. 151–173. Wiley, UK (Series in Computational Statistics) (2010)Google Scholar
  22. 22.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Øksendal B.: Optimal control of stochastic partial differential equations. Stoch. Anal. Appl. 23(1), 165–179 (2005)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Rosseel E., Boonen T., Vandewalle S.: Algebraic multigrid for stationary and time-dependent partial differential equations with stochastic coefficients. Numer. Linear Algebra Appl. 15, 141–163 (2008)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Vanmarcke E.: Random Fields: Analysis and Synthesis. MIT Press, USA (1983)zbMATHGoogle Scholar
  28. 28.
    Xiu D., Hesthaven J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comp. 27(3), 1118–1139 (2005)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany
  2. 2.Institut für Mathematik und Wissenschaftliches RechnenKarl-Franzens-Universität GrazGrazAustria

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