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Random Vectors and Random Fields in High Dimension: Parametric Model-Based Representation, Identification from Data, and Inverse Problems

  • Christian SoizeEmail author
Reference work entry

Abstract

The statistical inverse problem for the experimental identification of a non-Gaussian matrix-valued random field, that is, the model parameter of a boundary value problem, using some partial and limited experimental data related to a model observation, is a very difficult and challenging problem. A complete advanced methodology and the associated tools are presented for solving such a problem in the following framework: the random field that must be identified is a non-Gaussian matrix-valued random field and is not simply a real-valued random field; this non-Gaussian random field is in high stochastic dimension and is identified in a general class of random fields; some fundamental algebraic properties of this non-Gaussian random field must be satisfied such as symmetry, positiveness, invertibility in mean square, boundedness, symmetry class, spatial-correlation lengths, etc.; and the available experimental data sets correspond only to partial and limited data for a model observation of the boundary value problem.

The developments presented are mainly related to the elasticity framework, but the methodology is general and can be used in many areas of computational sciences and engineering. The developments are organized as follows. The first part is devoted to the definition of the statistical inverse problem that has to be solved in high stochastic dimension and is focussed on stochastic elliptic operators such that the ones that are encountered in the boundary value problems of the linear elasticity. The second one deals with the construction of two possible parameterized representations for a non-Gaussian positive-definite matrix-valued random field that models the model parameter of a boundary value problem. A parametric model-based representation is then constructed in introducing a statistical reduced model and a polynomial chaos expansion, first with deterministic coefficients and after with random coefficients. This parametric model-based representation is directly used for solving the statistical inverse problem. The third part is devoted to the description of all the steps of the methodology allowing the statistical inverse problem to be solved in high stochastic dimension. These steps are based on the identification of a prior stochastic model of the Non-Gaussian random field by using the maximum likelihood method and then, on the identification of a posterior stochastic model of the Non-Gaussian random field by using the Bayes method. The fourth part presents the construction of an algebraic prior stochastic model of the model parameter of the boundary value problem, for a non-Gaussian matrix-valued random field. The generator of realizations for such an algebraic prior stochastic model for a non-Gaussian matrix-valued random field is presented.

Keywords

Random vector Random field Random matrix High dimension High stochastic dimension Non-Gaussian Non-Gaussian random field Representation of random fields Polynomial chaos expansion Generator Maximum entropy principle Prior model Maximum likelihood method Bayesian method Identification Inverse problem Statistical inverse problem Random media Heterogeneous microstructure Composite materials Porous media 

References

  1. 1.
    Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2000)zbMATHGoogle Scholar
  2. 2.
    Arnst, M., Ghanem, R., Soize, C.: Identification of Bayesian posteriors for coefficients of chaos expansions. J. Comput. Phys. 229(9), 3134–3154 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batou, A., Soize, C.: Stochastic modeling and identification of an uncertain computational dynamical model with random fields properties and model uncertainties. Arch. Appl. Mech. 83(6), 831–848 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Batou, A., Soize, C.: Calculation of Lagrange multipliers in the construction of maximum entropy distributions in high stochastic dimension. SIAM/ASA J. Uncertain. Quantif. 1(1), 431–451 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burrage, K., Lenane, I., Lythe, G.: Numerical methods for second-order stochastic differential equations. SIAM J. Sci. Comput. 29, 245–264 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. Second Ser. 48(2), 385–392 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carlin, B.P., Louis, T.A.: Bayesian Methods for Data Analysis, 3rd edn. Chapman & Hall/CRC Press, Boca Raton (2009)zbMATHGoogle Scholar
  9. 9.
    Congdon, P.: Bayesian Statistical Modelling, 2nd edn. Wiley, Chichester (2007)zbMATHGoogle Scholar
  10. 10.
    Das, S., Ghanem, R.: A bounded random matrix approach for stochastic upscaling. Multiscale Model. Simul. 8(1), 296–325 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Das, S., Ghanem, R., Spall, J.C.: Asymptotic sampling distribution for polynomial chaos representation from data: a maximum entropy and fisher information approach. SIAM J. Sci. Comput. 30(5), 2207–2234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Das, S., Ghanem, R., Finette, S.: Polynomial chaos representation of spatio-temporal random field from experimental measurements. J. Comput. Phys. 228, 8726–8751 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Debusschere, B.J., Najm, H.N., Pebay, P.P., Knio, O.M., Ghanem, R., Le Maître, O.: Numerical challenges in the use of polynomial chaos representations for stochastic processes. SIAM J. Sci. Comput. 26(2), 698–719 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Desceliers, C., Ghanem, R., Soize, C.: Maximum likelihood estimation of stochastic chaos representations from experimental data. Int. J. Numer. Methods Eng. 66(6), 978–1001 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Desceliers, C., Soize, C., Ghanem, R.: Identification of chaos representations of elastic properties of random media using experimental vibration tests. Comput. Mech. 39(6), 831–838 (2007)CrossRefzbMATHGoogle Scholar
  16. 16.
    Desceliers, C., Soize, C., Naili, S., Haiat, G.: Probabilistic model of the human cortical bone with mechanical alterations in ultrasonic range. Mech. Syst. Signal Process. 32, 170–177 (2012)CrossRefGoogle Scholar
  17. 17.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1990)zbMATHGoogle Scholar
  18. 18.
    Doostan, A., Ghanem, R., Red-Horse, J.: Stochastic model reduction for chaos representations. Comput. Methods Appl. Mech. Eng. 196(37–40), 3951–3966 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ernst, O.G., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM Math. Model. Numer. Anal. 46(2), 317–339 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ghanem, R., Dham, S.: Stochastic finite element analysis for multiphase flow in heterogeneous porous media. Transp. Porous Media 32, 239–262 (1998)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ghanem, R., Doostan, R.: Characterization of stochastic system parameters from experimental data: a Bayesian inference approach. J. Comput. Phys. 217(1), 63–81 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991). See also the revised edition, Dover Publications, New York (2003)Google Scholar
  24. 24.
    Ghanem, R., Doostan, R., Red-Horse, J.: A probability construction of model validation. Comput. Methods Appl. Mech. Eng. 197(29–32), 2585–2595 (2008)CrossRefzbMATHGoogle Scholar
  25. 25.
    Ghosh, D., Ghanem, R.: Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions. Int. J. Numer. Methods Eng. 73(2), 162–184 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Guilleminot, J., Soize, C.: Non-Gaussian positive-definite matrix-valued random fields with constrained eigenvalues: application to random elasticity tensors with uncertain material symmetries. Int. J. Numer. Methods Eng. 88(11), 1128–1151 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Guilleminot, J., Soize, C.: Probabilistic modeling of apparent tensors in elastostatics: a MaxEnt approach under material symmetry and stochastic boundedness constraints. Probab. Eng. Mech. 28, 118–124 (2012)CrossRefGoogle Scholar
  28. 28.
    Guilleminot, J., Soize, C.: Stochastic model and generator for random fields with symmetry properties: application to the mesoscopic modeling of elastic random media. Multiscale Model. Simul. (SIAM Interdiscip. J.) 11(3), 840–870 (2013)Google Scholar
  29. 29.
    Guilleminot, J., Soize, C.: Itô SDE-based generator for a class of non-Gaussian vector-valued random fields in uncertainty quantification. SIAM J. Sci. Comput. 36(6), A2763–A2786 (2014)CrossRefzbMATHGoogle Scholar
  30. 30.
    Guilleminot, J., Soize, C., Kondo, D., Binetruy, C.: Theoretical framework and experimental procedure for modelling volume fraction stochastic fluctuations in fiber reinforced composites. Int. J. Solids Struct. 45(21), 5567–5583 (2008)CrossRefzbMATHGoogle Scholar
  31. 31.
    Guilleminot, J., Soize, C., Kondo, D.: Mesoscale probabilistic models for the elasticity tensor of fiber reinforced composites: experimental identification and numerical aspects. Mech. Mater. 41(12), 1309–1322 (2009)CrossRefGoogle Scholar
  32. 32.
    Guilleminot, J., Noshadravan, A., Soize, C., Ghanem, R.G.: A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Comput. Methods Appl. Mech. Eng. 200, 1637–1648 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Guilleminot, J., Soize, C., Ghanem, R.: Stochastic representation for anisotropic permeability tensor random fields. Int. J. Numer. Anal. Methods Geom. 36(13), 1592–1608 (2012)CrossRefGoogle Scholar
  34. 34.
    Guilleminot, J., Le, T.T., Soize, C.: Stochastic framework for modeling the linear apparent behavior of complex materials: application to random porous materials with interphases. Acta Mech. Sinica 29(6), 773–782 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  36. 36.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Springer, New York (2006)zbMATHGoogle Scholar
  37. 37.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630; 108(2), 171–190 (1957)Google Scholar
  38. 38.
    Kaipio, J., Somersalo, E.: Statistical and Computational Inverse Problems. Springer, New York (2005)zbMATHGoogle Scholar
  39. 39.
    Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  40. 40.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differentials Equations. Springer, Heidelberg (1992)CrossRefzbMATHGoogle Scholar
  41. 41.
    Krée, P., Soize, C.: Mathematics of Random Phenomena. Reidel, Dordrecht (1986)CrossRefzbMATHGoogle Scholar
  42. 42.
    Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  43. 43.
    Le Maitre, O.P., Knio, O.M., Najm, H.N.: Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197(1), 28–57 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lucor, D., Su, C.H., Karniadakis, G.E.: Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Eng. 60(3), 571–596 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Marzouk, Y.M., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228(6), 1862–1902 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Najm, H.H.: Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annu. Rev. Fluid Mech. 41, 35–52 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Nouy, A.: Proper generalized decomposition and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Eng. 16(3), 403–434 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Nouy, A., Soize, C.: Random fields representations for stochastic elliptic boundary value problems and statistical inverse problems. Eur. J. Appl. Math. 25(3), 339–373 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Perrin, G., Soize, C., Duhamel, D., Funfschilling, C.: Identification of polynomial chaos representations in high dimension from a set of realizations. SIAM J. Sci. Comput. 34(6), A2917–A2945 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Perrin, G., Soize, C., Duhamel, D., Funfschilling, C.: Karhunen-Loève expansion revisited for vector-valued random fields: scaling, errors and optimal basis. J. Comput. Phys. 242(1), 607–622 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Perrin, G., Soize, C., Duhamel, D., Funfschilling, C.: A posterior error and optimal reduced basis for stochastic processes defined by a set of realizations. SIAM/ASA J. Uncertain. Quantif. 2, 745–762 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Puig, B., Poirion, F., Soize, C.: Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms. Probab. Eng. Mech. 17(3), 253–264 (2002)CrossRefGoogle Scholar
  53. 53.
    Rozanov, Y.A.: Random Fields and Stochastic Partial Differential Equations. Kluwer Academic, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  54. 54.
    Serfling, R.J.: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980)CrossRefzbMATHGoogle Scholar
  55. 55.
    Soize, C.: The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions. World Scientific, Singapore (1994)CrossRefzbMATHGoogle Scholar
  56. 56.
    Soize, C.: Random-field model for the elasticity tensor of anisotropic random media. Comptes Rendus Mecanique 332, 1007–1012 (2004)CrossRefzbMATHGoogle Scholar
  57. 57.
    Soize, C., Ghanem, R.: Physical systems with random uncertainties: chaos representation with arbitrary probability measure. SIAM J. Sci. Comput. 26(2), 395–410 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Soize, C.: Non Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput. Methods Appl. Mech. Eng. 195(1–3), 26-64 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Soize, C.: Construction of probability distributions in high dimension using the maximum entropy principle. Applications to stochastic processes, random fields and random matrices. Int. J. Numer. Methods Eng. 76(10), 1583–1611 (2008)Google Scholar
  60. 60.
    Soize, C.: 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(2–3), 307–323 (2008)CrossRefGoogle Scholar
  61. 61.
    Soize, C.: 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), 2150–2164 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Soize, C.: 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), 3083–3099 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Soize, C.: Stochastic Models of Uncertainties in Computational Mechanics. American Society of Civil Engineers (ASCE), Reston (2012)CrossRefzbMATHGoogle Scholar
  64. 64.
    Soize, C.: Polynomial chaos expansion of a multimodal random vector. SIAM/ASA J. Uncertain. Quantif. 3(1), 34–60 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Soize, C., Desceliers, C.: Computational aspects for constructing realizations of polynomial chaos in high dimension. SIAM J. Sci. Comput. 32(5), 2820–2831 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Soize, C., Ghanem, R.: Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields. Comput. Methods Appl. Mech. Eng. 198(21–26), 1926–1934 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Spall, J.C.: Introduction to Stochastic Search and Optimization. Wiley, Hoboken (2003)CrossRefzbMATHGoogle Scholar
  68. 68.
    Stuart, A.M.: Inverse problems: a Bayesian perspective. Acta Numer. 19, 451–559 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Ta, Q.A., Clouteau, D., Cottereau, R.: Modeling of random anisotropic elastic media and impact on wave propagation. Eur. J. Comput. Mech. 19(1–2–3), 241–253 (2010)Google Scholar
  70. 70.
    Talay, D.: Simulation and numerical analysis of stochastic differential systems. In: Kree, P., Wedig, W. (eds.) Probabilistic Methods in Applied Physics. Lecture Notes in Physics, vol. 451, pp. 54–96. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  71. 71.
    Talay, D.: Stochastic Hamiltonian system: exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Process. Relat. Fields 8, 163–198 (2002)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005)CrossRefzbMATHGoogle Scholar
  73. 73.
    Tipireddy, R., Ghanem, R.: Basis adaptation in homogeneous chaos spaces. J. Comput. Phys. 259, 304–317 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Vanmarcke, E.: Random Fields, Analysis and Synthesis, Revised and Expanded New edn. World Scientific, Singapore (2010)CrossRefzbMATHGoogle Scholar
  75. 75.
    Walpole, L.J.: Elastic behavior of composite materials: theoretical foundations. Adv. Appl. Mech. 21, 169–242 (1981)CrossRefzbMATHGoogle Scholar
  76. 76.
    Walter, E., Pronzato, L.: Identification of Parametric Models from Experimental Data. Springer, Berlin (1997)zbMATHGoogle Scholar
  77. 77.
    Wan, X.L., Karniadakis, G.E.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Xiu, D.B., Karniadakis, G.E.: Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics, 6th edn. Elsevier/Butterworth-Heinemann, Amsterdam (2005)zbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Laboratoire Modélisation et Simulation Multi Echelle (MSME)Université Paris-EstMarne-la-ValleeFrance

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