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FMfI 2016: Agriculture as a Metaphor for Creativity in All Human Endeavors pp 147–171Cite as

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

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Part of the book series: Mathematics for Industry ((MFI,volume 28))

Abstract

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.

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Notes

  1. 1.

    In this example, we will assume that the coupling between \(F_1\) and \(F_2\) has been broken.

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. A.C. Atkinson, A.N. Donev. Optimum Experimental Designs (Oxford University Press, 1992)

    Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  8. M.A. Beaumont, W. Zhang, D.J. Balding, Approximate bayesian computation in population genetics. Genetics 162(4), 2025–2035 (2002)

    Google Scholar 

  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. K. Beven, A. Binley, The future of distributed models: model calibration and uncertainty prediction. Hydrol. Process. 6(3), 279–298 (1992)

    Article  Google Scholar 

  11. H.G. Bock, S. Körkel, J.P. Schlöder, Parameter Estimation and Optimum Experimental Design for Differential Equation Models (Springer, Berlin, 2013), pp. 1–30

    Google Scholar 

  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. 3–27

    Google Scholar 

  13. A. Bucklew, Introduction to Rare Event Simulation (Springer, 2004)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. H.-J. Bungartz, M. Griebel, Sparse grids. Acta Numer. 13, 147–269 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Chaloner, I. Verdinelli, Bayesian experimental design: a review. Stat. Sci. 10(3), 273–304, 08 (1995)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. P.G. Constantine, Active subspaces: emerging ideas for dimension reduction in parameter studies. SIAM (2015)

    Google Scholar 

  20. P.G. Constantine, M.S. Eldred, E.T. Phipps, Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229–232, 1–12 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  25. L. Devroye, L. Gyorfi, Nonparametric Density Estimation: The L \(_1\) View (Wiley, New York, 1985)

    Google Scholar 

  26. A. Doostan, H. Owhadi, A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Gautschi, A Survey of Gauss-Christoffel Quadrature Formulae Birkhäuser Basel, 1981), pp. 72–147

    Google Scholar 

  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. T. Gerstner, M. Griebel, Numerical integration using sparse grids. Numer. Algorithms 18(3–4), 209–232 (1998)

    Google Scholar 

  30. T. Gerstner, M. Griebel, Dimension-adaptive tensor-product quadrature. Computing 71(1), 65–87 (2003)

    Google Scholar 

  31. R.G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach (Springer, New York, NY, USA, 1991)

    Book  MATH  Google Scholar 

  32. M.B. Giles, Multilevel monte carlo methods. Acta Numer. 24, 259–328 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  35. J.B. Gregersen, P.J.A. Gijsbers, S.J.P. Westen, Openmi: open modelling interface. J. Hydroinform. 9(3), 175–191 (2007)

    Article  Google Scholar 

  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. 711–737

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. J. Hampton, A. Doostan, Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280, 363–386 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Hegland, G. Hooker, S. Roberts, Finite element thin plate splines in density estimation. ANZIAM J. 42, 712–734 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Hohenbichler, R. Rackwitz, First-order concepts in system reliability. Struct. Saf. 1(3), 177–188 (1982)

    Article  Google Scholar 

  44. L. Horesh, E. Haber, L. Tenorio, Optimal Experimental Design for the Large-Scale Nonlinear Ill-Posed Problem of Impedance Imaging (Wiley, 2010), pp. 273–290

    Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  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 Problems

    Google Scholar 

  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)

    Article  Google Scholar 

  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. J.D. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty. J. Comput. Phys. 229(12), 4648–4663 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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. 181–203

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  54. M.A. Janssen, The practice of archiving model code of agent-based models. J. Artif. Soc. Soc. Simul. 20(1), 2 (2017)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  57. J. Kaipio, E. Somersalo, Statistical and Computational Inverse Problems (Springer, 2005)

    Google Scholar 

  58. A. Der Kiureghian, H.Z. Lin, S.J. Hwang, Second-order reliability approximations. J. Eng. Mech. 113(8), 1208–1225 (1987)

    Article  Google Scholar 

  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 Analysis

    Google Scholar 

  60. R.J. Lempert, A new decision sciences for complex systems. Proc. Natl. Acad. Sci. 99(suppl 3), 7309–7313 (2002)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. J. Li, J. Li, D. Xiu, An efficient surrogate-based method for computing rare failure probability. J. Comput. Phys. 230(24), 8683–8697 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  64. Q. Liao, K. Willcox, A domain decomposition approach for uncertainty analysis. SIAM J. Sci. Comput. 37(1), A103–A133 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  65. H. Liu, J.D. Lafferty, L.A. Wasserman, Sparse nonparametric density estimation in high dimensions using the rodeo, in AISTATS (2007), pp. 283–290

    Google Scholar 

  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). 06

    Google Scholar 

  67. M.D. Morris, Factorial sampling plans for preliminary computational experiments. Technometrics 33(2), 161–174 (1991)

    Article  Google Scholar 

  68. A. Narayan, C. Gittelson, D. Xiu, A stochastic collocation algorithm with multifidelity models. SIAM J. Sci. Comput. 36(2), A495–A521 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  69. A. Narayan, J.D. Jakeman, T. Zhou, A Christoffel function weighted least squares algorithm for collocation approximations. Math. Comput. 86, 1913–1947 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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. 1852

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  73. B. Peherstorfer, K. Willcox, M. Gunzburger, Optimal model management for multifidelity monte carlo estimation. SIAM J. Sci. Comput. (2016). to appear

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 651–659

    Google Scholar 

  76. C.E. Rasmussen, C. Williams, Gaussian Processes for Machine Learning (MIT Press, 2006)

    Google Scholar 

  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)

    Article  Google Scholar 

  78. G.O. Roberts, J.S. Rosenthal, Optimal scaling for various metropolis-hastings algorithms. Stat. Sci. 16(4), 351–367 (2001). 11

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  80. A. Saltelli, R. Bolado, An alternative way to compute Fourier amplitude sensitivity test (fast). Comput. Stat. Data Anal. 26(4), 445–460 (1998)

    Article  MATH  Google Scholar 

  81. A. Saltelli, K. Chan, E. Scott, Sensitivity Analysis (Wiley, New York, 2004)

    Google Scholar 

  82. S. Sankararaman, S. Mahadevan, Likelihood-based approach to multidisciplinary analysis under uncertainty. J. Mech. Des. 134(3) (2012)

    Google Scholar 

  83. D.W. Scott, Multivariate density estimation: theory, practice, and visualization (Wiley, 2015)

    Google Scholar 

  84. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)

    MATH  Google Scholar 

  85. I.M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1(4), 407–414 (1993)

    MathSciNet  MATH  Google Scholar 

  86. M. Sobol’, B.V. Shukhman, Integration with quasi random sequences: numerical experience. Int. J. Mod. Phys. C 6(2), 263–275 (1995)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  88. R. Srinivasan, Importance Sampling: Applications in Communications and Detection (Springer, 2002)

    Google Scholar 

  89. A.H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, N.J., 1971)

    Google Scholar 

  90. A.M. Stuart, Inverse problems: a bayesian perspective. Acta Numer. 19, 451–559 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  91. B. Sudret, Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)

    Google Scholar 

  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)

    Article  Google Scholar 

  93. G.R. Terrell, D.W. Scott, Variable kernel density estimation. Ann. Stat. 20(3), 1236–1265 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  94. J.R. Thompson, R.A. Tapiam, Nonparametric function estimation, modeling, and simulation. SIAM (1990)

    Google Scholar 

  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)

    Article  Google Scholar 

  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). accepted

    Google Scholar 

  97. R.E. Wengert, A simple automatic derivative evaluation program. Commun. ACM 7(8), 463–464 (1964)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  100. D. Xiu, Numerical integration formulas of degree two. Appl. Numer. Math. 58(10), 1515–1520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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  1. Australian National University, Canberra, ACT, Australia

    A. J. Jakeman

  2. Sandia National Laboratories, Albuquerque, NM, USA

    J. D. Jakeman

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  1. Mathematics, Information and Statistics, CSIRO, Acton, Aust Capital Terr, Australia

    Robert S. Anderssen

  2. La Trobe University, Melbourne, Victoria, Australia

    Philip Broadbridge

  3. Kyushu University, Fukuoka, Japan

    Yasuhide Fukumoto

  4. Kyushu University, Fukuoka, Japan

    Kenji Kajiwara

  5. Queensland University of Technology, Brisbane, Queensland, Australia

    Matthew Simpson

  6. Queensland University of Technology, Brisbane, Queensland, Australia

    Ian Turner

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Jakeman, A.J., Jakeman, J.D. (2018). An Overview of Methods to Identify and Manage Uncertainty for Modelling Problems in the Water–Environment–Agriculture Cross-Sector. In: Anderssen, R., Broadbridge, P., Fukumoto, Y., Kajiwara, K., Simpson, M., Turner, I. (eds) Agriculture as a Metaphor for Creativity in All Human Endeavors. FMfI 2016. Mathematics for Industry, vol 28. Springer, Singapore. https://doi.org/10.1007/978-981-10-7811-8_15

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