Advertisement

Non-conforming Localized Model Reduction with Online Enrichment: Towards Optimal Complexity in PDE Constrained Optimization

  • Mario Ohlberger
  • Felix Schindler
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 200)

Abstract

We propose a new non-conforming localized model reduction paradigm for efficient solution of large scale or multiscale PDE constrained optimization problems. The new conceptual approach goes beyond the classical offline/online splitting of traditional projection based model order reduction approaches for the underlying state equation, such as the reduced basis method. Instead of first constructing a surrogate model that has globally good approximation quality with respect to the whole parameter range, we propose an iterative enrichment procedure that refines and locally adapts the surrogate model specifically for the parameters that are depicted during the outer optimization loop.

Keywords

Model reduction Reduced basis method LRBMS Optimization Control Online enrichment Discontinuous Galerkin 

MSC (2010):

35Q93 65K10 65N30 

References

  1. 1.
    Albrecht, F., Haasdonk, B., Kaulmann, S., Ohlberger, M.: The localized reduced basis multiscale method. In: Proceedings of Algoritmy 2012. Conference on Scientific Computing, Vysoke Tatry, Podbanske, 9–14 Sept. 2012, pp. 393–403. Slovak University of Technology in Bratislava, Publishing House of STU (2012)Google Scholar
  2. 2.
    Benedix, O., Vexler, B.: A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. Appl. 44(1), 3–25 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buhr, A., Engwer, C., Ohlberger, M., Rave, S.: ArbiLoMod, a simulation technique designed for arbitrary local modifications (2015). http://arxiv.org/abs/1512.07840
  4. 4.
    Dedè, L.: Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50(2), 287–305 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34, A937–A969 (2012). doi: 10.1137/10081157X MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29(2), 235–256 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grepl, M.A., Kärcher, M.: Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Math. Acad. Sci. Paris 349(15–16), 873–877 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Haasdonk, B.: Reduced basis methods for parametrized PDEs–A tutorial introduction for stationary and instationary problems. In: Model Reduction and Approximation: Theory and Algorithms. Benner, P., Cohen, A., Ohlberger, M., and Willcox, K. (eds.). SIAM, Philadelphia, PA (2017)Google Scholar
  9. 9.
    Hintermüller, M., Hinze, M., Hoppe, R.H.W.: Weak-duality based adaptive finite element methods for PDE-constrained optimization with pointwise gradient state-constraints. J. Comput. Math. 30(2), 101–123 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kaulmann, S., Ohlberger, M., Haasdonk, B.: A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems. C. R. Math. Acad. Sci. Paris 349(23–24), 1233–1238 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, W., Yan, N.: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15(1–4), 285–309 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ohlberger, M., Rave, S., Schindler, F.: Model reduction for multiscale lithium-ion battery simulation. In: Karasözen, B., et al. (ed.) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol. 112, pp. 317–331. Springer (2016)Google Scholar
  13. 13.
    Ohlberger, M., Schaefer, M.: A reduced basis method for parameter optimization of multiscale problems. In: Proceedings of Algoritmy 2012, Conference on Scientific Computing, Vysoke Tatry, Podbanske, 9-14 Sept. pp. 272–281 (2012)Google Scholar
  14. 14.
    Ohlberger, M., Schaefer, M.: Error control based model reduction for parameter optimization of elliptic homogenization problems. In: Le Gorrec, Y. (ed.) 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, CPDE 2013; Paris; France; 25 September 2013 through 27 Sept. 2013; Code 103235, vol. 1, pp. 251–256. International Federation of Automatic Control (IFAC) (2013)Google Scholar
  15. 15.
    Ohlberger, M., Schindler, F.: A-posteriori error estimates for the localized reduced basis multi-scale method. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, Springer Proceedings in Mathematics & Statistics, vol. 77, pp. 421–429. Springer International Publishing (2014)Google Scholar
  16. 16.
    Ohlberger, M., Schindler, F.: Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6), A2865–A2895 (2015)CrossRefzbMATHGoogle Scholar
  17. 17.
    Oliveira, I.B., Patera, A.T.: Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optim. Eng. 8(1), 43–65 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rösch, A., Wachsmuth, D.: A-posteriori error estimates for optimal control problems with state and control constraints. Numer. Math. 120(4), 733–762 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tröltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44(1), 83–115 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13(1–3), 231–252 (1999). (Computational optimization—a tribute to Olvi Mangasarian, Part II)Google Scholar
  21. 21.
    Vossen, G., Volkwein, S.: Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems. Numer. Algebr. Control Optim. 2(3), 465–485 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Applied Mathematics, Center for Nonlinear Sciences and Center for Multiscale Theory and Computation, University of MuensterMünsterGermany
  2. 2.Applied Mathematics and Center for Nonlinear Sciences, University of MuensterMünsterGermany

Personalised recommendations