Skip to main content
SpringerLink
Log in

A splitting/polynomial chaos expansion approach for stochastic evolution equations

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

In this paper, we combine deterministic splitting methods with a polynomial chaos expansion method for solving stochastic parabolic evolution problems. The stochastic differential equation is reduced to a system of deterministic equations that we solve efficiently by splitting methods. The method can be applied to a wide class of problems where the related stochastic processes are given uniquely in terms of stochastic polynomials. A comprehensive convergence analysis is provided and numerical experiments validate our approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series 55, U.S. Government Printing Office, Washington, D.C. (1964)

    MATH  Google Scholar 

  2. R. Askey: Orthogonal polynomials and special functions. Society for Industrial and Applied Mathematics, 1975.

  3. Babuska, I., Tempone, R., Zouraris, G. E.: Galerkin finite element approximations of stochastic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)

    Article  MathSciNet  Google Scholar 

  4. Barbu, V., Röckner, M.: A splitting algorithm for stochastic partial differential equations driven by linear multiplicative noise. Stoch. PDE: Anal. Comp. 5, 457–471 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bensoussan, A., Glowinski, R., Rǎcanu, A.: Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25, 81–106 (1992)

    Article  MathSciNet  Google Scholar 

  6. Cao, Y.: On convergence rate of Wiener–Itô expansion for generalized random variables. Stochastics 78(3), 179–187 (2006)

    Article  MathSciNet  Google Scholar 

  7. Carelli, E. Hausenblas, E., Prohl, A.: Time-splitting methods to solve the stochastic incompressible Stokes equation. SIAM J. Numer. Anal. 50(6), 2917–2939 (2012)

    Article  MathSciNet  Google Scholar 

  8. Chihara, T. S.: An introduction to orthogonal polynomials. Gordon and Breach, New York-London-Paris, 1978.

    MATH  Google Scholar 

  9. Chen, T., Rozovskii, B., Shu, C.-W.: Numerical solutions of stochastic PDEs driven by arbitrary type of noise. Stoch. Partial Differ. Equ. Anal. Comp. 7(1), 1–39 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Cohen, D., Dujardin, G.: Exponential integrators for nonlinear Schrödinger equations with white noise dispersion. Stoch. PDE: Anal. Comp. 5, 592–613 (2017)

    Article  Google Scholar 

  11. Constantine, P.: A primer on stochastic Galerkin methods. Stanford University, Preprint (2007)

    Google Scholar 

  12. Constantine, P. G., Doostan, A. Iaccarino, G.: A hybrid collocation/Galerkin scheme for convective heat transfer problems with stochastic boundary conditions. Internat. J. Numer. Methods Engrg. 80(6-7), 868–880 (2009)

    Article  MathSciNet  Google Scholar 

  13. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications 44. Cambridge University Press, Cambridge, (1992)

  14. Faou, E.: Analysis of splitting methods for reaction-diffusion problems using stochastic calculus. Math. Comp. , 267(78), 1467–1483 (2009)

    Article  MathSciNet  Google Scholar 

  15. Faou, E., Ostermann, A., Schratz, K.: Analysis of exponential splitting methods for inhomogeneous parabolic equations. IMA J. Numer. Anal. 35, 161–178 (2015)

    Article  MathSciNet  Google Scholar 

  16. Ghanem, R. G., Spanos, P. D.: Stochastic finite elements: a spectral approach. Springer-Verlag, New York (1991)

    Book  Google Scholar 

  17. Grecksch, W., Lisei, H.: Approximation of stochastic nonlinear equations of Schrödinger type by the splitting method. Stoch. Anal. Appl. 31(2), 314–335 (2013)

    Article  MathSciNet  Google Scholar 

  18. Gyöngy, I., Krylov, N.: On the splitting-up method and stochastic partial differential equations. Ann. Probab. 31, 564–591 (2003)

    Article  MathSciNet  Google Scholar 

  19. Hafizoglu, C., Lasiecka, I., Levajković, T., Mena, H., Tuffaha, A.: The stochastic linear quadratic control problem with singular estimates. SIAM J. Control Optim. 55(2), 595–626 (2017)

    Article  MathSciNet  Google Scholar 

  20. Huschto, T., Podolskij, M., Sager, S.: The asymptotic error of chaos expansion approximations for stochastic differential equations. Mod. Stoch. Theory Appl. 6(2), 145–165 (2019)

    Article  MathSciNet  Google Scholar 

  21. Hansen, E., Ostermann, A.: Exponential splitting for unbounded operators. Math. Comp. 78, 1485–1496 (2009)

    Article  MathSciNet  Google Scholar 

  22. Hansen, E., Ostermann, A.: High order splitting methods for analytic semigroups exist. BIT 49, 527–542 (2009)

    Article  MathSciNet  Google Scholar 

  23. Hansen, E., Ostermann, A.: Dimension splitting for evolution equations. Numer. Math. 108, 557–570 (2008)

    Article  MathSciNet  Google Scholar 

  24. Hell, T., Ostermann, A., Sandbichler, M.: Modification of dimension-splitting methods–overcoming the order reduction due to corner singularities. IMA J. Numer. Anal. 35(3), 1078–1091 (2015)

  25. Hida, T., Kuo, H.-H., Pathoff, J., Streit, L.: White noise–An infinite dimensional calculus. Mathematics and its Applications 253. Kluwer Academic Publishers, Dordrecht (1993)

  26. Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. A modeling, white noise functional approach. Second edition. Universitext, Springer, New York (2010)

  27. Hou, T. Y., Luo, W., Rozovskii, B., Zhou, H.-M.: Wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys. 216, 687–706 (2006)

    Article  MathSciNet  Google Scholar 

  28. Hu, Y.: Some recent progress on stochastic heat equations. Acta Math. Sci. Ser. B 39 (3), 874–914 (2019)

    Article  MathSciNet  Google Scholar 

  29. Hundsdorfer, W., Verwer, J.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Computational Mathematics 33, Springer-Verlag, Berlin (2003)

  30. Itô, K.: Multiple Wiener integral. J. Math. Soc. Japan 3, 157–169 (1951)

    Article  MathSciNet  Google Scholar 

  31. Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40, 735–744 (2000)

    Article  MathSciNet  Google Scholar 

  32. Le Maître, O. P., Knio, O. M.: Spectral methods for uncertainty quantification. With applications to computational fluid dynamics. Scientific Computation, Springer, New York (2010)

  33. Levajković, T., Mena, H.: Equations involving Malliavin calculus operators: Applications and numerical approximation. SpringerBriefs in Mathematics, Springer, Cham (2017)

  34. Levajković, T., Mena, H., Pfurtscheller, L.: Solving stochastic LQR problems by polynomial chaos. IEEE Control Systems Letters 2(4), 641–646 (2018)

    Article  MathSciNet  Google Scholar 

  35. Levajković, T., Mena, H., Tuffaha, A.: The stochastic linear quadratic control problem: A chaos expansion approach. Evol. Equ. Control Theory 5(1), 105–134 (2016)

    Article  MathSciNet  Google Scholar 

  36. Levajković, T., Pilipović, S., Seleši, D., Žigić, M.: Stochastic evolution equations with multiplicative noise. Electron. J. Probab. 20(19), 1–23 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Levajković, T., Pilipović, S., Seleši, D., Žigić, M.: Stochastic evolution equations with Wick-polynomial nonlinearities. Electron. J. Probab. 23(116), 1–25 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Lototsky, S. V., Rozovskii, B. L.: Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab., 34(2), 638–662 (2006)

    Article  MathSciNet  Google Scholar 

  39. Lototsky, S., Mikulevicius, R., Rozovskii, B.: Nonlinear filtering revisited: a spectral approach. SIAM J. Control Optim. 35, 435–461 (1997)

    Article  MathSciNet  Google Scholar 

  40. Lunardi, A.: Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser, Basel (1995)

  41. Mikulevicius, R., Rozovskii, B. L.: Stochastic Navier-Stokes equations. Propagation and chaos and statistical moments, in Optimal Control and Partial Differential Equations: In Honour of Alain Bensoussan, (J. L. Menaldi, E. Rofman and A. Sulem, eds.), IOS Press, Amsterdam, 258–267 (2001)

  42. Mikulevicius, R., Rozovskii, B. L.: On distribution free Skorokhod-Malliavin calculus. Stoch. Partial Differ. Equ. Anal. Comput. 4(2), 319–360 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Nualart, D., Rozovskii, B. L.: Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise. J. Funct. Anal. 149, 200–225 (1997)

    Article  MathSciNet  Google Scholar 

  44. Ostermann, A., Schratz, K.: Error analysis of splitting methods for inhomogeneous evolution equations. Appl. Numer. Math. 62(10), 1436–1446 (2012)

    Article  MathSciNet  Google Scholar 

  45. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences 44, Springer, New York (1983)

  46. Schoutens, W.: Stochastic processes and orthogonal polynomials. Lecture Notes in Statistics 146, Springer-Verlag, New York (2000)

  47. Szegö, G.: Orthogonal polynomials. American Mathematical Society Colloquium Publications 23, American Mathematical Society, New York (1939)

  48. Trefethen, L. N.: Spectral methods in Matlab. SIAM, Philadelphia, PA (2000)

    Book  Google Scholar 

  49. Wan, X., Rozovskii, B.: The Wick–Malliavin approximation of elliptic problems with log-normal random coefficients. SIAM J. Sci. Comput. 35, A2370–A2392 (2013)

    Article  MathSciNet  Google Scholar 

  50. Xiu, D., Karniadakis, G. E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  MathSciNet  Google Scholar 

  51. Xiu, D, Karniadakis, G. E.: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Engrg. 191(43), 4927–4948 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially supported by a Research grant for Austrian graduates granted by the Office of the Vice Rector for Research of University of Innsbruck. The computational results presented have been partially achieved using the HPC infrastructure LEO of the University of Innsbruck. A. Kofler was supported by the program Nachwuchsförderung 2014 at University of Innsbruck. H. Mena was supported by the Austrian Science Fund—Project Id: P27926. We thank the anonymous referee for his/her valuable comments that helped greatly to improve this work.

Author information

Authors and Affiliations

  1. Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany

    Andreas Kofler

  2. Institute of Stochastics and Business Mathematics, Vienna University of Technology, Vienna, Austria

    Tijana Levajković

  3. Department of Mathematics, Yachay Tech University, Urcuquí, Ecuador

    Hermann Mena

  4. Department of Mathematics, University of Innsbruck, Innsbruck, Austria

    Hermann Mena & Alexander Ostermann

Authors
  1. Andreas Kofler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tijana Levajković
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hermann Mena
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alexander Ostermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tijana Levajković.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kofler, A., Levajković, T., Mena, H. et al. A splitting/polynomial chaos expansion approach for stochastic evolution equations. J. Evol. Equ. 21, 1345–1381 (2021). https://doi.org/10.1007/s00028-020-00627-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-020-00627-5

Mathematics Subject Classification

Keywords

Search

Navigation