An Overview of Methods to Identify and Manage Uncertainty for Modelling Problems in the Water–Environment–Agriculture Cross-Sector

  • A. J. Jakeman
  • J. D. JakemanEmail author
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 28)


Uncertainty pervades the representation of systems in the water–environment–agriculture cross-sector. Successful methods to address uncertainties have largely focused on standard mathematical formulations of biophysical processes in a single sector, such as partial or ordinary differential equations. More attention to integrated models of such systems is warranted. Model components representing the different sectors of an integrated model can have less standard, and different, formulations to one another, as well as different levels of epistemic knowledge and data informativeness. Thus, uncertainty is not only pervasive but also crosses boundaries and propagates between system components. Uncertainty assessment (UA) cries out for more eclectic treatment in these circumstances, some of it being more qualitative and empirical. Here, we discuss the various sources of uncertainty in such a cross-sectoral setting and ways to assess and manage them. We have outlined a fast-growing set of methodologies, particularly in the computational mathematics literature on uncertainty quantification (UQ), that seem highly pertinent for uncertainty assessment. There appears to be considerable scope for advancing UA by integrating relevant UQ techniques into cross-sectoral problem applications. Of course this will entail considerable collaboration between domain specialists who often take first ownership of the problem and computational methods experts.


Mathematics-for-Industry Water resources Uncertainty assessment Uncertainty quantification 



Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. The article is a contribution to the NSF-funded National Socio-Environmental Synthesis Center project on Effective core practices for model-based integrated water resources management.


  1. 1.
    S. Amaral, D. Allaire, K. Willcox, A decomposition-based approach to uncertainty analysis of feed-forward multicomponent systems. Int. J. Numer. Methods Eng. 100(13), 982–1005 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Amaral, D. Allaire, K. Willcox, Optimal \(l_2\)-norm empirical importance weights for the change of probability measure. Stat. Comput. 27(3), 625–643 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Arnst, R. Ghanem, E. Phipps, J. Red-Horse, Dimension reduction in stochastic modeling of coupled problems. Int. J. Numer. Methods Eng. 92(11), 940–968 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Arnst, R. Ghanem, E. Phipps, J. Red-Horse, Measure transformation and efficient quadrature in reduced-dimensional stochastic modeling of coupled problems. Int. J. Numer. Methods Eng. 92(12), 1044–1080 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A.C. Atkinson, A.N. Donev. Optimum Experimental Designs (Oxford University Press, 1992)Google Scholar
  6. 6.
    J. Ball, M. Babister, R. Nathan, W. Weeks, P.E. Weinmann, M. Retallick, I. Testoni (eds.), Australian Rainfall and Runoff: A Guide to Flood Estimation. Commonwealth of Australia (Geoscience Australia, 2016)Google Scholar
  7. 7.
    I. Bauer, H.G. Bock, S. Krkel, J.P. Schlder, Numerical methods for optimum experimental design in DAE systems. J. Comput. Appl. Math. 120(12), 1–25 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M.A. Beaumont, W. Zhang, D.J. Balding, Approximate bayesian computation in population genetics. Genetics 162(4), 2025–2035 (2002)Google Scholar
  9. 9.
    N.D. Bennett, B.F.W. Croke, G. Guariso, J.H.A. Guillaume, S.H. Hamilton, A.J. Jakeman, S. Marsili-Libelli, L.T.H. Newham, J.P. Norton, C. Perrin, S.A. Pierce, B. Robson, R. Seppelt, A.A. Voinov, B.D. Fath, V. Andreassian, Characterising performance of environmental models. Env. Model. Softw. 40, 1–20 (2013)Google Scholar
  10. 10.
    K. Beven, A. Binley, The future of distributed models: model calibration and uncertainty prediction. Hydrol. Process. 6(3), 279–298 (1992)CrossRefGoogle Scholar
  11. 11.
    H.G. Bock, S. Körkel, J.P. Schlöder, Parameter Estimation and Optimum Experimental Design for Differential Equation Models (Springer, Berlin, 2013), pp. 1–30Google Scholar
  12. 12.
    G.P. Bonneau, H.C. Hege, C.R. Johnson, M.M. Oliveira, K. Potter, P. Rheingans, T. Schultz, Overview and state-of-the-art of uncertainty visualization, in Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization, ed. by C.D. Hansen, M. Chen, C.R. Johnson, A.E. Kaufman, H. Hagen (Springer, London, 2014), pp. 3–27Google Scholar
  13. 13.
    A. Bucklew, Introduction to Rare Event Simulation (Springer, 2004)Google Scholar
  14. 14.
    T. Bui-Thanh, O. Ghattas, J. Martin, G. Stadler, A computational framework for infinite-dimensional bayesian inverse problems part i: The linearized case, with application to global seismic inversion. SIAM J. Sci. Comput. 35(6), A2494–A2523 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    H.-J. Bungartz, M. Griebel, Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    K. Chaloner, I. Verdinelli, Bayesian experimental design: a review. Stat. Sci. 10(3), 273–304, 08 (1995)Google Scholar
  17. 17.
    Y. Chen, J.D. Jakeman, C. Gittelson, D. Xiu, Local polynomial chaos expansion for linear differential equations with high dimensional random inputs. SIAM J. Sci. Comput. 37(1), A79–A102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. Chkifa, A. Cohen, G. Migliorati, F. Nobile, R. Tempone, Discrete least squares polynomial approximation with random evaluations application to parametric and stochastic elliptic PDEs. ESAIM: M2AN 49(3), 815–837 (2015)Google Scholar
  19. 19.
    P.G. Constantine, Active subspaces: emerging ideas for dimension reduction in parameter studies. SIAM (2015)Google Scholar
  20. 20.
    P.G. Constantine, M.S. Eldred, E.T. Phipps, Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229–232, 1–12 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P.G. Constantine, E.T. Phipps, T.M. Wildey, Efficient uncertainty propagation for network multiphysics systems. Int. J. Numer. Methods Eng. 99(3), 183–202 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    T. Cui, J. Martin, Y.M. Marzouk, A. Solonen, A. Spantini, Likelihood-informed dimension reduction for nonlinear inverse problems. Inverse Probl. 30(11), 114015 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    R.I. Cukier, H.B. Levine, K.E. Shuler, Nonlinear sensitivity analysis of multi-parameter model systems. J. Comput. Phys. 26, 1–42 (1978)CrossRefzbMATHGoogle Scholar
  24. 24.
    O. David, J.C. Ascough, W. Lloyd, T.R. Green, K.W. Rojas, G.H. Leavesley, L.R. Ahuja, A software engineering perspective on environmental modeling framework design. Environ. Model. Softw. 39, 201–213 (2013)CrossRefGoogle Scholar
  25. 25.
    L. Devroye, L. Gyorfi, Nonparametric Density Estimation: The L \(_1\) View (Wiley, New York, 1985)Google Scholar
  26. 26.
    A. Doostan, H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    W. Gautschi, A Survey of Gauss-Christoffel Quadrature Formulae Birkhäuser Basel, 1981), pp. 72–147Google Scholar
  28. 28.
    G. Geraci, M.S. Eldred, G. Iaccarino, A multifidelity multilevel monte carlo method for uncertainty propagation in aerospace applications, in 19th AIAA Non-Deterministic Approaches Conference (AIAA SciTech Forum, 2017)Google Scholar
  29. 29.
    T. Gerstner, M. Griebel, Numerical integration using sparse grids. Numer. Algorithms 18(3–4), 209–232 (1998)Google Scholar
  30. 30.
    T. Gerstner, M. Griebel, Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003)Google Scholar
  31. 31.
    R.G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach (Springer, New York, NY, USA, 1991)CrossRefzbMATHGoogle Scholar
  32. 32.
    M.B. Giles, Multilevel monte carlo methods. Acta Numer. 24, 259–328 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    W.R. Gilks, S. Richardson, D. Spiegelhalter, Markov Chain Monte Carlo in Practice. Chapman & Hall/CRC Interdisciplinary Statistics (Taylor & Francis, 1995)Google Scholar
  34. 34.
    A. Gorodetsky, Y. Marzouk, Mercer kernels and integrated variance experimental design: Connections between gaussian process regression and polynomial approximation. SIAM/ASA J. Uncertain. Quantif. 4(1), 796–828 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    J.B. Gregersen, P.J.A. Gijsbers, S.J.P. Westen, Openmi: open modelling interface. J. Hydroinform. 9(3), 175–191 (2007)CrossRefGoogle Scholar
  36. 36.
    J.H.A. Guillaume, R.J. Hunt, A. Comunian, R.S. Blakers, B. Fu, Methods for exploring uncertainty in groundwater management predictions, in Integrated Groundwater Management: Concepts, Approaches and Challenges, ed. by A.J. Jakeman, O. Barreteau, R.J. Hunt, J.D. Rinaudo, A. Ross (Springer International Publishing, Cham, 2016), pp. 711–737Google Scholar
  37. 37.
    J.H.A. Guillaume, M. Arshad, A.J. Jakeman, M. Jalava, M. Kummu, Robust discrimination between uncertain management alternatives by iterative reflection on crossover point scenarios: Principles, design and implementations. Environ. Model. Softw. 83, 326–343 (2016)CrossRefGoogle Scholar
  38. 38.
    E. Haber, Z. Magnant, C. Lucero, L. Tenorio, Numerical methods for a-optimal designs with a sparsity constraint for ill-posed inverse problems. Comput. Optim. Appl. 52(1), 293–314 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    P. Hall, S.J. Sheather, M.C. Jones, J.S. Marron, On optimal data-based bandwidth selection in kernel density estimation. Biometrika 78(2), 263–269 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    J.H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    J. Hampton, A. Doostan, Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280, 363–386 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    M. Hegland, G. Hooker, S. Roberts, Finite element thin plate splines in density estimation. ANZIAM J. 42, 712–734 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    M. Hohenbichler, R. Rackwitz, First-order concepts in system reliability. Struct. Saf. 1(3), 177–188 (1982)CrossRefGoogle Scholar
  44. 44.
    L. Horesh, E. Haber, L. Tenorio, Optimal Experimental Design for the Large-Scale Nonlinear Ill-Posed Problem of Impedance Imaging (Wiley, 2010), pp. 273–290Google Scholar
  45. 45.
    G.M. Hornberger, R.C. Spear, An approach to the preliminary analysis of environmental systems. J. Environ. Manag. 12, 8–18 (1981)Google Scholar
  46. 46.
    R.W. Hut, N.C. van de Giesen, N. Drost, Comment on most computational hydrology is not reproducible, so is it really science?, in Let Hydrologists Learn the Latest Computer Science by Working with Research Software Engineers (rses) and not Reinvent the Waterwheel Ourselves, ed. by C. Hutton et al. (Water Resources Research, 2017)Google Scholar
  47. 47.
    C. Hutton, T. Wagener, J. Freer, D. Han, C. Duffy, B. Arheimer, Most computational hydrology is not reproducible, so is it really science? Water Resour. Res. 52(10), 7548–7555 (2016)CrossRefGoogle Scholar
  48. 48.
    A.J. Jakeman, R.A. Letcher, Integrated assessment and modelling: features, principles and examples for catchment management. Environ. Model. Softw. 18(6), 491 – 501, 2003. Applying Computer Research to Environmental ProblemsGoogle Scholar
  49. 49.
    A.J. Jakeman, R.A. Letcher, J.P. Norton, Ten iterative steps in development and evaluation of environmental models. Environ. Model. Softw. 21(5), 602–614 (2006)CrossRefGoogle Scholar
  50. 50.
    A.J. Jakeman, O. Barreteau, R.J. Hunt, J.D. Rinaudo, A. Ross (eds.), Integrated Groundwater Management: Concepts, Approaches and Challenges (Springer International Publishing, 2016)Google Scholar
  51. 51.
    J.D. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty. J. Comput. Phys. 229(12), 4648–4663 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    J.D. Jakeman, S.G. Roberts, Local and dimension adaptive stochastic collocation for uncertainty quantification, in Sparse Grids and Applications, vol. 88, Lecture Notes in Computational Science and Engineering, ed. by J. Garcke, M. Griebel (Springer, Berlin Heidelberg, 2013), pp. 181–203CrossRefGoogle Scholar
  53. 53.
    J.D. Jakeman, T. Wildey, Enhancing adaptive sparse grid approximations and improving refinement strategies using adjoint-based a posteriori error estimates. J. Comput. Phys. 280, 54–71 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    M.A. Janssen, The practice of archiving model code of agent-based models. J. Artif. Soc. Soc. Simul. 20(1), 2 (2017)MathSciNetCrossRefGoogle Scholar
  55. 55.
    J.L. Jennifer, J.M. Gilbert, P.G. Constantine, R.M. Maxwell, Reprint of: active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model. Comput. Geosci. 90, 78–89 (2016)Google Scholar
  56. 56.
    M.C. Jones, J.S. Marron, S.J. Sheather, A brief survey of bandwidth selection for density estimation. J. Am. Stat. Assoc. 91(433), 401–407 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems (Springer, 2005)Google Scholar
  58. 58.
    A. Der Kiureghian, H.Z. Lin, S.J. Hwang, Second-order reliability approximations. J. Eng. Mech. 113(8), 1208–1225 (1987)CrossRefGoogle Scholar
  59. 59.
    J.H. Kwakkel, E. Pruyt, Exploratory modeling and analysis, an approach for model-based foresight under deep uncertainty. Technol. Forecast. Soci. Change 80(3), 419 – 431, 2013. Future-Oriented Technology AnalysisGoogle Scholar
  60. 60.
    R.J. Lempert, A new decision sciences for complex systems. Proc. Natl. Acad. Sci. 99(suppl 3), 7309–7313 (2002)CrossRefGoogle Scholar
  61. 61.
    R.J. Lempert, D.G. Groves, S.W. Popper, S.C. Bankes, A general, analytic method for generating robust strategies and narrative scenarios. Manag. Sci. 52(4), 514–528 (2006)CrossRefGoogle Scholar
  62. 62.
    R.A. Kelly (Letcher), A.J. Jakeman, O. Barreteau, M.E. Borsuk, S. ElSawah, S.H. HAmilton, H.J. Henriksen, S. Kuikka, H.R. Maier, A.E. Rizzoli, H. van Delden, A.A. Voinov, Selecting among five common modelling approaches for integrated environmental assessment and management. Environ. Model. Softw. 47, 159–181 (2013)Google Scholar
  63. 63.
    J. Li, J. Li, D. Xiu, An efficient surrogate-based method for computing rare failure probability. J. Comput. Phys. 230(24), 8683–8697 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Q. Liao, K. Willcox, A domain decomposition approach for uncertainty analysis. SIAM J. Sci. Comput. 37(1), A103–A133 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    H. Liu, J.D. Lafferty, L.A. Wasserman, Sparse nonparametric density estimation in high dimensions using the rodeo, in AISTATS (2007), pp. 283–290Google Scholar
  66. 66.
    J.C. Mattingly, N.S. Pillai A.M. Stuart, Diffusion limits of the random walk metropolis algorithm in high dimensions. Ann. Appl. Probab. 22(3), 881–930 (2012). 06Google Scholar
  67. 67.
    M.D. Morris, Factorial sampling plans for preliminary computational experiments. Technometrics 33(2), 161–174 (1991)CrossRefGoogle Scholar
  68. 68.
    A. Narayan, C. Gittelson, D. Xiu, A stochastic collocation algorithm with multifidelity models. SIAM J. Sci. Comput. 36(2), A495–A521 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    A. Narayan, J.D. Jakeman, T. Zhou, A Christoffel function weighted least squares algorithm for collocation approximations. Math. Comput. 86, 1913–1947 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    L.W.-T. Ng, M. Eldred, Multifidelity uncertainty quantification using non-intrusive polynomial chaos and stochastic collocation, in 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 20th AIAA/ASME/AHS Adaptive Structures Conference 14th AIAA (2012), p. 1852Google Scholar
  71. 71.
    F. Nobile, R. Tempone, C.G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309–2345 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    S.D. Peckham, E.W.H. Hutton, B. Norris, A component-based approach to integrated modeling in the geosciences. Comput. Geosci. 53, 3–12 (2013)CrossRefGoogle Scholar
  73. 73.
    B. Peherstorfer, K. Willcox, M. Gunzburger, Optimal model management for multifidelity monte carlo estimation. SIAM J. Sci. Comput. (2016). to appearGoogle Scholar
  74. 74.
    D. Pflüger, B. Peherstorfer, H.-J. Bungartz, Spatially adaptive sparse grids for high-dimensional data-driven problems. J. Complex. 26(5), 508–522 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    C.E. Rasmussen, Gaussian processes to speed up hybrid monte carlo for expensive bayesian integrals, in Bayesian Statistics, ed. by J.M Bernardo, A.P. Dawid, J.O Berger, M. West, D. Heckerman, M.J. Bayarri, F.M.A. Smith, vol. 7 (Oxford University Press, 2003), pp. 651–659Google Scholar
  76. 76.
    C.E. Rasmussen, C. Williams, Gaussian Processes for Machine Learning (MIT Press, 2006)Google Scholar
  77. 77.
    J.C. Refsgaard, J.P. van der Sluijs, A.L. Hjberg, P.A. Vanrolleghem, Uncertainty in the environmental modelling process a framework and guidance. Environ. Model. Softw. 22(11), 1543–1556 (2007)CrossRefGoogle Scholar
  78. 78.
    G.O. Roberts, J.S. Rosenthal, Optimal scaling for various metropolis-hastings algorithms. Stat. Sci. 16(4), 351–367 (2001). 11Google Scholar
  79. 79.
    J.O. Royset, R.J.-B. Wets, Fusion of hard and soft information in nonparametric density estimation. Eur. J. Oper. Res. 247(2), 532–547 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    A. Saltelli, R. Bolado, An alternative way to compute Fourier amplitude sensitivity test (fast). Comput. Stat. Data Anal. 26(4), 445–460 (1998)CrossRefzbMATHGoogle Scholar
  81. 81.
    A. Saltelli, K. Chan, E. Scott, Sensitivity Analysis (Wiley, New York, 2004)Google Scholar
  82. 82.
    S. Sankararaman, S. Mahadevan, Likelihood-based approach to multidisciplinary analysis under uncertainty. J. Mech. Des. 134(3) (2012)Google Scholar
  83. 83.
    D.W. Scott, Multivariate density estimation: theory, practice, and visualization (Wiley, 2015)Google Scholar
  84. 84.
    S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)zbMATHGoogle Scholar
  85. 85.
    I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1(4), 407–414 (1993)MathSciNetzbMATHGoogle Scholar
  86. 86.
    M. Sobol’, B.V. Shukhman, Integration with quasi random sequences: numerical experience. Int. J. Mod. Phys. C 6(2), 263–275 (1995)CrossRefzbMATHGoogle Scholar
  87. 87.
    R.C. Spear, T.M. Grieb, N. Shang, Parameter uncertainty and interaction in complex environmental models. Water Resour. Res. 30(11), 3159–3169 (1994)CrossRefGoogle Scholar
  88. 88.
    R. Srinivasan, Importance Sampling: Applications in Communications and Detection (Springer, 2002)Google Scholar
  89. 89.
    A.H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, N.J., 1971)Google Scholar
  90. 90.
    A.M. Stuart, Inverse problems: a bayesian perspective. Acta Numer. 19, 451–559 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    B. Sudret, Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)Google Scholar
  92. 92.
    D.G. Tarboton, A. Sharma, U. Lall, Disaggregation procedures for stochastic hydrology based on nonparametric density estimation. Water Resour. Res. 34(1), 107–119 (1998)CrossRefGoogle Scholar
  93. 93.
    G.R. Terrell, D.W. Scott, Variable kernel density estimation. Ann. Stat. 20(3), 1236–1265 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  94. 94.
    J.R. Thompson, R.A. Tapiam, Nonparametric function estimation, modeling, and simulation. SIAM (1990)Google Scholar
  95. 95.
    J.P. Van Der Sluijs, M. Craye, S. Funtowicz, P. Kloprogge, J. Ravetz, J. Risbey, Combining quantitative and qualitative measures of uncertainty in model-based environmental assessment: The nusap system. Risk Anal. 25(2), 481–492 (2005)CrossRefGoogle Scholar
  96. 96.
    S. Walsh, T. Wildey, J.D. Jakeman, A consistent bayesian formulation for stochastic inverse problems based on push-forward measures. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part B: Mech. Eng. (2016). acceptedGoogle Scholar
  97. 97.
    R.E. Wengert, A simple automatic derivative evaluation program. Commun. ACM 7(8), 463–464 (1964)CrossRefzbMATHGoogle Scholar
  98. 98.
    D. Xiu, J.S. Hesthaven, High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  99. 99.
    D. Xiu, G.E. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  100. 100.
    D. Xiu, Numerical integration formulas of degree two. Appl. Numer. Math. 58(10), 1515–1520 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Australian National UniversityCanberraAustralia
  2. 2.Sandia National LaboratoriesAlbuquerqueUSA

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