Computing and Visualization in Science

, Volume 13, Issue 6, pp 249–264 | Cite as

Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables

  • Harbir Antil
  • Matthias Heinkenschloss
  • Ronald H. W. Hoppe
  • Danny C. Sorensen
Regular Article

Abstract

We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization variables are related to spatially localized quantities. Our approach uses domain decomposition applied to the optimality system to isolate the subsystem that explicitly depends on the optimization variables from the remaining linear optimality subsystem. We apply balanced truncation model reduction to the linear optimality subsystem. The resulting coupled reduced optimality system can be interpreted as the optimality system of a reduced optimization problem. We derive estimates for the error between the solution of the original optimization problem and the solution of the reduced problem. The approach is demonstrated numerically on an optimal control problem and on a shape optimization problem.

Keywords

Optimal control Shape optimization Domain decomposition Model reduction 

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References

  1. 1.
    Akçelik V., Biros G., Ghattas O., Long K.R., van Bloemen Waanders B.: A variational finite element method for source inversion for convective-diffusive transport. Finite Elem. Anal. Des. 39(8), 683–705 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Antil, H., Heinkenschloss, M., Hoppe, R.H.W.: Domain Decomposition and Balanced Truncation Model Reduction for Shape Optimization of the Stokes System. Technical Report TR09–24, Department of Computational and Applied Mathematics, Rice University (2009). Optim. Meth. Softw. (in press)Google Scholar
  3. 3.
    Antoulas A.C.: Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, vol. 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005)Google Scholar
  4. 4.
    Benner, P., Mehrmann, V., Sorensen, D.C. (eds.): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Dedé L., Quarteroni A.: Optimal control and numerical adaptivity for advection-diffusion equations. ESAIM: Math. Model. Numer. Anal. 39, 1019–1040 (2005)MATHCrossRefGoogle Scholar
  6. 6.
    Dullerud G.E., Paganini F.: A Course in Robust Control Theory. Texts in Applied Mathematics, vol. 36. Springer, Berlin (2000)Google Scholar
  7. 7.
    Fatone L., Gervasio P., Quarteroni A.: Multimodels for incompressible flows. J. Math. Fluid Mech. 2(2), 126–150 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fatone L., Gervasio P., Quarteroni A.: Multimodels for incompressible flows: iterative solutions for the Navier–Stokes/Oseen coupling. M2AN Math. Model. Numer. Anal. 35(3), 549–574 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Formaggia L., Gerbeau J.F., Nobile F., Quarteroni A.: On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191(6–7), 561–582 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Glover K.: All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds. Int. J. Control 39(6), 1115–1193 (1984)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Griewank, A., Walther, A.: Evaluating Derivatives, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008). Principles and techniques of algorithmic differentiationGoogle Scholar
  12. 12.
    Gugercin S., Sorensen D.C., Antoulas A.C.: A modified low-rank Smith method for large-scale Lyapunov equations. Numer. Algorithms 32(1), 27–55 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Heinkenschloss, M., Reis, T., Antoulas, A.C.: Balanced Truncation Model Reduction for Systems with Inhomogeneous Initial Conditions. Technical Report TR09-29, Department of Computational and Applied Mathematics, Rice University (2009)Google Scholar
  14. 14.
    Hinze M., Volkwein S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control. In: Benner, P., Mehrmann, V., Sorensen, D.C. (eds) Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol. 45, pp. 261–306. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Kelley C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999)MATHGoogle Scholar
  16. 16.
    Lall S., Marsden J.E., Glavaški S.: A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 12(6), 519–535 (2002)MATHCrossRefGoogle Scholar
  17. 17.
    Lucia D.J., Beran P.S., Silva W.A.: Reduced-order modeling: new approaches for computational physics. Prog. Aerosp. Sci. 40(1–2), 51–117 (2004)CrossRefGoogle Scholar
  18. 18.
    Lucia D.J., King P.I., Beran P.S.: Domain decomposition for reduced-order modeling of a flow with moving shocks. AIAA J. 40, 2360–2362 (2002)CrossRefGoogle Scholar
  19. 19.
    Moore B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automat. Control 26(1), 17–32 (1981)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Quarteroni A., Tuveri M., Veneziani A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2(4), 163–197 (2000)MATHCrossRefGoogle Scholar
  21. 21.
    Rump, S.M.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer, Dordrecht (1999). http://www.ti3.tu-harburg.de/rump/
  22. 22.
    Smith B., Bjørstad P., Gropp W.: Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  23. 23.
    Sun, K.: Domain Decomposition and Model Reduction for Large-Scale Dynamical Systems. Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, Houston (2008)Google Scholar
  24. 24.
    Sun K., Glowinski R., Heinkenschloss M., Sorensen D.C.: Domain decomposition and model reduction of systems with local nonlinearities. In: Kunisch, K., Of, G., Steinbach, O. (eds) Numerical Mathematics and Advanced Applications. ENUMATH 2007., pp. 389–396. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  25. 25.
    Toselli A., Widlund O.: Domain Decomposition Methods—Algorithms and Theory. Computational Mathematics, vol. 34. Springer, Berlin (2004)Google Scholar
  26. 26.
    Zhou K., Doyle J.C., Glover K.: Robust and Optimal Control. Prentice Hall, Englewood Cliffs (1996)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Harbir Antil
    • 1
  • Matthias Heinkenschloss
    • 1
  • Ronald H. W. Hoppe
    • 2
    • 3
  • Danny C. Sorensen
    • 1
  1. 1.Department of Computational and Applied Mathematics, MS-134Rice UniversityHoustonUSA
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA
  3. 3.Institute of MathematicsUniversity of AugsburgAugsburgGermany

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