Sparse Representations for Uncertainty Quantification of a Coupled Field-Circuit Problem

  • Roland PulchEmail author
  • Sebastian Schöps
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 30)


We consider a model of an electric circuit, where differential algebraic equations for a circuit part are coupled to partial differential equations for an electromagnetic field part. An uncertainty quantification is performed by changing physical parameters into random variables. A random quantity of interest is expanded into the (generalised) polynomial chaos using orthogonal basis polynomials. We investigate the determination of sparse representations, where just a few basis polynomials are required for a sufficiently accurate approximation. Furthermore, we apply model order reduction with proper orthogonal decomposition to obtain a low-dimensional representation in an alternative basis.



The research of the second author is supported by the Excellence Initiative of the German Federal and State Governments and by the Graduate School of Computational Engineering at Technische Universität Darmstadt.


  1. 1.
    Antoulas, A.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (2005)CrossRefGoogle Scholar
  2. 2.
    Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ho, C.W., Ruehli, A., Brennan, P.: The modified nodal approach to network analysis. IEEE Trans. Circ. Syst. 22(6), 504–509 (1975)CrossRefGoogle Scholar
  5. 5.
    Jakeman, J.D., Narayan, A., Zhou, T.: A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions. SIAM J. Sci. Comput. 39(3), A1114–A1144 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Pulch, R.: Model order reduction and low-dimensional representations for random linear dynamical systems. Math. Comput. Simulat. 144, 1–20 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Pulch, R.: Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest. Math. Comput. Simulat. 166, 76–92 (2019)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Schöps, S.: Multiscale Modeling and Multirate Time-Integration of Field/Circuit Coupled Problems. VDI Verlag. Fortschritt-Berichte VDI, Reihe 21, Nr. 398 (2011)Google Scholar
  9. 9.
    Schöps, S., De Gersem, H., Weiland, T.: Winding functions in transient magnetoquasistatic field-circuit coupled simulations. COMPEL 29(2), 2063–2083 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Stroud, A.J.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)zbMATHGoogle Scholar
  11. 11.
    Tsukerman, I.A., Konrad, A., Meunier, G., Sabonnadiere, J.C.: Coupled field-circuit problems: trends and accomplishments. IEEE Trans. Magn. 29(2), 1701–1704 (1993)CrossRefGoogle Scholar
  12. 12.
    Xiu, D.: Numerical Methods for Stochastic Computations: a Spectral Method Approach. Princeton University Press, Princeton (2010)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Universität GreifswaldInstitute of Mathematics and Computer ScienceGreifswaldGermany
  2. 2.Technische Universität DarmstadtCentre for Computational EngineeringDarmstadtGermany

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