A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension

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.

[1]  Bradley P. Carlin,et al.  Bayesian Methods for Data Analysis , 2008 .

[2]  O. L. Maître,et al.  Uncertainty propagation in CFD using polynomial chaos decomposition , 2006 .

[3]  B. Kedem,et al.  Bayesian Prediction of Transformed Gaussian Random Fields , 1997 .

[4]  Faming Liang,et al.  Statistical and Computational Inverse Problems , 2006, Technometrics.

[5]  Stephen J. Ganocy,et al.  Bayesian Statistical Modelling , 2002, Technometrics.

[6]  G. Karniadakis,et al.  Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures , 2006, SIAM J. Sci. Comput..

[7]  Christian Soize,et al.  Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes, random fields and random matrices , 2008 .

[8]  Habib N. Najm,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005, SIAM J. Sci. Comput..

[9]  Christian Soize.  Random matrix theory for modeling uncertainties in computational mechanics , 2005 .

[10]  C. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[11]  Christian Soize,et al.  Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size , 2008 .

[12]  Habib N. Najm,et al.  Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems , 2008, J. Comput. Phys..

[13]  Roger G. Ghanem,et al.  Asymptotic Sampling Distribution for Polynomial Chaos Representation of Data: A Maximum Entropy and Fisher information approach , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[14]  Allan Benjamin,et al.  A probabilistic approach to uncertainty quantification with limited information , 2003, Reliab. Eng. Syst. Saf..

[15]  Z. Q. John Lu,et al.  Bayesian methods for data analysis, third edition , 2010 .

[16]  Roger G. Ghanem,et al.  Identification of Bayesian posteriors for coefficients of chaos expansions , 2010, J. Comput. Phys..

[17]  Christian Soize,et al.  Theoretical framework and experimental procedure for modelling mesoscopic volume fraction stochastic fluctuations in fiber reinforced composites , 2008 .

[18]  Nicholas Zabaras,et al.  Hierarchical Bayesian models for inverse problems in heat conduction , 2005 .

[19]  Fabio Nobile,et al.  A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data , 2008, SIAM J. Numer. Anal..

[20]  O. L. Maître,et al.  A Newton method for the resolution of steady stochastic Navier–Stokes equations , 2009 .

[21]  Nicolas Moës,et al.  An extended stochastic finite element method for solving stochastic partial differential equations on random domains , 2008 .

[22]  Baskar Ganapathysubramanian,et al.  A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach , 2008, J. Comput. Phys..

[23]  Christian Soize,et al.  Reduced Chaos decomposition with random coefficients of vector-valued random variables and random fields , 2009 .

[24]  M. Hazelton.  Variable kernel density estimation , 2003 .

[25]  I. Babuska,et al.  Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation , 2005 .

[26]  S. SIAMJ..  SPARTAN GIBBS RANDOM FIELD MODELS FOR GEOSTATISTICAL APPLICATIONS∗ , 2003 .

[27]  Herbert K. H. Lee,et al.  Efficient models for correlated data via convolutions of intrinsic processes , 2005 .

[28]  Roger Ghanem,et al.  Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media , 1998 .

[29]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..

[30]  Roger G. Ghanem,et al.  Polynomial chaos representation of spatio-temporal random fields from experimental measurements , 2009, J. Comput. Phys..

[31]  Christian Soize,et al.  Identification of Chaos Representations of Elastic Properties of Random Media Using Experimental Vibration Tests , 2007 .

[32]  Christian Soize,et al.  Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data , 2010 .

[33]  Eric Walter,et al.  Identification of Parametric Models: from Experimental Data , 1997 .

[34]  G. Karniadakis,et al.  Solving elliptic problems with non-Gaussian spatially-dependent random coefficients , 2009 .

[35]  G. I. Schuëller,et al.  On the treatment of uncertainties in structural mechanics and analysis , 2007 .

[36]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[37]  BabuskaIvo,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007 .

[38]  Nicholas Zabaras,et al.  A Bayesian inference approach to the inverse heat conduction problem , 2004 .

[39]  Christian Soize.  Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators , 2006 .

[40]  N NajmHabib,et al.  Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes , 2005 .

[41]  Stefan Finsterle,et al.  Robust estimation of hydrogeologic model parameters , 1998 .

[42]  G. Tian,et al.  Bayesian Missing Data Problems: EM, Data Augmentation and Noniterative Computation , 2009 .

[43]  A. Nouy.  Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms , 2008 .

[44]  Roger G. Ghanem,et al.  On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data , 2006, J. Comput. Phys..

[45]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[46]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2007, SIAM Rev..

[47]  J. A. Vargas-Guzmán,et al.  The successive linear estimator: a revisit , 2002 .

[48]  Habib N. Najm,et al.  Stochastic spectral methods for efficient Bayesian solution of inverse problems , 2005, J. Comput. Phys..

[49]  Christian Soize,et al.  Computational Aspects for Constructing Realizations of Polynomial Chaos in High Dimension , 2010, SIAM J. Sci. Comput..

[50]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[51]  Nicholas Zabaras,et al.  An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method , 2009 .

[52]  Charbel Farhat,et al.  Strain and stress computations in stochastic finite element methods , 2008 .

[53]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[54]  James C. Spall,et al.  Introduction to stochastic search and optimization - estimation, simulation, and control , 2003, Wiley-Interscience series in discrete mathematics and optimization.

[55]  Christian Soize,et al.  Maximum likelihood estimation of stochastic chaos representations from experimental data , 2006 .

[56]  Christian Soize.  Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions , 2010 .

[57]  Habib N. Najm,et al.  Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics , 2009 .

[58]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[59]  Richard D. Deveaux,et al.  Applied Smoothing Techniques for Data Analysis , 1999, Technometrics.

[60]  J. Beck,et al.  Updating Models and Their Uncertainties. I: Bayesian Statistical Framework , 1998 .

[61]  Roger Ghanem,et al.  A probabilistic construction of model validation , 2008 .

[62]  Mike West,et al.  Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media , 2002, Technometrics.

[63]  Christian Soize,et al.  Probabilistic approach for model and data uncertainties and its experimental identification in structural dynamics: Case of composite sandwich panels , 2006 .