Random field representations for stochastic elliptic boundary value problems and statistical inverse problems

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of non-Gaussian positive-definite matrix-valued random fields, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analysed. A minimal parameterisation of discretised random fields belonging to this general class is proposed. Using this parameterisation of the general class, a complete identification procedure is proposed. New results of the mathematical and numerical analyses of the parameterised stochastic elliptic boundary value problem are presented. The numerical solution of this parametric stochastic problem provides an explicit approximation of the application that maps the parameterised general class of random fields to the corresponding set of random solutions. This approximation can be used during the identification procedure in order to avoid the solution of multiple forward stochastic problems. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a particular mathematical analysis is developed and dedicated approximation methods are introduced. In order to obtain an algorithm for constructing the approximation of a very high-dimensional map, complexity reduction methods are introduced and are based on the use of sparse or low-rank approximation methods that exploit the tensor structure of the solution which results from the parameterisation of the general class of random fields.

[1]  S. G. Mikhlin,et al.  Integral equations―a reference text , 1975 .

[2]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Alexandre Clément,et al.  Identification of random shapes from images through polynomial chaos expansion of random level set functions , 2009 .

[4]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

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

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

[7]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

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

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

[10]  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..

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

[12]  Roger Ghanem,et al.  Stochastic model reduction for chaos representations , 2007 .

[13]  Gene H. Golub,et al.  Matrix Computations, Third Edition , 1996 .

[14]  R. Cottereau,et al.  Modeling of random anisotropic elastic media and impact on wave propagation , 2010 .

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

[16]  Eigenvalue and singular value estimates for integral operators: a unifying approach , 2012 .

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

[18]  Christoph Schwab,et al.  Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients , 2007 .

[19]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[20]  Gene H. Golub,et al.  Matrix computations , 1983 .

[21]  A. Nouy.  Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations , 2009 .

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

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

[24]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[25]  Christian Soize,et al.  A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension , 2011, Computer Methods in Applied Mechanics and Engineering.

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

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

[28]  Christian Soize,et al.  Stochastic Model and Generator for Random Fields with Symmetry Properties: Application to the Mesoscopic Modeling of Elastic Random Media , 2013, Multiscale Model. Simul..

[29]  Albert Cohen,et al.  Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs , 2010, Found. Comput. Math..

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

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

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

[35]  George Em Karniadakis,et al.  Generalized polynomial chaos and random oscillators , 2004 .

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

[37]  O. Ernst,et al.  ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS , 2011 .

[38]  Julia Charrier,et al.  Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients , 2012, SIAM J. Numer. Anal..

[39]  Gianluca Iaccarino,et al.  A least-squares approximation of partial differential equations with high-dimensional random inputs , 2009, J. Comput. Phys..

[40]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[41]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[42]  Virginie Ehrlacher,et al.  Convergence of a greedy algorithm for high-dimensional convex nonlinear problems , 2010, 1004.0095.

[43]  Christian Soize,et al.  A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures , 2011 .

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

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

[46]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[47]  Antonio Falcó,et al.  Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces , 2011, Numerische Mathematik.

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

[49]  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.

[50]  P. Absil,et al.  Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation , 2004 .

[51]  A. Nouy.  Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems , 2010 .

[52]  A. Nouy.  A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations , 2007 .

[53]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

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

[55]  Omar M. Knio,et al.  Spectral Methods for Uncertainty Quantification , 2010 .

[56]  Anthony T. Patera,et al.  A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient , 2009 .

[57]  Boris N. Khoromskij,et al.  Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs , 2011, SIAM J. Sci. Comput..

[58]  A. M. Stuart,et al.  Sparse deterministic approximation of Bayesian inverse problems , 2011, 1103.4522.

[59]  Claude Jeffrey Gittelson,et al.  Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* , 2011, Acta Numerica.

[60]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[61]  C. J. Gittelson.  STOCHASTIC GALERKIN DISCRETIZATION OF THE LOG-NORMAL ISOTROPIC DIFFUSION PROBLEM , 2010 .

[62]  Antonio Falcó,et al.  A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach , 2011 .

[63]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[64]  Hermann G. Matthies,et al.  Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations , 2005 .

[65]  Hermann G. Matthies,et al.  Solving stochastic systems with low-rank tensor compression , 2012 .

[66]  Reinhold Schneider,et al.  Approximation rates for the hierarchical tensor format in periodic Sobolev spaces , 2014, J. Complex..

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

[68]  Christian Soize,et al.  Non‐Gaussian positive‐definite matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries , 2011 .

[69]  Roger Ghanem,et al.  Numerical solution of spectral stochastic finite element systems , 1996 .

[70]  Albert Cohen,et al.  Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs , 2015 .

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

[72]  Juan Galvis,et al.  Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity , 2009, SIAM J. Numer. Anal..

[73]  J. Reade.  Eigenvalues of integral operators with smooth positive definite kernels , 2005 .