Skip to main content
SpringerLink
Log in

True Error Control for the Localized Reduced Basis Method for Parabolic Problems

  • Chapter
  • First Online:
Model Reduction of Parametrized Systems

Part of the book series: MS&A ((MS&A,volume 17))

Abstract

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques (Makridakis and Nochetto, SIAM J. Numer. Anal. 41(4):1585–1594, 2003. doi:10.1137/S0036142902406314; Lakkis and Makridakis, Math. Comput. 75(256):1627–1658, 2006. doi:10.1090/S0025-5718-06-01858-8; Demlow et al., SIAM J. Numer. Anal. 47(3):2157–2176, 2009. doi:10.1137/070708792; Georgoulis et al., SIAM J. Numer. Anal. 49(2):427–458, 2011. doi:10.1137/080722461). In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method (Ohlberger and Schindler, SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015. doi:10.1137/151003660). Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.

This work has been supported by the German Federal Ministry of Education and Research (BMBF) under contract number 05M13PMA.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    http://www.spe.org/web/csp/index.html.

  2. 2.

    http://pymor.org.

  3. 3.

    http://github.com/dune-community/dune-gdt.

References

  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, September 9–14, 2012, pp. 393–403. Slovak University of Technology in Bratislava, Publishing House of STU (2012)

    Google Scholar 

  2. Ali, M., Steih, K., Urban, K.: Reduced basis methods with adaptive snapshot computations. Adv. Comput. Math. 43(2), 257–294 (2017). doi:10.1007/s10444-016-9485-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Demlow, A., Lakkis, O., Makridakis, C.: A posteriori error estimates in the maximum norm for parabolic problems. SIAM J. Numer. Anal. 47(3), 2157–2176 (2009). doi:10.1137/070708792

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Georgoulis, E.H., Lakkis, O., Virtanen, J.M.: A posteriori error control for discontinuous Galerkin methods for parabolic problems. SIAM J. Numer. Anal. 49(2), 427–458 (2011). doi:10.1137/080722461

    Article  MathSciNet  MATH  Google Scholar 

  6. Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. M2AN Math. Model. Numer. Anal. 42(2), 277–302 (2008). doi:10.1051/m2an:2008001

    Google Scholar 

  7. Hesthaven, J., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer, Cham (2016). doi:10.1007/978-3-319-22470-1

    Book  MATH  Google Scholar 

  8. Kaulmann, S., Flemisch, B., Haasdonk, B., Lie, K.A., Ohlberger, M.: The localized reduced basis multiscale method for two-phase flows in porous media. Int. J. Numer. Methods Eng. 102(5), 1018–1040 (2015). doi:10.1002/nme.4773

    Article  MathSciNet  MATH  Google Scholar 

  9. Lakkis, O., Makridakis, C.: Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 75(256), 1627–1658 (2006). doi:10.1090/S0025-5718-06-01858-8

    Article  MathSciNet  MATH  Google Scholar 

  10. Makridakis, C., Nochetto, R.H.: Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM J. Numer. Anal. 41(4), 1585–1594 (2003). doi:10.1137/S0036142902406314

    Article  MathSciNet  MATH  Google Scholar 

  11. Milk, R., Rave, S., Schindler, F.: pyMOR – generic algorithms and interfaces for model order reduction. SIAM J. Sci. Comput. 38(5), S194–S216 (2016). doi:10.1137/15m1026614

    Google Scholar 

  12. Ohlberger, M., Rave, S., Schindler, F.: Model reduction for multiscale lithium-ion battery simulation. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2015, pp. 317–331. Springer International Publishing, Cham (2016). doi:10.1007/978-3-319-39929-4_31

    Chapter  Google Scholar 

  13. 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, Cham (2014). doi:10.1007/978-3-319-05684-5_41

    Google Scholar 

  14. Ohlberger, M., Schindler, F.: Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6), A2865–A2895 (2015). doi:10.1137/151003660

    Article  MathSciNet  MATH  Google Scholar 

  15. Patera, A.T., Rozza, G.: Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations, version 1.0. Technical Repotr, Copyright MIT 2006–2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering (2006)

    Google Scholar 

  16. Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. La Matematica per il 3+2. Springer, Cham (2016). doi:10.1007/978-3-319-15431-2

    MATH  Google Scholar 

  17. Verfürth, R.: A posteriori error estimation techniques for finite element methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). doi:10.1093/acprof:oso/9780199679423.001.0001

    Book  MATH  Google Scholar 

  18. Yano, M.: A minimum-residual mixed reduced basis method: exact residual certification and simultaneous finite-element reduced-basis refinement. ESAIM: Math. Model. Numer. Anal. 50, 163–185 (2015). doi:10.1051/m2an/2015039

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Computational and Applied Mathematics, University of Münster, Einsteinstrasse 62, 48149, Münster, Germany

    Mario Ohlberger, Stephan Rave & Felix Schindler

Authors
  1. Mario Ohlberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stephan Rave
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Felix Schindler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Felix Schindler .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Ohlberger, M., Rave, S., Schindler, F. (2017). True Error Control for the Localized Reduced Basis Method for Parabolic Problems. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_11

Download citation

Publish with us

Policies and ethics

Search

Navigation