Polynomial Chaos Methods

  • Mass Per Pettersson
  • Gianluca Iaccarino
  • Jan Nordström
Part of the Mathematical Engineering book series (MATHENGIN)


In this chapter we review methods for formulating partial differential equations based on the random field representations outlined in Chap.  2 These include the stochastic Galerkin method, which is the predominant choice in this book, as well as other methods that frequently occur in the literature, e.g., stochastic collocation methods and spectral projection. We also briefly discuss methods that are not polynomial chaos methods themselves but are viable alternatives.


Quadrature Point Spectral Projection Polynomial Chaos Stochastic Collocation 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.


  1. 1.
    Abgrall R (2008) A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems. Rapport de recherche.
  2. 2.
    Abgrall R, Congedo PM, Corre C, Galera S (2010) A simple semi-intrusive method for uncertainty quantification of shocked flows, comparison with a non-intrusive polynomial chaos method. In: ECCOMAS CFD, LisbonGoogle Scholar
  3. 3.
    Babuška IM, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005–1034CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bäck J, Nobile F, Tamellini L, Tempone R (2011) Implementation of optimal Galerkin and collocation approximations of PDEs with random coefficients. ESAIM Proc 33:10–21. doi: 10.1051/proc/201133002,
  5. 5.
    Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element: a non intrusive approach by regression. Eur J Comput Mech 15:81–92zbMATHGoogle Scholar
  6. 6.
    Blatman G, Sudret B (2008) Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. Comptes Rendus Mécanique 336(6):518–523CrossRefzbMATHGoogle Scholar
  7. 7.
    Clenshaw CW, Curtis AR (1960) A method for numerical integration on an automatic computer. Numerische Mathematik 2:197CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Doostan A, Owhadi H (2011) A non-adapted sparse approximation of PDEs with stochastic inputs. J Comput Phys 230(8):3015–3034. doi: 10.1016/,
  9. 9.
    Ganapathysubramanian B, Zabaras N (2007) Sparse grid collocation schemes for stochastic natural convection problems. J Comput Phys 225(1):652–685. doi: 10.1016/,
  10. 10.
    Gautschi W (1982) On generating orthogonal polynomials. SIAM J Sci Stat Comput 3:289–317. doi: 10.1137/0903018 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New YorkCrossRefzbMATHGoogle Scholar
  12. 12.
    Golub GH, Welsch JH (1967) Calculation of Gauss quadrature rules. Tech rep, StanfordGoogle Scholar
  13. 13.
    Hosder S, Walters R, Balch M (2007) Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables. In: AIAA-2007-1939, 9th AIAA non-deterministic approaches conference, HonoluluGoogle Scholar
  14. 14.
    Keese A, Matthies H (2003) Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations. Tech rep, Institute of Scientific Computing TU Braunschweig, BrunswickGoogle Scholar
  15. 15.
    Mathelin L, Hussaini MY (2003) A stochastic collocation algorithm for uncertainty analysis. Tech Rep 2003-212153, NASA Langley Research CenterGoogle Scholar
  16. 16.
    Mathelin L, Hussaini MY, Zang TA, Bataille F (2003) Uncertainty propagation for turbulent, compressible flow in a quasi-1D nozzle using stochastic methods. In: AIAA-2003-4240, 16TH AIAA CFD conference, Orlando, pp 23–26Google Scholar
  17. 17.
    Migliorati G, Nobile F, von Schwerin E, Tempone R (2013) Approximation of quantities of interest in stochastic PDEs by the random discrete L 2 projection on polynomial spaces. SIAM J Sci Comput 35(3):A1440–A1460CrossRefzbMATHGoogle Scholar
  18. 18.
    Reagan MT, Najm HN, Ghanem RG, Knio OM (2003) Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combust Flame 132(3):545–555CrossRefGoogle Scholar
  19. 19.
    Tuminaro RS, Phipps ET, Miller CW, Elman HC (2011) Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int J Uncertain Quantif 1(1):19–33CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Wan X, Karniadakis GE (2005) An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J Comput Phys 209:617–642. doi:,
  21. 21.
    Wan X, Karniadakis GE (2006) Long-term behavior of polynomial chaos in stochastic flow simulations. Comput Methods Appl Math Eng 195:5582–5596CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Wan X, Karniadakis GE (2006) Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J Sci Comput 28(3):901–928. doi: 10.1137/050627630,
  23. 23.
    Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Commun Comput Phys 2(2):293–309zbMATHMathSciNetGoogle Scholar
  24. 24.
    Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach. Princeton University Press, Princeton.
  25. 25.
    Xiu D, Hesthaven JS (2005) High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput 27:1118–1139. doi: 10.1137/040615201,

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Mass Per Pettersson
    • 1
  • Gianluca Iaccarino
    • 2
  • Jan Nordström
    • 3
  1. 1.Uni ResearchBergenNorway
  2. 2.Department of Mechanical Engineering and Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  3. 3.Department of Mathematics Computational MathematicsLinköping UniversityLinköpingSweden

Personalised recommendations