Intrusive Polynomial Chaos Methods for Forward Uncertainty Propagation

  • Bert DebusschereEmail author
Reference work entry


Polynomial chaos (PC)-based intrusive methods for uncertainty quantification reformulate the original deterministic model equations to obtain a system of equations for the PC coefficients of the model outputs. This system of equations is larger than the original model equations, but solving it once yields the uncertainty information for all quantities in the model. This chapter gives an overview of the literature on intrusive methods, outlines the approach on a general level, and then applies it to a system of three ordinary differential equations that model a surface reaction system. Common challenges and opportunities for intrusive methods are also highlighted.


Galerkin projection Intrusive spectral projection Polynomial chaos 


  1. 1.
    Augustin, F., Rentrop, P.: Stochastic Galerkin techniques for random ordinary differential equations. Numer. Math. 122(3), 399–419 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Babuška, I., Tempone, R., Zouraris, G.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babuška, I., Tempone, R., Zouraris, G.: Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput Methods Appl. Mech. Eng. 194, 1251–1294 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beran, P.S., Pettit, C.L., Millman, D.R.: Uncertainty quantification of limit-cycle oscillations. J. Comput. Phys. 217(1), 217–47 (2006). doi:10.1016/ Scholar
  5. 5.
    Chen, Q.Y., Gottlieb, D., Hesthaven, J.: Uncertainty analysis for the steady-state flows in a dual throat nozzle. J. Comput. Phys. 204, 378–398 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Deb, M.K., Babuška, I., Oden, J.: Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190, 6359–6372 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Debusschere, B., Najm, H., Matta, A., Knio, O., Ghanem, R., Le Maître, O.: Protein labeling reactions in electrochemical microchannel flow: numerical simulation and uncertainty propagation. Phys. Fluids 15(8), 2238–2250 (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Debusschere, B., Najm, H., Pébay, P., Knio, O., Ghanem, R., Le Maître, O.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26(2), 698–719 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Debusschere, B., Sargsyan, K., Safta, C., Chowdhary, K.: UQ Toolkit. (2015)
  10. 10.
    Elman, H.C., Miller, C.W., Phipps, E.T., Tuminaro, R.S.: Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int. J. Uncertain. Quantif. 1(1), 19–33 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ernst, O., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM: Math. Model. Numer. Anal. 46, 317–339 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ghanem, R., Dham, S.: Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media 32, 239–262 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  14. 14.
    Knio, O., Le Maître, O.: Uncertainty propagation in CFD using polynomial chaos decomposition. Fluid Dyn. Res. 38(9), 616–40 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Le Maître, O., Knio, O., Najm, H., Ghanem, R.: A stochastic projection method for fluid flow I. Basic formulation. J. Comput. Phys. 173, 481–511 (2001)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Le Maître, O., Reagan, M., Najm, H., Ghanem, R., Knio, O.: A stochastic projection method for fluid flow II. Random process. J. Comput. Phys. 181, 9–44 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Le Maître, O., Knio, O., Debusschere, B., Najm, H., Ghanem, R.: A multigrid solver for two-dimensional stochastic diffusion equations. Comput. Methods Appl Mech. Eng. 192, 4723–4744 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Le Maître, O., Ghanem, R., Knio, O., Najm, H.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197(1), 28–57 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Le Maître, O., Najm, H., Ghanem, R., Knio, O.: Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys. 197, 502–531 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Le Maître, O., Reagan, M., Debusschere, B., Najm, H., Ghanem, R., Knio, O.: Natural convection in a closed cavity under stochastic, non-Boussinesq conditions. SIAM J. Sci. Comput. 26(2), 375–394 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Le Maître, O., Najm, H., Pébay P, Ghanem, R., Knio, O.: Multi-resolution analysis scheme for uncertainty quantification in chemical systems. SIAM J. Sci. Comput. 29(2), 864–889 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Le Maitre, O.P., Mathelin, L., Knio, O.M., Hussaini, M.Y.: Asynchronous time integration for polynomial chaos expansion of uncertain periodic dynamics. Discret. Contin. Dyn. Syst. 28(1), 199–226 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lucor, D., Karniadakis, G.: Noisy inflows cause a shedding-mode switching in flow past an oscillating cylinder. Phys. Rev. Lett. 92(15), 154501 (2004)CrossRefGoogle Scholar
  24. 24.
    Ma, X., Zabaras, N.: A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media. J. Comput. Phys. 227(18), 8448–8471 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Makeev, A.G., Maroudas, D., Kevrekidis, I.G.: “Coarse” stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples. J. Chem. Phys. 116(23), 10,083 (2002)CrossRefGoogle Scholar
  26. 26.
    Marzouk, Y.M., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228(6), 1862–1902 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194, 1295–1331 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Millman, D., King, P., Maple, R., Beran, P., Chilton, L.: Uncertainty quantification with a B-spline stochastic projection. AIAA J. 44(8), 1845–1853 (2006)CrossRefGoogle Scholar
  29. 29.
    Najm, H.: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Ann. Rev. Fluid Mech. 41(1), 35–52 (2009). doi:10.1146/annurev.fluid.010908.165248MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Najm, H., Valorani, M.: Enforcing positivity in intrusive PC-UQ methods for reactive ODE systems. J. Comput. Phys. 270, 544–569 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Narayanan, V., Zabaras, N.: Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations. J. Comput. Phys. 202(1), 94–133 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Pawlowski, R.P., Phipps, E.T., Salinger, A.G.: Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part I: Template-based generic programming. Sci. Program. 20(2), 197–219 (2012). doi:10.3233/SPR-2012-0350, arXiv:1205.3952v1Google Scholar
  33. 33.
    Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Owen, S.J., Siefert, C.M., Staten, M.L.: Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part II: application to partial differential equations. Sci. Program. 20(3), 327–345 (2012). doi:10.3233/SPR-2012-0351, arXiv:1205.3952v1Google Scholar
  34. 34.
    Perez, R., Walters, R.: An implicit polynomial chaos formulation for the euler equations. In: Paper AIAA 2005-1406, 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno (2005)Google Scholar
  35. 35.
    Pettersson, M.P., Iaccarino, G., Nordström, J.: Polynomial Chaos Methods for Hyperbolic Partial Differential Equations. Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  36. 36.
    Pettersson, P., Nordström, J., Iaccarino, G.: Boundary procedures for the time-dependent Burgers’ equation under uncertainty. Acta Math. Sci. 30(2), 539–550 (2010). doi:10.1016/S0252-9602(10)60061-6MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pettersson, P., Iaccarino, G., Nordström, J.: A stochastic Galerkin method for the Euler equations with Roe variable transformation. J. Comput. Phys. 257(PA), 481–500 (2014)Google Scholar
  38. 38.
    Pettit, C.L., Beran, P.S.: Spectral and multiresolution wiener expansions of oscillatory stochastic processes. J. Sound Vib. 294(4/5):752–779 (2006). doi:10.1016/j.jsv.2005.12.043CrossRefGoogle Scholar
  39. 39.
    Phipps, E.: Stokhos. (2015). Accessed 9 Sept 2015
  40. 40.
    Phipps, E., Hu, J., Ostien, J.: Exploring emerging manycore architectures for uncertainty quantification through embedded stochastic Galerkin methods. Int. J. Comput. Math. 1–23 (2013). doi:10.1080/00207160.2013.840722Google Scholar
  41. 41.
    Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29(2), 350–375 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Reagan, M., Najm, H., Debusschere, B., Le Maître O, Knio, O., Ghanem, R.: Spectral stochastic uncertainty quantification in chemical systems. Combust. Theory Model. 8, 607–632 (2004)CrossRefzbMATHGoogle Scholar
  43. 43.
    Sargsyan, K., Debusschere, B., Najm, H., Marzouk, Y.: Bayesian inference of spectral expansions for predictability assessment in stochastic reaction networks. J. Comput. Theor. Nanosci. 6(10), 2283–2297 (2009)CrossRefGoogle Scholar
  44. 44.
    Schwab, C., Todor, R.: Sparse finite elements for stochastic elliptic problems. Numer. Math. 95, 707–734 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sonday, B., Berry, R., Najm, H., Debusschere, B.: Eigenvalues of the Jacobian of a Galerkin-projected uncertain ODE system. SIAM J. Sci. Comput. 33, 1212–1233 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Todor, R., Schwab, C.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27, 232–261 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229(18), 6485–6511 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Roe solver with entropy corrector for uncertain hyperbolic systems. J. Comput. Appl. Math. 235(2), 491–506 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Tryoen, J., Maître, O.L., Ern, A.: Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws. SIAM J. Sci. Comput. 34(5), A2459–A2481 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Vigil, R., Willmore, F.: Oscillatory dynamics in a heterogeneous surface reaction: Breakdown of the mean-field approximation. Phys. Rev. E. Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), 1225–1231 (1996)Google Scholar
  51. 51.
    Villegas, M., Augustin, F., Gilg, A., Hmaidi, A., Wever, U.: Application of the Polynomial Chaos Expansion to the simulation of chemical reactors with uncertainties. Math. Comput. Simul. 82(5), 805–817 (2012). doi:10.1016/j.matcom.2011.12.001MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Wan, X., Karniadakis, G.: Long-term behavior of polynomial chaos in stochastic flow simulations. Comput. Methods Appl. Mech. Eng. 195(2006), 5582–5596 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Wan, X., Karniadakis, G.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209, 617–642 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Wan, X., Xiu, D., Karniadakis, G.: Stochastic solutions for the two-dimensional advection-diffusion equation. SIAM J. Sci. Comput. 26(2), 578–590 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936. doi:10.2307/2371268 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002). doi:10.1137/S1064827501387826MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Xiu, D., Karniadakis, G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Xiu, D., Karniadakis, G.: A new stochastic approach to transient heat conduction modeling with uncertainty. Int. J. Heat Mass Transf. 46(24), 4681–4693 (2003)CrossRefzbMATHGoogle Scholar
  60. 60.
    Xiu, D., Lucor, D., Su, C.H., Karniadakis, G.: Stochastic modeling of flow-structure interactions using generalized polynomial chaos. ASME J. Fluids Eng. 124, 51–59 (2002)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland (outside the USA) 2017

Authors and Affiliations

  1. 1.Mechanical EngineeringSandia National LaboratoriesLivermoreUSA
  2. 2.Reacting Flow Research DepartmentSandia National LaboratoriesLivermoreUSA

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