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Reduced-Order Modeling and ROM-Based Optimization of Batch Chromatography

  • Peter Benner
  • Lihong Feng
  • Suzhou Li
  • Yongjin ZhangEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

A reduced basis method is applied to batch chromatography and the underlying optimization problem is solved efficiently based on the resulting reduced model. A technique of adaptive snapshot selection is proposed to reduce the complexity and runtime of generating the reduced basis. With the help of an output-oriented error bound, the construction of the reduced model is managed automatically. Numerical examples demonstrate the performance of the adaptive technique in reducing the offline time. The ROM-based optimization is successful in terms of the accuracy and the runtime for getting the optimal solution.

Keywords

Posteriori Error Estimation Reduce Basis Parabolic Partial Differential Equation Reduce Basis Method Random Sample Point 
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.

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References

  1. 1.
    M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Math. Acad. Sci. Paris Ser. I 339(9), 667–672 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    M. Drohmann, B. Haasdonk, M. Ohlberger, Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), 937–969 (2012)CrossRefMathSciNetGoogle Scholar
  3. 3.
    J.L. Eftang, D.J. Knezevic, A.T. Patera, An hp certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. Syst. 17(4), 395–422 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    W. Gao, S. Engell, Iterative set-point optimization of batch chromatography. Comput. Chem. Eng. 29(6), 1401–1409 (2005)CrossRefGoogle Scholar
  5. 5.
    M.A. Grepl, Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations, Ph.D. thesis, Massachusetts Institute of Technology, 2005Google Scholar
  6. 6.
    D. Gromov, S. Li, J. Raisch, A hierarchical approach to optimal control of a hybrid chromatographic batch process. Adv. Control Chem. Process. 7, 339–344 (2009)Google Scholar
  7. 7.
    G. Guiochon, A. Felinger, D.G. Shirazi, A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography (Academic, Boston, 2006)Google Scholar
  8. 8.
    B. Haasdonk, M. Dihlmann, M. Ohlberger, A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. Syst. 17(4), 423–442 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    B. Haasdonk, M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM Math. Model. Numer. Anal. 42(2), 277–302 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    S.G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt
  11. 11.
    A.K. Noor, J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 145–161 (1980)Google Scholar
  12. 12.
    A.T. Patera, G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering. Cambridge, MA (2007). Available from http://augustine.mit.edu/methodology/methodology_book.htm
  13. 13.
    Y. Zhang, L. Feng, S. Li, P. Benner, Accelerating PDE Constrained Optimization by the Reduced Basis Method: Application to Batch Chromatography, preprint MPIMD/14-09, Max Planck Institute Magdeburg Preprints, 2014Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Peter Benner
    • 1
  • Lihong Feng
    • 1
  • Suzhou Li
    • 1
  • Yongjin Zhang
    • 1
    Email author
  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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