Wiener Calculus for Differential Equations with Uncertainties

  • Florian Augustin
  • Peter Rentrop
  • Utz Wever
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 17)


In technical applications uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of N. Wiener was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from industry and academics. For each application chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.


Sparse Grid Tunnel Diode Polynomial Chaos Polynomial Chaos Expansion Stochastic Collocation Method 
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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  1. 1.
    Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Wever, U.: Polynomial chaos for the approximation of uncertainties: Chances and limits. Eur. J. Appl. Math. 19(2), 149–190 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Augustin, F., Rentrop, P.: Stochastic Galerkin techniques for random ordinary differential equations. Numerische Mathematik (submitted 2010)Google Scholar
  3. 3.
    Babuska, I.M., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Babuska, I.M., Tempone, R., Zouraris, G.E.: Galerkin finite element approximation of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chauvière, C., Hesthaven, J.S., Lurati, L.: Computational modeling of uncertainty in time-domain electromagnetics. SIAM J. Sci. Comput. 28(2), 751–775 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ernst, O.G., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions (2010). (Preprint 60) DFG SPP 1324
  7. 7.
    Feldmann, U., Denk, G.: Private communication, Infineon AG (1993)Google Scholar
  8. 8.
    Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithm. 18(3–4), 209–232 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ghanem, R.G., Spanos, P.D.: Stochastic finite elements: A spectral approach. Springer, New York (1991)zbMATHCrossRefGoogle Scholar
  10. 10.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations 1. Springer, Berlin, Heidelberg (1993)Google Scholar
  11. 11.
    Kampowski, W., Rentrop, P., Schmidt, W.: Classification and numerical simulation of electric circuits. Surv. Math. Ind. 2, 23–65 (1992)Google Scholar
  12. 12.
    Klenke, A.: Probability theory. Universitext. Springer, London (2008)CrossRefGoogle Scholar
  13. 13.
    Le Maître, O.P., Knio, O.: Spectral methods for uncertainty quantification. Scientific Computation. Springer, Dordrecht, Heidelberg, London, New York (2010)zbMATHCrossRefGoogle Scholar
  14. 14.
    Matthies, H.G.: Stochastic finite elements: computational approaches to stochastic partial differential equations. ZAMM 88(11), 849–873 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Øksendal, B.: Stochastic differential equations, 5th edn. Universitext. Springer, Berlin, Heidelberg, New York (2000)Google Scholar
  16. 16.
    Pulch, R.: Polynomial chaos for multirate partial differential algebraic equations with random parameters. Appl. Numer. Math. 59(10), 2610–2624 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wan, X., Karniadakis, G.E.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Xiu, D.: Efficient collocation approach for parametric uncertainty analysis. Comm. Comput. Phys. 2(2), 293–309 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics (M2)Technische Universität MünchenGarchingGermany
  2. 2.SIEMENS AGMunichGermany

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