Advertisement

Adaptive method for indirect identification of the statistical properties of random fields in a Bayesian framework

  • Guillaume PerrinEmail author
  • Christian Soize
Original paper
  • 54 Downloads

Abstract

This work considers the challenging problem of identifying the statistical properties of random fields from indirect observations. To this end, a Bayesian approach is introduced, whose key step is the nonparametric approximation of the likelihood function from limited information. When the likelihood function is based on the evaluation of an expensive computer code, this work also proposes a method to select iteratively new design points to reduce the uncertainties on the results that are due to the approximation of the likelihood. Two applications are finally presented to illustrate the efficiency of the proposed procedure: a first one based on analytic data, and a second one dealing with the identification of the random elasticity field of an heterogeneous microstructure.

Keywords

Bayesian framework Uncertainty quantification Statistical inference Stochastic process Kernel density estimation 

Notes

References

  1. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723MathSciNetzbMATHCrossRefGoogle Scholar
  2. Arnst M, Ghanem R, Soize C (2010) Identification of bayesian posteriors for coefficients of chaos expansions. J Comput Phys 229(9):3134–3154MathSciNetzbMATHCrossRefGoogle Scholar
  3. Atwell J, King B (2001) Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math Comput Modell 33(1–3):1–19MathSciNetzbMATHCrossRefGoogle Scholar
  4. Auffray Y, Barbillon P, Marin JM (2012) Maximin design on non hypercube domains and kernel interpolation. Stat Comput 22(3):703–712MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bilionis I, Zabaras N (2015) Bayesian uncertainty propagation using gaussian processes. In: Ghanem R, Higdon D, Owhadi H (eds) Handbook of uncertainty quantification. Springer, New YorkzbMATHGoogle Scholar
  6. Box G, Jenkins G (1970) Time series analysis: forecasting and control. Holden-Day, San FranciscozbMATHGoogle Scholar
  7. Chen P, Schwab C (2015) Sparse-grid, reduced basis bayesian inversion. Comput Methods Appl Mech Eng 297:84–115MathSciNetzbMATHCrossRefGoogle Scholar
  8. Clouteau D, Cottereau R, Lombaert G (2013) Dynamics of structures coupled with elastic media—a review of numerical models and methods. J Sound Vib 332:2415–2436CrossRefGoogle Scholar
  9. Conrad PR, Marzouk YM, Pillai NS, Smith A (2016) Accelerating asymptotically exact MCMC for computationally intensive models via local approximations. J Am Stat Assoc 111:1591–1607MathSciNetCrossRefGoogle Scholar
  10. Conrad PR, Davis A, Marzouk YM, Pillai NS, Smith A (2018) Parallel local approximation MCMC for expensive models. SIAM/ASA J Uncertain Quantif 6(1):39–373MathSciNetzbMATHCrossRefGoogle Scholar
  11. Damblin G, Barbillonz P, Keller M, Pasanisi A, Parent E (2013) Adaptive numerical designs for the calibration of computer codes. SIAM/ASA J Uncertain Quantif 6(1):151–179MathSciNetzbMATHCrossRefGoogle Scholar
  12. Draguljić D, Santner TJ, Dean AM (2012) Noncollapsing space-filling designs for bounded nonrectangular regions. Technometrics 54(2):169–178MathSciNetCrossRefGoogle Scholar
  13. Emery J, Grigoriu M, Field RF Jr (2016) Bayesian methods for characterizing unknown parameters of material models. Appl Math Model 13–14:6395–6411MathSciNetCrossRefGoogle Scholar
  14. Fang K, Li R, Sudjianto A (2006) Design and modeling for computer experiments. Computer science and data analysis series. Chapman & Hall, LondonzbMATHGoogle Scholar
  15. Fang K, Lin D (2003) Uniform experimental designs and their applications in industry. Handb Stat 22:131–178MathSciNetCrossRefGoogle Scholar
  16. Fielding M, Nott DJ, Liong SY (2011) Efficient MCMC schemes for computationally expensive posterior distributions. Technometrics 53(1):16–28MathSciNetCrossRefGoogle Scholar
  17. Ghanem R, Spanos PD (2003) Stochastic finite elements: a spectral approach, rev edn. Dover Publications, New YorkzbMATHGoogle Scholar
  18. Guilleminot J, Soize C (2013) On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties. J Elast 111:109–130MathSciNetzbMATHCrossRefGoogle Scholar
  19. Higdon D, Lee H, Holloman C (2003) Markov chain monte carlo based approaches for inference in computationally intensive inverse problems. Bayesian Stat 7:181–197MathSciNetGoogle Scholar
  20. Higdon D, Gattiker J, Williams B, Rightley M (2008) Computer model calibration using high-dimensional output. J Am Stat Assoc 103(482):570–583MathSciNetzbMATHCrossRefGoogle Scholar
  21. Joseph VR, Gul E, Ba S (2015) Maximum projection designs for computer experiments. Biometrika 102(2):371–380MathSciNetzbMATHCrossRefGoogle Scholar
  22. Kennedy M, O’Hagan A (2001) Bayesian calibration of computer models. J R Stat Soc 63:425–464MathSciNetzbMATHCrossRefGoogle Scholar
  23. Lai WM, Rubin D, Krempl E (2010) Introduction to continuum mechanics. Elsevier, Inc, AmsterdamzbMATHGoogle Scholar
  24. Le Maître O, Knio O (2010) Spectral methods for uncertainty quantification. Springer, New YorkzbMATHCrossRefGoogle Scholar
  25. Lekivetz R, Jones B (2015) Fast flexible space-filling designs for nonrectangular regions. Qual Reliab Eng Int 31(5):829–837CrossRefGoogle Scholar
  26. Li J, Marzouk YM (2014) Adaptive construction of surrogates for the bayesian solution of inverse problems. SIAM J Sci Comput 36:A1163–A1186MathSciNetzbMATHCrossRefGoogle Scholar
  27. Mak S, Joseph VR (2018) Minimax and minimax projection designs using clustering. J Comput Graph Stat 27(1):166–178.  https://doi.org/10.1080/10618600.2017.1302881 MathSciNetCrossRefGoogle Scholar
  28. Marin JM, Robert CP (2007) Bayesian core. Springer, New YorkzbMATHGoogle Scholar
  29. Marzouk YM, Najm HN (2009) Dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems. J Comput Phys 228(6):1862–1902MathSciNetzbMATHCrossRefGoogle Scholar
  30. Marzouk YM, Xiu D (2009) A stochastic collocation approach to bayesian inference in inverse problems. Commun Comput Phys 6:826–847MathSciNetzbMATHCrossRefGoogle Scholar
  31. Matthies H, Zander E, Rosi B, Litvinenko A (2016) Parameter estimation via conditional expectation: a Bayesian inversion. Adv Model Simul Eng Sci 3(1):24CrossRefGoogle Scholar
  32. McKay M, Beckman R, Conover W (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21:239–245MathSciNetzbMATHGoogle Scholar
  33. Nguyen MT, Desceliers C, Soize C, Allain J, Gharbi H (2015) Multiscale identification of random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations. J Multiscale Comput Eng 13(4):281–295CrossRefGoogle Scholar
  34. Nouy A (2010) Proper generalized decomposition and separated representations for the numerical solution of high dimensional stochastic problems. Arch Comput Methods Eng 17:403–434MathSciNetzbMATHCrossRefGoogle Scholar
  35. Nouy A, Soize C (2014) Random fields representations for stochastic elliptic boundary value problems and statistical inverse problems. Eur J Appl Math 25:339–373MathSciNetzbMATHCrossRefGoogle Scholar
  36. Perrin G (2019) Adaptive calibration of a computer code with time-series output. Reliab Eng Syst Saf.  https://doi.org/10.1016/j.ress.2019.106728
  37. Perrin G, Cannamela C (2017) A repulsion-based method for the definition and the enrichment of opotimized space filling designs in constrained input spaces. J Soc Fr Stat 158(1):37–67zbMATHGoogle Scholar
  38. Perrin G, Soize C, Duhamel D, Funfschilling C (2012) Identification of polynomial chaos representations in high dimension from a set of realizations. SIAM J Sci Comput 34(6):2917–2945MathSciNetzbMATHCrossRefGoogle Scholar
  39. Perrin G, Soize C, Duhamel D, Funfschilling C (2013) Karhunen–loéve expansion revisited for vector-valued random fields: scaling, errors and optimal basis. J Comput Phys 242:607–622MathSciNetzbMATHCrossRefGoogle Scholar
  40. Perrin G, Soize C, Duhamel D, Funfschilling C (2014) A posteriori error and optimal reduced basis for stochastic processes defined by a finite set of realizations. SIAM/ASA J Uncertain Quantif 2:745–762MathSciNetzbMATHCrossRefGoogle Scholar
  41. Perrin G, Soize C, Marque-Pucheu S, Garnier J (2017) Nested polynomial trends for the improvement of gaussian process-based predictors. J Comput Phys 346:389–402MathSciNetzbMATHCrossRefGoogle Scholar
  42. Perrin G, Soize C, Ouhbi N (2018) Data-driven kernel representations for sampling with an unknown block dependence structure under correlation constraints. J Comput Stat Data Anal 119:139–154MathSciNetzbMATHCrossRefGoogle Scholar
  43. Rasmussen CE (2003) Gaussian processes to speed up hybrid monte carlo for expensive bayesian integrals. Bayesian Stat 7:651–659MathSciNetGoogle Scholar
  44. Rubinstein RT, Kroese D (2008) Simulation and the Monte Carlo method. Wiley, HobokenzbMATHGoogle Scholar
  45. Santner TJ, Williams B, Notz W (2003) The design and analysis of computer experiments. Springer, New YorkzbMATHCrossRefGoogle Scholar
  46. Scott DW, Sain SR (2004) Multidimensional density estimation. In: Handbook of statistics, vol 24, pp 229–261Google Scholar
  47. Sinsbeck M, Nowak W (2017) Sequential design of computer experiments for the solution of Bayesian inverse. SIAM/ASA J Uncertain Quantif 5:640–664MathSciNetzbMATHCrossRefGoogle Scholar
  48. Soize C (2006) Non-gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput Methods Appl Mech Eng 195:26–64MathSciNetzbMATHCrossRefGoogle Scholar
  49. Soize C (2008) Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probab Eng Mech 23:307–323CrossRefGoogle Scholar
  50. Soize C (2010) Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data. Comput Methods Appl Mech Eng 199(33–36):2150–2164MathSciNetzbMATHCrossRefGoogle Scholar
  51. Soize C (2011) A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension. Comput Methods Appl Mech Eng 200(45–46):3083–3099MathSciNetzbMATHCrossRefGoogle Scholar
  52. Soize C, Ghanem R (2016) Data-driven probability concentration and sampling on manifold. J Comput Phys 321(September 2015):242–258MathSciNetzbMATHCrossRefGoogle Scholar
  53. Soize C, Ghanem R (2017) Probabilistic learning on manifold for optimization under uncertainties. Proc Uncecomp 2017:1–15Google Scholar
  54. Stinstra E, den Hertog D, Stehouwer P, Vestjens A (2003) Constrained maximin designs for computer experiments. Technometrics 45(4):340–346MathSciNetCrossRefGoogle Scholar
  55. Stinstra E, den Hertog D, Stehouwer P, Vestjens A (2010) Uniform designs over general input domains with applications to target region estimation in computer experiments. Comput Stat Data Anal 51(1):219–232MathSciNetGoogle Scholar
  56. Stuart AM (2010) Inverse problems: a bayesian perspective. Acta Numerica 19:451–559MathSciNetzbMATHCrossRefGoogle Scholar
  57. Tian M, Li D, Cao Z, Phoon K, Wang Y (2016) Bayesian identification of random field model using indirect test data. Eng Geol 210:197–211CrossRefGoogle Scholar
  58. Tsilifis P, Ghanem RG, Hajali P (2017) Efficient Bayesian experimentation using an expected information gain lower bound. SIAM/ASA J Uncertain Quantif 5:30–62MathSciNetzbMATHCrossRefGoogle Scholar
  59. Wan J, Zabaras N (2011) A Bayesian approach to multiscale inverse problems using the sequential Monte Carlo method. Inverse Probl 27:105004MathSciNetzbMATHCrossRefGoogle Scholar
  60. Wand MP, Jones MC (1995) Kernel smoothing. Encycl Stat Behav Sci 60(60):212MathSciNetzbMATHGoogle Scholar
  61. Whittle P (1951) Hypothesis testing in time series, Ph.D. thesis. University of UppsalaGoogle Scholar
  62. Whittle P (1983) Prediction and regulation by linear least-square methods. University of Minnesota Press, MinneapoliszbMATHGoogle Scholar
  63. Williams M (2011) The eigenfunctions of the Karhunen–Loeve integral equation for a spherical system. Propab Eng Mech 26:202–207CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CEA/DAM/DIFArpajonFrance
  2. 2.MSME UMR 8208 CNRSUniversité Paris-EstMarne-la-ValléeFrance

Personalised recommendations