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A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. (English) Zbl 1230.76049
Summary: An overview of a comprehensive framework is given for estimating the predictive uncertainty of scientific computing applications. The framework is comprehensive in the sense that it treats both types of uncertainty (aleatory and epistemic), incorporates uncertainty due to the mathematical form of the model, and it provides a procedure for including estimates of numerical error in the predictive uncertainty. Aleatory (random) uncertainties in model inputs are treated as random variables, while epistemic (lack of knowledge) uncertainties are treated as intervals with no assumed probability distributions. Approaches for propagating both types of uncertainties through the model to the system response quantities of interest are briefly discussed. Numerical approximation errors (due to discretization, iteration, and computer round off) are estimated using verification techniques, and the conversion of these errors into epistemic uncertainties is discussed. Model form uncertainty is quantified using (a) model validation procedures, i.e., statistical comparisons of model predictions to available experimental data, and (b) extrapolation of this uncertainty structure to points in the application domain where experimental data do not exist. Finally, methods for conveying the total predictive uncertainty to decision makers are presented. The different steps in the predictive uncertainty framework are illustrated using a simple example in computational fluid dynamics applied to a hypersonic wind tunnel.

MSC:
76M99 Basic methods in fluid mechanics
76K05 Hypersonic flows
00A72 General theory of simulation
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[1] Oberkampf, W.L.; Roy, C.J., Verification and validation in scientific computing, (2010), Cambridge University Press Cambridge · Zbl 1211.68499
[2] Morgan, M.G.; Henrion, M., Uncertainty: A guide to dealing with uncertainty in quantitative risk and policy analysis, (1990), Cambridge University Press Cambridge, UK
[3] Cullen, A.C.; Frey, H.C., Probabilistic techniques in exposure assessment: A handbook for dealing with variability and uncertainty in models and inputs, (1999), Plenum Press New York
[4] Vose, D., Risk analysis: A quantitative guide, (2008), Wiley New York
[5] Haimes, Y.Y., Risk modeling, assessment, and management, (2009), John Wiley New York · Zbl 1105.91001
[6] Bedford, T.; Cooke, R., Probabilistic risk analysis: foundations and methods, (2001), Cambridge University Press Cambridge, UK · Zbl 0977.60002
[7] Ghosh, J.K.; Delampady, M.; Samanta, T., An introduction to Bayesian analysis: theory and methods, (2006), Springer Berlin · Zbl 1135.62002
[8] Sivia, D.; Skilling, J., Data analysis: A Bayesian tutorial, (2006), Oxford University Press Oxford · Zbl 1102.62001
[9] AIAA, Guide for the verification and validation of computational fluid dynamics simulations, American Institute of Aeronautics and Astronautics, AIAA-G-077-1998, Reston, VA, 1998.
[10] ASME, Guide for Verification and Validation in Computational Solid Mechanics, American Society of Mechanical Engineers, ASME Standard V&V 10-2006, New York, NY, 2006.
[11] Coleman, H.W.; Stern, F., Uncertainties and CFD code validation, J. fluids engrg., 119, 795-803, (1997)
[12] Stern, F.; Wilson, R.V.; Coleman, H.W.; Paterson, E.G., Comprehensive approach to verification and validation of CFD simulations – part 1: methodology and procedures, J. fluids engrg., 123, 793-802, (2001)
[13] Ferson, S.; Oberkampf, W.L.; Ginzburg, L., Model validation and predictive capability for the thermal challenge problem, Comput. methods appl. mech. engrg., 197, 2408-2430, (2008) · Zbl 1388.74029
[14] Ferson, S.; Ginzburg, L.R., Different methods are needed to propagate ignorance and variability, Reliab. engrg. syst. safety, 54, 133-144, (1996)
[15] Ferson, S.; Hajagos, J.G., Arithmetic with uncertain numbers: rigorous and (often) best possible answers, Reliab. engrg. syst. safety, 85, 135-152, (2004)
[16] C.J. Roy, Review of discretization error estimators in scientific computing, in: AIAA Paper 2010-0126, 2010.
[17] Roache, P.J., Verification and validation in computational science and engineering, (1998), Hermosa Publishers Albuquerque, New Mexico
[18] Bank, R.E., Hierarchical bases and the finite element method, Acta numer., 5, 1-45, (1996) · Zbl 0865.65078
[19] Zienkiewicz, O.C.; Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates, part 2: error estimates and adaptivity, Int. J. num. meth. engrg., 33, 1365-1382, (1992) · Zbl 0769.73085
[20] Cavallo, P.A.; Sinha, N., Error quantification for computational aerodynamics using an error transport equation, J. aircraft, 44, 1954-1963, (2007)
[21] T.I-P. Shih, B.R. Williams, Development and evaluation of an a posteriori method for estimating and correcting grid-induced errors in solutions of the Navier-Stokes equations, in: AIAA Paper 2009-1499, 2009.
[22] Skeel, R.D., Thirteen ways to estimate global error, Numer. math., 48, 1-20, (1986) · Zbl 0562.65050
[23] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, Comput. methods appl. mech. engrg., 142, 1-88, (1997) · Zbl 0895.76040
[24] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), Wiley Interscience New York · Zbl 1008.65076
[25] Pierce, N.A.; Giles, M.B., Adjoint recovery of superconvergent functionals from PDE approximations, SIAM review, 42, 247-264, (2000) · Zbl 0948.65119
[26] Roy, C.J., Review of code and solution verification procedures for computational simulation, J. comput. phys., 205, 131-156, (2005) · Zbl 1072.65118
[27] Roache, P.J., Perspective: a method for uniform reporting of grid refinement studies, J. fluids engrg., 116, 405-413, (1994)
[28] Shafer, G., A mathematical theory of evidence, (1976), Princeton University Press · Zbl 0359.62002
[29] Ghanem, R.; Spanos, P., Stochastic finite elements: A spectral approach, (1991), Springer-Verlag · Zbl 0722.73080
[30] Najm, H., Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, Ann. rev. fluid mech., 41, 35-52, (2009) · Zbl 1168.76041
[31] M.S. Eldred, L.P. Swiler, Efficient algorithms for mixed aleatory-epistemic uncertainty quantification with application to radiation-hardened electronics. Part I: Algorithms and Benchmark Results, Sandia National Laboratories, SAND2009-5805, Albuquerque, NM, 2009.
[32] M.S. Eldred, C.G. Webster, P. Constantine, Evaluation of non-intrusive approaches for Wiener-Askey generalized polynomial chaos, in: AIAA Paper 2008-1892, 2008.
[33] M.S. Eldred, Recent advances in non-intrusive polynomial chaos and stochastic collocation methods for uncertainty analysis and design, in: AIAA Paper 2009-2274, 2009.
[34] Giunta, A.A.; McFarland, J.M.; Swiler, L.P.; Eldred, M.S., The promise and peril of uncertainty quantification using response surface approximations, Struct. infrastruct. engrg., 2, 175-189, (2006)
[35] Coleman, H.W.; Stern, F., Uncertainties and CFD code validation, J. fluids engrg., 119, 795-803, (1997)
[36] Oberkampf, W.L.; Trucano, T.G., Verification and validation in computational fluid dynamics, Prog. aerospace sci., 38, 209-272, (2002)
[37] Oberkampf, W.L.; Barone, M.F., Measures of agreement between computation and experiment: validation metrics, J. comput. phys., 217, 5-36, (2006) · Zbl 1147.76606
[38] W.L. Oberkampf, S. Ferson, Model validation under both aleatory and epistemic uncertainty, in: NATO/RTO Symposium on Computational Uncertainty in Military Vehicle Design, NATO, AVT-147/RSY-022, Athens, Greece, 2007.
[39] Babuska, I.; Nobile, F.; Tempone, R., A systematic approach to model validation based on Bayesian updates and prediction related rejection criteria, Comput. methods appl. mech. engrg., 197, 2517-2539, (2008) · Zbl 1139.74012
[40] T.G. Trucano, M. Pilch, W.L. Oberkampf, General Concepts for Experimental Validation of ASCI Code Applications, Sandia National Laboratories, SAND2002-0341, Albuquerque, NM, 2002.
[41] Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066
[42] van Leer, B., Towards the ultimate conservative difference scheme, V.A second order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223
[43] Helton, J.C., Uncertainty and sensitivity analysis in the presence of stochastic and subjective uncertainty, J. statist. comput. simul., 57, 3-76, (1997) · Zbl 0937.62004
[44] Sallaberry, C.J.; Helton, J.C.; Hora, S.C., Extension of Latin hypercube samples with correlated variables, Reliab. engrg. syst. safety, 93, 1047-1059, (2008)
[45] S. Ferson, W.T. Tucker, Sensitivity in Risk Analyses with Uncertain Numbers, Sandia National Laboratories, SAND2006-2801, Albuquerque, NM, 2006.
[46] Kreinovich, V.; Beck, J.; Ferregut, C.; Sanchez, A.; Keller, G.R.; Averill, M.; Starks, S.A., Monte-Carlo-type techniques for processing interval uncertainty, and their potential engineering applications, Reliab. comput., 13, 25-69, (2007) · Zbl 1106.65043
[47] Devore, J., Probability and statistics for engineering and the sciences, (2009), Brooks/Cole Belmont, CA
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