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Reduced Basis Approximations for Maxwell’s Equations in Dispersive Media

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Model Reduction of Parametrized Systems

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

Abstract

Simulation of electromagnetic and optical wave propagation in, e.g. water, fog or dielectric waveguides requires modeling of linear, temporally dispersive media. Using a POD-greedy and ID-greedy sampling driven by an error indicator, we seek to generate a reduced model which accurately captures the dynamics over a wide range of parameters, modeling the dispersion. The reduced basis model reduction reduces the model order by a factor of more than 20, while maintaining an approximation error of significantly less than 1% over the whole parameter range.

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Notes

  1. 1.

    All computations were done on a Intel(R) Core(TM)2 Quad CPU Q6700 @ 2.66GHz desktop machine with 8GB RAM, running Ubuntu 12.04.5 LTS and MATLAB R2012b.

References

  1. Banks, H.T., Bokil, V.A., Gibson, N.L.: Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media. Numer. Methods Partial Differ. Equ. 25(4), 885–917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bidégaray-Fesquet, B.: Stability of FDTD schemes for Maxwell-Debye and Maxwell-Lorentz equations. SIAM J. Numer. Anal. 46(5), 2551–2566 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calvo, M.P., Sanz-Serna, J.M.: Order conditions for canonical Runge-Kutta-Nyström methods. BIT Numer. Math. 32(1), 131–142 (1992)

    Article  MATH  Google Scholar 

  5. Cheng, H., Gimbutas, Z., Martinsson, P.G., Rokhlin, V.: On the compression of low rank matrices. SIAM J. Sci. Comput. 26(4), 1389–1404 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Goswami, C., Mukherjee, S., Karmakar, S., Pal, M., Ghatek, R.: FDTD modeling of Lorentzian DNG metamaterials by auxiliary differential equation method. J. Electromagn. Anal. Appl. 6(5), 106–114 (2014)

    Google Scholar 

  7. Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Math. Model. Numer. Anal. 42(2), 277–302 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Henneron, T., Clenet, S.: Model order reduction of non-linear magnetostatic problems based on POD and DEI methods. IEEE Trans. Magn. 50(2), 33–36 (2014). doi:10.1109/TMAG.2013.2283141

    Article  Google Scholar 

  9. Jin, J.M.: Theory and Computation of Electromagnetic Fields. Wiley, New York (2011)

    Google Scholar 

  10. Jung, N., Patera, A., Haasdonk, B., Lohmann, B.: Model order reduction and error estimation with an application to the parameter-dependent eddy current equation. Math. Comput. Modell. Dyn. Syst. 17, 561–582 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Oughstun, K.E.: Electromagnetic and Optical Pulse Propagation 1: Spectral Representations in Temporally Dispersive Media. Electromagnetic and Optical Pulse Propagation. Springer, New York (2006)

    Google Scholar 

  12. Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Meth. Eng. 15, 229–275 (2008)

    Article  MATH  Google Scholar 

  13. Schmidthaeusler, D., Clemens, M.: Low-order electroquasistatic field simulations based on proper orthogonal decomposition. IEEE Trans. Magn. 48, 567–570 (2012)

    Article  Google Scholar 

  14. Schneebeli, A.: An H(curl;Ω)-conforming FEM: Nédélec’s elements of first type. Technical Report (2003)

    Google Scholar 

  15. Zhang, Y., Feng, L., Li, S., Benner, P.: Accelerating PDE constrained optimization by the reduced basis method: application to batch chromatography. Int. J. Numer. Methods Eng. 104(11), 983–1007 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank Prof. Jan Hesthaven (EPFL) for helpful advice and fruitful discussion.

This work is supported by the collaborative project nanoCOPS, Nanoelectronic COupled Problems Solutions, supported by the European Union in the FP7-ICT-2013-11 Program under Grant Agreement Number 619166.

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Authors and Affiliations

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

    Peter Benner

  2. Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg, Sandtorstr. 1, 39106, Magdeburg, Germany

    Martin Hess

Authors
  1. Peter Benner
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  2. Martin Hess
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Corresponding author

Correspondence to Martin Hess .

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

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Benner, P., Hess, M. (2017). Reduced Basis Approximations for Maxwell’s Equations in Dispersive Media. 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_7

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