Stochastic Partial Differential Equations

  • H. P. Langtangen
  • H. Osnes
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 33)


The purpose of this chapter is to give an introduction to stochastic partial differential equations from a computational point of view. The presented tools provide a consistent quantitative way of relating uncertainty in input to uncertainty in output for PDE-based models. We first give an analytical treatment of some stochastic differential equation model problems. These problems concerns deflection of a beam with random loading and material strength, heat conduction in random media, and pollution transport with random advection. Later, we develop Diffpack simulators for solving the model problems numerically. Two numerical solution methods are addressed: Monte Carlo simulations and perturbation methods. The main tools for generating and estimating stochastic variables and fields are outlined, and we show a suggested design of stochastic PDE simulators, which makes it easy to equip a standard sequential Diffpack simulator with stochastic treatment of uncertain input data.


Random Field Random Number Generator Base Class Stochastic Variable Maximum Deflection 
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  1. 1.
    A. Bellin, S. Salandin, and A. Rinaldo. Simulation of dispersion in heteroge-neous porous formations: Statistics, first-order theories, convergence of computations. Water Resour. Res., 28:2211–2227, 1992.CrossRefGoogle Scholar
  2. 2.
    P. P. Benham, R. J. Crawford, and C. G. Armstrong. Mechanics of Engineering Materials. Longman, 2nd edition, 1996.Google Scholar
  3. 3.
    G. K. Bhattacharyya and R. A. Johnson. Statistical Concepts and Methods. Wiley, 1977.Google Scholar
  4. 4.
    G. Dagan. Stochastic modeling of groundwater flow by unconditional and conditional probabilities. i: Conditional simulation and the direct problem. Water Resour. Res., 18:813–833, 1982.CrossRefGoogle Scholar
  5. 5.
    G. Dagan. Solute transport in heterogeneous porous formations. J. Fluid Mech., 145:151–177, 1984.CrossRefzbMATHGoogle Scholar
  6. 6.
    G. Dagan. Theory of solute transport by groundwater. Ann. Rev. Fluid Mech., 19:183–215, 1987.CrossRefzbMATHGoogle Scholar
  7. 7.
    G. Deodatis and M. Shinozuka. The weighted integral method. ii: Response variability and reliability. ASCE J. Engrg. Mech., 117:1865–1877, 1991.CrossRefGoogle Scholar
  8. 8.
    M. D. Dettinger and J. L. Wilson. First order analysis of uncertainty in numerical models of groundwater flow, Part 1. Mathematical development. Water Resour. Res.,17:149–161, 1981.CrossRefGoogle Scholar
  9. 9.
    R. G. Ghanem and P. D. Spanos. Finite Elements: A Spectral Approach. Springer-Verlag, 1991.Google Scholar
  10. 10.
    S. Gran. A Course in Ocean Engineering. Elsevier, 1992.Google Scholar
  11. 11.
    R. C. Hibbeler. Mechanics of Materials. Prentice Hall, 4th edition, 1999.Google Scholar
  12. 12.
    H. Holden and N. H. Risebro. Stochastic properties of the scalar buckley-leverett equation. SIAM J. Appl. Math.,51:1472–1488, 1991.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    E. H. Isaaks and R. M. Srivastava. An Introduction to Applied Geostatistics. Oxford University Press, 1989.Google Scholar
  14. 14.
    H. P. Langtangen. Tips and frequently asked questions about Diffpack. Numerical Objects Report Series, Numerical Objects A.S., 2002. Scholar
  15. 15.
    H. P. Langtangen. Computational Partial Differential Equations-Numerical Methods and Diffpack Programming. Texts in Computational Science and Engineering. Springer, 2nd edition, 2003.Google Scholar
  16. 16.
    W. L. Liu, A. Mani, and T. Belytschko. Finite element methods in probabilistic mechanics. Probab. Engrg. Mech.,2:201–213, 1987.CrossRefGoogle Scholar
  17. 17.
    H. O. Madsen, S. Krenk, and N. C. Lind. Methods of Structural Safety. Prentice-Hall, 1986.Google Scholar
  18. 18.
    D. E. Newland. An Introduction to Random Vibrations and Spectral Analysis. Longman, 1984.Google Scholar
  19. 19.
    H. Omre, K. Sølna, and H. Tjelmeland. Simulation of random functions on large lattices. In Proceedings Geostatistics Tróia’ 92, pages 179–199, Tróia, Italy, 1992.Google Scholar
  20. 20.
    H. Osnes and H. P. Langtangen. An improved probabilistic finite element method for stochastic groundwater flow. Advances in Water Resources,22:185–195, 1998.CrossRefGoogle Scholar
  21. 21.
    H. Osnes and H. P. Langtangen. A study of some finite difference schemes for a unidirectional stochastic transport equation. SIAM J. Sci. Comput., 19(3):799–812, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. M. Ross. Introduction to Probability Models. Academic Press, 6th edition, 1997.Google Scholar
  23. 23.
    L. Smith and R. A. Freeze. Stochastic analysis of steady state groundwater flow in a bounded domain. 2. Two-dimensional simulations. Water Resour. Res., 15(6):1543–1559, 1979.CrossRefGoogle Scholar
  24. 24.
    A. F. B. Tompson, R. Ababou, and L. W. Gelhar. Implementation of the three-dimensional turning bands random field generator. Water Resour. Res., 25:2227–2243, 1989.CrossRefGoogle Scholar
  25. 25.
    E. Vanmarcke. Random Fields. MIT Press, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • H. P. Langtangen
    • 1
    • 2
  • H. Osnes
    • 3
    • 4
  1. 1.Simula Research LaboratoryNorway
  2. 2.Dept. of InformaticsUniversity of OsloNorway
  3. 3.Dept. of MathematicsUniversity of OsloNorway
  4. 4.Det norske VeritasNorway

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