Quadrature Methods with Adjusted Grids for Stochastic Models of Coupled Problems

  • Roland PulchEmail author
  • Andreas Bartel
  • Sebastian Schöps
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 22)


We consider coupled problems with uncertain parameters modelled as random variables. Due to the largely differing behaviour of subsystems in coupled problems, we introduce a strategy of adjusted grids defined in the parameter domain for resolving the stochastic model. This allows us to adapt quadrature grids to each subsystem. The communication between the different grids requires global approximations of coupling variables in the random space. Since implicit time integration methods are typically included, we investigate dynamic iteration schemes to realise this approach. Numerical results for a thermal-electric test circuit outline the feasibility of the method.


Coupled problems Stochastic modeling Thermal-electric circuit Uncertain parameters 



This work is a part of the project ‘Nanoelectronic Coupled Problems Solutions’ (NANOCOPS) funded by the European Union within FP7-ICT-2013 (grant no. 619166).


  1. 1.
    Bartel, A., Pulch, R.: A concept for classification of partial differential algebraic equations in nanoelectronics. In: Bonilla, L.L., Moscoso, M., Platero, G., Vega, J.M. (eds.) Progress in Industrial Mathematics at ECMI 2006. Mathematics in Industry, vol. 12, pp. 506–511. Springer, Berlin (2007)CrossRefGoogle Scholar
  2. 2.
    Bartel, A., Günther, M., Schulz, M.: Modeling and discretization of a thermal-electric test circuit. In: Antreich, K. (eds.) Modeling, Simulation and Optimization of Integrated Circuits. ISNM, vol. 146, pp. 187–201. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  3. 3.
    Bartel, A., Brunk, M., Günther, M., Schöps, S.: Dynamic iteration for coupled problems of electric circuits and distributed devices. SIAM J. Sci. Comput. 35(2), B315–B335 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chauvière, C., Hesthaven, J.S., Lurati, L.: Computational modeling of uncertainty in time-domain electromagnetics. SIAM J. Sci. Comput. 28(2), 751–775 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Pulch, R.: Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations. J. Comput. Appl. Math. 262, 281–291 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Roland Pulch
    • 1
    Email author
  • Andreas Bartel
    • 2
  • Sebastian Schöps
    • 3
  1. 1.Institute for Mathematics and Computer ScienceErnst-Moritz-Arndt-Universität GreifswaldGreifswaldGermany
  2. 2.Chair of Applied Mathematics and Numerical AnalysisBergische Universität WuppertalWuppertalGermany
  3. 3.Graduate School Computational ElectromagneticsTechnische Universität DarmstadtDarmstadtGermany

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