A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension. (English) Zbl 1230.74241
Summary: This paper is devoted to the identification of Bayesian posteriors for the random coefficients of the high-dimension polynomial chaos expansions of non-Gaussian tensor-valued random fields using partial and limited experimental data. The experimental data sets correspond to an observation vector which is the response of a stochastic boundary value problem depending on the tensor-valued random field which has to be identified. So an inverse stochastic problem must be solved to perform the identification of the random field. A complete methodology is proposed to solve this very challenging problem in high dimension, which consists in using the first four steps introduced in a previous paper, followed by the identification of the posterior model. The steps of the methodology are the following: (1) introduction of a family of prior algebraic stochastic model (PASM), (2) identification of an optimal PASM in the constructed family using the partial experimental data, (3) construction of a statistical reduced-order optimal PASM, (4) construction, in high dimension, of the polynomial chaos expansion with deterministic vector-valued coefficients of the reduced-order optimal PASM, (5) substitution of these deterministic vector-valued coefficients by random vector-valued coefficients in order to extend the capability of the polynomial chaos expansion to represent the experimental data and for which the joint probability distribution must be identified, (6) construction of the prior probability model of these random vector-valued coefficients and finally, (7) identification of the posterior probability model of these random vector-valued coefficients using partial and limited experimental data, through the stochastic boundary value problem. Two methods are proposed to carry out the identification of the posterior model. The first one is based on the use of the classical Bayesian method. The second one is a new approach derived from the Bayesian method, which is more efficient in high dimension. An application is presented for which several millions of random coefficients are identified.
MSC:
| 74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |
| 74G75 | Inverse problems in equilibrium solid mechanics |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
References:
| [1] | Arnst, M.; Ghanem, R.; Soize, C., Identification of Bayesian posteriors for coefficients of chaos expansion, J. Comput. Phys., 229, 9, 3134-3154 (2010) · Zbl 1184.62034 |
| [2] | Babuska, I.; Tempone, R.; Zouraris, G. E., Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Comput. Methods Appl. Mech. Engrg., 194, 12-16, 1251-1294 (2005) · Zbl 1087.65004 |
| [3] | Babuska, I.; Nobile, F.; Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45, 3, 1005-1034 (2007) · Zbl 1151.65008 |
| [4] | Beck, J. L.; Katafygiotis, L. S., Updating models and their uncertainties. I: Bayesian statistical framework, J. Engrg. Mech., 124, 4, 455-461 (1998) |
| [5] | Bernardo, J. M.; Smith, A. F.M., Bayesian Theory (2000), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0943.62009 |
| [6] | Bowman, A. W.; Azzalini, A., Applied Smoothing Techniques for Data Analysis (1997), Oxford University Press · Zbl 0889.62027 |
| [7] | Carlin, B. P.; Louis, T. A., Bayesian Methods for Data Analysis (2009), Chapman & Hall/ CRC Press: Chapman & Hall/ CRC Press Boca Raton |
| [8] | Chen, C.; Duhamel, D.; Soize, C., Probabilistic approach for model and data uncertainties and its experimental identification in structural dynamics: case of composite sandwich panels, J. Sound Vib., 294, 1-2, 64-81 (2006) |
| [9] | Congdon, P., Bayesian Statistical Modelling (2007), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 1161.62336 |
| [10] | Das, S.; Ghanem, R.; Spall, J., Asymptotic sampling distribution for polynomial chaos representation of data: a maximum-entropy and fisher information approach, SIAM J. Sci. Comput., 30, 5, 2207-2234 (2008) · Zbl 1181.62004 |
| [11] | Das, S.; Ghanem, R.; Finette, S., Polynomial chaos representation of spatio-temporal random field from experimental measurements, J. Comput. Phys., 228, 8726-8751 (2009) · Zbl 1177.65019 |
| [12] | Debusschere, B. J.; Najm, H. N.; Pebay, P. P., Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput., 26, 2, 698-719 (2004) · Zbl 1072.60042 |
| [13] | De Oliveira, V.; Kedem, B.; Short, D. A., Bayesian prediction of transformed Gaussian random fields, J. Am. Statist. Assoc., 92, 440, 1422-1433 (1997) · Zbl 0919.62020 |
| [14] | Desceliers, C.; Ghanem, R.; Soize, C., Maximum likelihood estimation of stochastic chaos representations from experimental data, Int. J. Numer. Methods Engrg., 66, 6, 978-1001 (2006) · Zbl 1110.74826 |
| [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) · Zbl 1161.74016 |
| [16] | Finsterle, S.; Najita, J., Robust estimation of hydrogeologic model parameters, Water Resour. Res., 34, 11, 2939-2947 (1998) |
| [17] | Ghanem, R.; Spanos, P. D., Stochastic Finite Elements: A spectral Approach (1991), Springer-verlag: Springer-verlag New-York, (Revised edition, Dover Publications, New York, 2003) · Zbl 0722.73080 |
| [18] | Ghanem, R.; Dham, S., Stochastic finite element analysis for multiphase flow in heterogeneous porous media, Transp. Porous Media, 32, 239-262 (1998) |
| [19] | Ghanem, R.; Doostan, R., Characterization of stochastic system parameters from experimental data: a Bayesian inference approach, J. Comput. Phys., 217, 1, 63-81 (2006) · Zbl 1102.65004 |
| [20] | Ghanem, R.; Doostan, R.; Red-Horse, J., A probability construction of model validation, Comput. Methods Appl. Mech. Engrg., 197, 29-32, 2585-2595 (2008) · Zbl 1213.74007 |
| [21] | Ghosh, D.; Farhat, C., Strain and stress computation in stochastic finite element methods, Int. J. Numer. Methods Engrg., 74, 8, 1219-1239 (2008) · Zbl 1159.74425 |
| [22] | Guilleminot, J.; Soize, C.; Kondo, D.; Benetruy, C., Theoretical framework and experimental procedure for modelling volume fraction stochastic fluctuations in fiber reinforced composites, Int. J. Solid Struct., 45, 21, 5567-5583 (2008) · Zbl 1177.74113 |
| [23] | Hristopulos, D. T., Spartan Gibbs random field models for geostatistical applications, SIAM J. Sci. Comput., 24, 6, 2125-2162 (2003) · Zbl 1043.62078 |
| [24] | Kaipio, J.; Somersalo, E., Statistical and Computational Inverse Problems (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1068.65022 |
| [25] | Knio, O. M.; Le Maitre, O. P., Uncertainty propagation in CFD using polynomial chaos decomposition, Fluid Dynam. Res., 38, 9, 616-640 (2006) · Zbl 1178.76297 |
| [26] | Lawson, C. L.; Hanson, R. J., Solving Least Squares Problems (1974), Prentice-Hall · Zbl 0185.40701 |
| [27] | Lee, H. K.H.; Higdon, D. M.; Bi, Z. O.; Ferreira, M. A.R.; West, M., Markov random field models for high-dimensional parameters in simulations of fluid flow in porous media, Technometrics, 44, 3, 230-241 (2002) |
| [28] | Lee, H. K.H.; Higdon, D. M.; Calder, C. A.; Holloman, C. H.H., Efficient models for correlated data via convolutions of intrinsic processes, Statist. Model., 5, 1, 53-74 (2005) · Zbl 1071.62089 |
| [29] | Le Maitre, O. P., A Newton method for the resolution of steady stochastic Navier-Stokes equations, Comput. Fluids, 38, 8, 1566-1579 (2009) · Zbl 1242.76287 |
| [30] | Le-Maitre, O. P.; Knio, O. M., Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics (2010), Springer: Springer Heidelberg · Zbl 1193.76003 |
| [31] | Ma, X.; Zabaras, N., An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method, Inverse Probl., 25, 3 (2009), (Article Number 035013) · Zbl 1161.62011 |
| [32] | Marzouk, Y. M.; Najm, H. N.; Rahn, L. A., Stochastic spectral methods for efficient Bayesian solution of inverse problems, J. Comput. Phys., 224, 2, 560-586 (2007) · Zbl 1120.65306 |
| [33] | 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) · Zbl 1161.65308 |
| [34] | Najm, H. N., Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, J. Rev. Fluid Mech., 41, 35-52 (2009) · Zbl 1168.76041 |
| [35] | Nouy, A., Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms, Comput. Methods Appl. Mech. Engrg., 197, 51-52, 4718-4736 (2008) · Zbl 1194.74458 |
| [36] | Nouy, A.; Clement, A.; Schoefs, F., An extended stochastic finite element method for solving stochastic partial differential equations on random domains, Comput. Methods Appl. Mech. Engrg., 197, 51-52, 4663-4682 (2008) · Zbl 1194.74457 |
| [37] | Red-Horse, J. R.; Benjamin, A. S., A probabilistic approach to uncertainty quantification with limited information, Reliab. Engrg. Syst. Safety, 85, 183190 (2004) |
| [38] | Schueller, G. I., Computational methods in stochastic mechanics and reliability analysis, Comput. Methods Appl. Mech. Engrg., 194, 12-16, 1251-1795 (2005) |
| [39] | Schueller, G. I., On the treatment of uncertainties in structural mechanics and analysis, Comput. Struct., 85, 5, 235-243 (2007) |
| [40] | Serfling, R. J., Approximation Theorems of Mathematical Statistics (1980), John Wiley & Sons · Zbl 0423.60030 |
| [41] | Soize, C.; Ghanem, R., Physical systems with random uncertainties: chaos representation with arbitrary probability measure, SIAM J. Sci. Comput., 26, 2, 395-410 (2004) · Zbl 1075.60084 |
| [42] | Soize, C., Random matrix theory for modeling uncertainties in computational mechanics, Comput. Methods Appl. Mech. Engrg., 194, 12-16, 1333-1366 (2005) · Zbl 1083.74052 |
| [43] | Soize, C., Non Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Comput. Methods Appl. Mech. Engrg., 195, 26-64 (2006) · Zbl 1093.74065 |
| [44] | Soize, C., Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size, Probab. Engrg. Mech., 23, 307-323 (2008) |
| [45] | 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 Engrg., 76, 10, 1583-1611 (2008) · Zbl 1195.74311 |
| [46] | Soize, C.; Ghanem, R., Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields, Comput. Methods Appl. Mech. Engrg., 198, 21-26, 1926-1934 (2009) · Zbl 1227.65010 |
| [47] | Soize, C., Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions, Int. J. Numer. Methods Engrg., 81, 8, 939-970 (2010) · Zbl 1183.74381 |
| [48] | 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. Engrg., 199, 33-36, 2150-2164 (2010) · Zbl 1231.74501 |
| [49] | Soize, C.; Desceliers, C., Computational aspects for constructing realizations of polynomial chaos in high dimension, SIAM J. Sci. Comput., 32, 5, 2820-2831 (2010) · Zbl 1225.60118 |
| [50] | Spall, J. C., Introduction to Stochastic Search and Optimization (2003), John Wiley and Sons: John Wiley and Sons Hoboken, New Jersey · Zbl 1088.90002 |
| [51] | Tan, M. T.; Tian, G.-L.; Ng, K. W., Bayesian Missing Data Problems, EM, Data Augmentation and Noniterative Computation (2010), Chapman & Hall/ CRC Press: Chapman & Hall/ CRC Press Boca Raton · Zbl 1216.62042 |
| [52] | Terrell, G. R.; Scott, D. W., Variable kernel density estimation, Ann. Statist., 20, 3, 1236-1265 (1992) · Zbl 0763.62024 |
| [53] | Vargas-Guzman, J. A.; Yeh, T.-C. J., The successive linear estimator: a revisit, Adv. Water Resour., 25, 7, 773-781 (2002) |
| [54] | Walter, E.; Pronzato, L., Identification of Parametric Models from Experimental Data (1997), Springer · Zbl 0864.93014 |
| [55] | Wan, X. L.; Karniadakis, G. E., Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28, 3, 901-928 (2006) · Zbl 1128.65009 |
| [56] | Wan, X. L.; Karniadakis, G. E., Solving elliptic problems with non-Gaussian spatially-dependent random coefficients, Comput. Methods Appl. Mech. Engrg., 198, 21-26, 1985-1995 (2009) · Zbl 1227.65014 |
| [57] | Wang, J. B.; Zabaras, N., A Bayesian inference approach to the inverse heat conduction problem, Int. J. Heat Mass Transf., 47, 17-18, 3927-3941 (2004) · Zbl 1070.80002 |
| [58] | Wang, J. B.; Zabaras, N., Hierarchical Bayesian models for inverse problems in heat conduction, Inverse Probl., 21, 1, 183-206 (2005) · Zbl 1060.62036 |
| [59] | Zabaras, N.; Ganapathysubramanian, B., A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, J. Comput. Phys., 227, 9, 4697-4735 (2008) · Zbl 1142.65008 |
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