Accurate uncertainty quantification using inaccurate computational models

SIAM Journal on Scientific Computing

Volumn 31, Issue 5, 2009, Pages 3274-3300

  Koutsourelakis, Phaedon Stelios ^a  

^a School of Chemical and Molecular Engineering   (United States)

Author keywords

Bayesian; Monte Carlo; Nonparametric regression; Uncertainty quantification

Indexed keywords

COMPUTATION THEORY; COMPUTATIONAL METHODS; UNCERTAINTY ANALYSIS;

BAYESIAN; BAYESIAN FORMULATION; COARSENINGS; COMPUTATIONAL MODELLING; MONTE CARLO; MONTE CARLO SAMPLING; NON-GAUSSIAN UNCERTAINTIES; NON-PARAMETRIC REGRESSION; STATE-OF-THE-ART TECHNIQUES; UNCERTAINTY QUANTIFICATIONS;

MONTE CARLO METHODS;

EID: 68949186013     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/080733565     Document Type: Article

References (44)

1. C. ANDRIEU, N. DE FREITAS, AND A. DouCET, Robust full Bayesian learning for radial basis networks, Neural Comput., 13 (2001), pp. 2359-2407.
2. S.K. Au AND J. BECK, Estimation of small failure probabilities in high dimensions by subset simulation, Probab. Eng. Mech., 16 (2001), pp. 263-277.
3. G.I. Barenblatt, The mathematical theory of equilibrium of cracks in brittle fracture,Adv. Appl. Mech., 7 (1962), pp. 55-129.
4. G.T. CAmACho AND M. ORTIz, Computational modeling ofimpact damage in brittle materials, Internat. J. Solids Structures, 33 (1996), pp. 2899-2938.
5. Y.Z. CAo, M.Y. HuSSAINI, AND T.A. ZANg, Exploitation ofsensitivity derivatives for improving sampling methods, AIAA J., 42 (2004), pp. 815-822.
6. Y.Z. Cao, M.Y. Hussaini, T. Zang, and A. Zatezalo, A variance reduction method based on sensitivity derivatives, Appl. Numer. Math., 56 (2006), pp. 800-813.
7. N. Chopin, Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference, Ann. Statist., 32 (2004), pp. 2385-2411.
8. P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, New York, 2004.
9. P. Del Moral, A. DoucET, AND A. JASRA, Sequential Monte Carlo for Bayesian computation (with discussion), in Bayesian Statistics 8, Oxford University Press, London, 2006.
10. P. Del Moral, A. DoucET, AND A. JASRAu, Sequential Monte Carlo Samplers,J. Royal Statist. Soc. Ser. B, 68 (2006), pp. 411-436.
11. A. DoucET, M. BRIERS, AND S. Senecal, Efficient block sampling strategies for sequential Monte Carlo, J. Comput. Graph. Statist., 15 (2006), pp. 693-711.
12. A. DoucET, J.F.G. DE FREITAS, AND N.J. Gordon, EDS., Sequential Monte Carlo Methods in Practice, Springer, New York, 2001.
13. D.S. DuGDALE, Yielding of steel sheets containing clits, J. Mech. Phys. Solids, 8 (1960), pp. 100104.
14. K.T. FANG, R. Li, AND A. SuDJIANTo, Design and Modeling for Computer Experiments, Chapman and Hall/CRC, London, 2005.
15. B. GANAPATHYSuBRAMANIAN AND N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comput. Phys., 225 (2007), pp. 652-685.
16. A. Gelman, J.B. CARLIN, H.S. STERN, AND D.B. Rubin, Bayesian Data Analysis,2nd ed., Chapman and Hall/CRC, London, 2003.
17. A. GELMAN AND X.L. Meng, Simulating normalizing constants: From importance sampling to bridge sampling to path sampling, Statist. Sci., 13 (1998), pp. 163-185.
18. R. GHANEM AND P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, New York, 1991.
19. P.J. GREEN, Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika, 82 (1995), pp. 711-732.
20. C.C. Holmes AND K. Held, Bayesian auxiliary variable models for binary and polychotomous regression, Bayesian Anal., 1 (2006), pp. 145-168.
21. W. Jefferys AND J. BERGER, Ockhams razor and Bayesian analysis, Amer. Sci., 80 (1992), pp. 64-72.
22. M.C. KENNEDY AND A. O'HAGAN, Predicting the output from a complex computer code when fast approximations are available, Biometrika, 87 (2000), pp. 1-13.
23. G.S. Kimeldorf AND G. WAHBA, A correspondence between Bayesian estimation on stochastic processes and smoothing by splines, Ann. Math. Statist., 41 (1971), pp. 495-502.
24. G. Klopfer, M.Y. Hussaini, P. Ngnepieba, and A. Zatezalo, Quantification of uncertainty associated with low-fidelity simulations, in Proceedings of the Sixth International Conference on Engineering Computational Technology, Scotland, 2008.
25. P.S. KouTSouRELAKIS, Design of complex systems in the presence of large uncertainties: A statistical approach, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 4092-4103.
26. P.S. KouTSouRELAKIS, Uncertainty Quantification in Complex Systems using Approximate Solvers, http://arxiv.org/abs/0808.3416.
27. M.S. Lewicki AND T.J. SEJNowSKI, Learning overcomplete representations, Neural Comput., 12 (2000), pp. 337-365.
28. F. LIANG, M. Liao, M. MuKHERJEE, AND S. NAD West, Nonparametric Bayesian Kernel Models, Technical report, ISDS discussion paper, Duke University, Durham, NC, 2006.
29. J.S. Liu, Monte Carlo Strategies in Scientific Computing, Springer Ser. Statist. 16, Springer, New York, 2001.
30. S.N. MacEachern, M. CLYDE, AND J.S. Liu, Sequential importance sampling for nonpara- metric Bayes models: The next generation, Canad. J. Statist., 27 (1998), pp. 251-267.
31. L. MATHELIN, M.Y. Hussaini, AND T.A. Zang, Stochastic approaches to uncertainty quantification in cfd simulations, Numer. Algorithms, 38 (2005), pp. 209-236.
32. I. Murray AND Z. GHAHRAMANI, A note on the evidence and Bayesian Occams razor, Technical report, Gatsby unit technical teport GCNU-TR 2005-003, University College London, London, 2005.
33. R.M. NEAL, Annealed importance sampling, Stat. Comput., 1 (2001), pp. 125-139.
34. M. ORTIZ AND A. PANDOLFI, Finite-deformation irreversible cohesive elements for three- dimensional crack propagation analysis, Internat. J. Numer. Methods Engrg., 44 (1999), pp. 1267-1282.
35. C.E. RASMUSSEN AND Z. GHAHRAMANI, Occam's razor, Neural Inf. Process. Syst. 13, 2001, pp. 294-300.
36. C.E. Rasmussen and Z. Ghahramani, Bayesian Monte Carlo, Adv. Neural Inf. Process. Sys. 15, MIT Press, Cambridge, MA, 2003.
37. C.P. Robert AND G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, New York, 2004.
38. J. Sacks, W.J. Welch, T. Mitchell, AND H.P. Wynn, Design and analysis of computer experiments, Statist. Sci., 4 (1989), pp. 409-423.
39. T.J. SANTNER, B.J. WILLIAMS, AND W.I. Notz, The Design and Analysis of Computer Experiments, Springer, New York, 2003.
40. M.E. TIppINg, Sparse Bayesian learning and the relevance vector machine, J. Mach. Learn.Res., 1 (2001), pp. 211-244.
41. N. WIENER, The homogeneous chaos, Amer. J. Math., 60 (1938), pp. 897-936.
42. D. XIU AND J.S. Hesthaven, High-order collocation methods for the differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118-1139.
43. X.P. Xu AND A. NEEDLEMAN, Numerical simulations of fast crack growth in brittle solids,J. Mech. Phys. Solids, 42 (1994), pp. 1397-1434.
44. C.L.Y. YEONG AND S. TORQUATO, Reconstructing random media I and II, Phys. Rev. E (3), 58 (1998), pp. 224-233.