Numerical solutions of stochastic PDEs driven by arbitrary type of noise

  • Tianheng Chen
  • Boris Rozovskii
  • Chi-Wang Shu


So far the theory and numerical practice of stochastic partial differential equations (SPDEs) have dealt almost exclusively with Gaussian noise or Lévy noise. Recently, Mikulevicius and Rozovskii (Stoch Partial Differ Equ Anal Comput 4:319–360, 2016) proposed a distribution-free Skorokhod–Malliavin calculus framework that is based on generalized stochastic polynomial chaos expansion, and is compatible with arbitrary driving noise. In this paper, we conduct systematic investigation on numerical results of these newly developed distribution-free SPDEs, exhibiting the efficiency of truncated polynomial chaos solutions in approximating moments and distributions. We obtain an estimate for the mean square truncation error in the linear case. The theoretical convergence rate, also verified by numerical experiments, is exponential with respect to polynomial order and cubic with respect to number of random variables included.


Distribution-free Stochastic PDE Stochastic polynomial chaos Wick product Skorokhod integral 



The authors would like to thank Michael Tretyakov and Zhongqiang Zhang for helpful discussions on the relation between commutativity and K-version convergence.


  1. 1.
    Askey, R., Wilson, J.A.: Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials. American Mathematical Society, Providence (1985)zbMATHGoogle Scholar
  2. 2.
    Benth, F.E., Gjerde, J.: A remark on the equivalence between Poisson and Gaussian stochastic partial differential equations. Potential Anal. 8, 179–193 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Debusschere, B.J., Najm, H.N., Pébay, P.P., Knio, O.M., Ghanem, R.O., Le Maıtre, O.P.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26, 698–719 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Da Prato, G., Debussche, A., Temam, R.: Stochastic Burgers’ equation. Nonlinear Differ. Equ. Appl. 1, 389–402 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Di Nunno, G., Øksendal, B., Proske, F.: Malliavin Calculus for Lévy Processes with Applications to Finance. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hou, T.Y., Luo, W., Rozovskii, B., Zhou, H.: Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys. 216, 687–706 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Itô, K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, H.: On spectral approximations of stochastic partial differential equations driven by Poisson noise. Ph.D. Thesis, University of Southern California (2007)Google Scholar
  12. 12.
    Lototsky, S., Mikulevicius, R., Rozovskii, B.: Nonlinear filtering revisited: a spectral approach. SIAM J. Control Optim. 35, 435–461 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lototsky, S., Rozovskii, B.: Passive scalar equation in a turbulent incompressible Gaussian velocity field. Russ. Math. Surv. 59, 297–312 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lototsky, S., Rozovskii, B.: Stochastic differential equations: a Wiener chaos approach. In: Kabanov, Y., Liptser, R., Stoyanov, J. (eds.) From Stochastic Calculus to Mathematical Finance, pp. 433–506. Springer, Berlin (2006)CrossRefGoogle Scholar
  15. 15.
    Lototsky, S., Rozovskii, B.: Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. 34, 638–662 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lototsky, S., Rozovskii, B.: Stochastic Partial Differential Equations. Springer, Berlin (2017)CrossRefzbMATHGoogle Scholar
  17. 17.
    Luo, W.: Wiener chaos expansion and numerical solutions of stochastic partial differential equations. Ph.D. Thesis, California Institute of Technology (2006)Google Scholar
  18. 18.
    Malliavin, P.: Stochastic calculus of variation and hypoelliptic operators. In: Proceedings of International Symposium, SDE Kyoto 1976, Kinokuniya, pp. 195–263 (1978)Google Scholar
  19. 19.
    Mikulevicius, R., Rozovskii, B.: Separation of observations and parameters in nonlinear filtering. In: Proceedings of the 32nd IEEE Conference on Decision and Control, pp. 1564–1569. IEEE (1993)Google Scholar
  20. 20.
    Mikulevicius, R., Rozovskii, B.: Linear parabolic stochastic PDE and Wiener chaos. SIAM J. Math. Anal. 29, 452–480 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mikulevicius, R., Rozovskii, B.: On unbiased stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 154, 787–834 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mikulevicius, R., Rozovskii, B.: On distribution free Skorokhod–Malliavin calculus. Stoch. Partial Differ. Equ. Anal. Comput. 4, 319–360 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Milstein, G., Tretyakov, M.: Solving parabolic stochastic partial differential equations via averaging over characteristics. Math. Comput. 78, 2075–2106 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Milstein, G., Tretyakov, M.: Stochastic Numerics for Mathematical Physics. Springer, Berlin (2013)zbMATHGoogle Scholar
  25. 25.
    Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: An Introduction. Springer, Berlin (2015)zbMATHGoogle Scholar
  26. 26.
    Skorokhod, A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, 219–233 (1976)CrossRefzbMATHGoogle Scholar
  27. 27.
    Venturi, D., Wan, X., Mikulevicius, R., Rozovskii, B., Karniadakis, G.: Wick-Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulations. Proc. R. Soc. A R. Soc. 469, 1–20 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wan, X., Rozovskii, B.: The Wick–Malliavin approximation of elliptic problems with log-normal random coefficients. SIAM J. Sci. Comput. 35, A2370–A2392 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wick, G.C.: The evaluation of the collision matrix. Phys. Rev. 80, 268–272 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Xiu, D., Karniadakis, G.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Xiu, D., Karniadakis, G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Xiu, D., Karniadakis, G.: Supersensitivity due to uncertain boundary conditions. Int. J. Numer. Methods Eng. 61, 2114–2138 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  34. 34.
    Zhang, Z., Rozovskii, B., Tretyakov, M., Karniadakis, G.: A multistage Wiener chaos expansion method for stochastic advection-diffusion-reaction equations. SIAM J. Sci. Comput. 34, A914–A936 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zhang, Z., Tretyakov, M., Rozovskii, B., Karniadakis, G.: A recursive sparse grid collocation method for differential equations with white noise. SIAM J. Sci. Comput. 36, A1652–A1677 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, Z., Tretyakov, M., Rozovskii, B., Karniadakis, G.: Wiener chaos versus stochastic collocation methods for linear advection-diffusion-reaction equations with multiplicative white noise. SIAM J. Numer. Anal. 53, 153–183 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zhang, Z., Karniadakis, G.: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Springer, Berlin (2017)CrossRefzbMATHGoogle Scholar
  38. 38.
    Zheng, M., Rozovskii, B., Karniadakis, G.: Adaptive Wick–Malliavin approximation to nonlinear SPDEs with discrete random variables. SIAM J. Sci. Comput. 37, A1872–A1890 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations