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

Published online by Cambridge University Press:  21 March 2014

A. NOUY  and
C. SOIZE
A. NOUY
Affiliation:
Ecole Centrale de Nantes, LUNAM Université, GeM UMR CNRS 6183, 1 rue de la Noe, 44321 Nantes, France email: anthony.nouy@ec-nantes.fr
C. SOIZE
Affiliation:
Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Valle, France email: christian.soize@univ-paris-est.fr
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Abstract

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.

Keywords

60G6065M3260H3562F9935J70
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 3 , June 2014 , pp. 339 - 373
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Absil, P.-A., Mahony, R. & Sepulchre, R. (2004) Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Acta Applicandae Math. 80, 199–220.CrossRefGoogle Scholar
[2]Arnst, M., Ghanem, R. & Soize, C. (2010) Identification of Bayesian posteriors for coefficients of chaos expansion. J. Comput. Phys. 229 (9), 3134–3154.CrossRefGoogle Scholar
[3]Babuska, I., Tempone, R. & Zouraris, G. E. (2004) Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 (2), 800–825.CrossRefGoogle Scholar
[4]Babuska, I., Tempone, R. & Zouraris, G. E. (2005) Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods Appl. Mech. Eng. 194 (12–16), 1251–1294.Google Scholar
[5]Babuska, I., Nobile, F. & Tempone, R. (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45 (3), 1005–1034.Google Scholar
[6]Babuska, I., Nobile, F. & Tempone, R. (2010) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52 (2), 317–355.Google Scholar
[7]Boyaval, S., Le Bris, C., Maday, Y., Nguyen, N. C. & Patera, A. T. (2009) A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable robin coefficient. Comput. Methods Appl. Mech. Eng. 198 (41–44), 3187–3206.CrossRefGoogle Scholar
[8]Cameron, R. H. & Martin, W. T. (1947) The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math., Second Series 48 (2), 385–392.Google Scholar
[9]Cances, E., Ehrlacher, V. & Lelievre, T. (2011) Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math. Models Methods Appl. Sci. 21 (12), 2433–2467.Google Scholar
[10]Charrier, J. (2012) Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50 (1), 216–246.Google Scholar
[11]Cohen, A., DeVore, R. & Schwab, C. (2010) Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10 (6), 615–646.Google Scholar
[12]Cohen, A., Chkifa, A. & Schwab, Ch. (2013) Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, in review, SAM Report 2013–25, Seminar for Applied Mathematics, ETH Zurich, Zurich.Google Scholar
[13]Das, S., Ghanem, R. & Spall, J. (2008) Asymptotic sampling distribution for polynomial chaos representation of data: A maximum-entropy and fisher information approach. SIAM J. Sci. Comput. 30 (5), 2207–2234.CrossRefGoogle Scholar
[14]Das, S., Ghanem, R. & Finette, S. (2009) Polynomial chaos representation of spatio-temporal random field from experimental measurements. J. Comput. Phys. 228, 8726–8751.CrossRefGoogle Scholar
[15]Deb, M., Babuska, I. & Oden, J. T. (2001) Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190, 6359–6372.Google Scholar
[16]De Oliveira, V., Kedem, B. & Short, D. A. (1997) Bayesian prediction of transformed gaussian random fields. J. Am. Stat.AP Assoc. 92 (440), 1422–1433.Google Scholar
[17]Desceliers, C., Ghanem, R. & Soize, C. (2006) Maximum likelihood estimation of stochastic chaos representations from experimental data. Int. J. Numer. Methods Eng. 66 (6), 978–1001.CrossRefGoogle Scholar
[18]Doostan, A., Ghanem, R. & Red-Horse, J. (2007) Stochastic model reduction for chaos representations. Comput. Methods Appl. Mech. Eng. 196 (37–40), 3951–3966.CrossRefGoogle Scholar
[19]Doostan, A. & Iaccarino, G. (2009) A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 228 (12), 4332–4345.Google Scholar
[20]Edelman, A., Arias, T. A. & Smith, S. T. (1998) The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl., 20 (2), 303–353.Google Scholar
[21]Ernst, O. G., Mugler, A., Starkloff, H. J. & Ullmann, E. (2012) On the convergence of generalized polynomial chaos expansions. ESAIM: Math. Modelling Numer. Anal. 46 (2), 317–339.Google Scholar
[22]Falco, A. & Nouy, A. (2011) A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Anal. Appl. 376 (2), 469–480.Google Scholar
[23]Falco, A. & Nouy, A. (2012) Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numer. Math. 121 (3), 503–530.Google Scholar
[24]Frauenfelder, P., Schwab, C. & Todor, R. A. (2005) Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194 (2–5), 205–228.CrossRefGoogle Scholar
[25]Galvis, J. & Sarkis, M. (2009) Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity. SIAM J. Numer. Anal. 47 (5), 3624–3651.CrossRefGoogle Scholar
[26]Ghanem, R. & Spanos, P. D. (1991) Stochastic Finite Elements: A spectral Approach, Spinger-Verlag, New York (revised edition, Dover Publications, New York, 2003).CrossRefGoogle Scholar
[27]Ghanem, R. & Kruger, R. M. (1996) Numerical solution of spectral stochastic finite element systems. Comput. Methods Appl. Mech. Eng. 129 (3), 289–303.Google Scholar
[28]Ghanem, R. & Doostan, R. (2006) Characterization of stochastic system parameters from experimental data: A Bayesian inference approach. J. Comput. Phys. 217 (1), 63–81.Google Scholar
[29]Gittelson, C. J. (2010) Stochastic Galerkin discretization of the log-normal isotropic diffusion problem. Math. Models Methods Appl. Sci. 20 (2), 237–263.Google Scholar
[30]Golub, G. H. & Van Loan, C. F. (1996) Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore.Google Scholar
[31]Guilleminot, J., Noshadravan, A., Soize, C. & Ghanem, R. (2011) A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Comput. Methods Appl. Mech. Eng. 200 (17–20), 1637–1648.Google Scholar
[32]Guilleminot, J. & Soize, C. (2011) 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.Google Scholar
[33]Guilleminot, J. & Soize, C. (2013) Stochastic model and generator for random fields with symmetry properties: Application to the mesoscopic modeling of elastic random media. Multiscale Model. Simul. (A SIAM Interdisciplinary Journal), 11 (3), 840–870.CrossRefGoogle Scholar
[34]Hackbusch, W. (2012) Tensor Spaces and Numerical Tensor Calculus, Volume 42 of Springer series in computational mathematics, Springer, Heidelberg.Google Scholar
[35]Hristopulos, D. T. (2003) Spartan Gibbs random field models for geostatistical applications. SIAM J. Sci. Comput. 24 (6), 2125–2162.Google Scholar
[36]Kaipio, J. & Somersalo, E. (2005) Statistical and Computational Inverse Problems, Springer-Verlag, New York.CrossRefGoogle Scholar
[37]Khoromskij, B. N. & Schwab, C. (2011) Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33 (1), 364–385.Google Scholar
[38]Kühn, T. (1987) Eigenvalues of integral operators with smooth positive definite kernels. Archiv der Mathematik 49, 525–534.Google Scholar
[39]Le Maitre, O. P., Knio, O. M. & Najm, H. N. (2004) Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys. 197 (1), 28–57.Google Scholar
[40]Le Maitre, O. P. & Knio, O. M. (2010) Spectral Methods for Uncertainty Quantification with Applications to Computational Fluid Dynamics, Springer, Heidelberg.Google Scholar
[41]Lucor, D., Su, C. H. & Karniadakis, G. E. (2004) Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Eng. 60 (3), 571–596.Google Scholar
[42]Marzouk, Y. M. & Najm, H. N. (2009) Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228 (6), 1862–1902.CrossRefGoogle Scholar
[43]Matthies, H. G. & Keese, A. (2005) Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194 (12–16), 1295–1331.Google Scholar
[44]Matthies, H. G. & Zander, E. (2012) Solving stochastic systems with low-rank tensor compression. Linear Algebr. Appl. 436 (10), 3819–3838.Google Scholar
[45]Menegatto, V. A. & Oliveira, C. P. (2012) Eigenvalue and singular value estimates for integral operators: A unifying approach. Mathematische Nachrichten 285 (17–18), 2222–2232.CrossRefGoogle Scholar
[46]Mugler, A. & Starkloff, H.-J. (2011) On elliptic partial differential equations with random coefficients. Studia Universitatis Babes-Bolyai – Series Mathematica 56 (2), 473–487.Google Scholar
[47]Nobile, F., Tempone, R. & Webster, C. G. (2008) A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46 (5), 2309–2345.Google Scholar
[48]Nouy, A. (2007) A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 196 (45–48), 4521–4537.Google Scholar
[49]Nouy, A. (2008) Generalized spectral decomposition method for solving stochastic finite element equations: Invariant subspace problem and dedicated algorithms. Comput. Methods Appl. Mech. Eng. 197 (51–52), 4718–4736.Google Scholar
[50]Nouy, A. (2009) Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch. Comput. Methods Eng. 16 (3), 251–285.CrossRefGoogle Scholar
[51]Nouy, A. (2010) Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch. Comput. Methods Eng. 17 (4), 403–434.CrossRefGoogle Scholar
[52]Schneider, R. & Uschmajew, A. (2013) Approximation rates for the hierarchical tensor format in periodic Sobolev spaces. J. Complexity 30 (2), 56–71.Google Scholar
[53]Schwab, C. & Gittelson, C. (2011) Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numerica 20, 291–467.Google Scholar
[54]Schwab, C. & Stuart, A. M. (2012) Sparse deterministic approximation of Bayesian inverse problems. Inverse Problems 28 (4)045003.Google Scholar
[55]Serfling, R. J. (1980) Approximation Theorems of Mathematical Statistics, John Wiley & Sons, New York.CrossRefGoogle Scholar
[56]Soize, C. & Ghanem, R. (2004) Physical systems with random uncertainties: Chaos representation with arbitrary probability measure. SIAM J. Sci. Comput. 26 (2), 395–410.Google Scholar
[57]Soize, C. (2006) Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators. Comput. Methods Appl. Mech. Eng. 195, 26–64.Google Scholar
[58]Soize, C. & Ghanem, R. (2009) Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields. Comput. Methods Appl. Mech. Eng. 198 (21–26), 1926–1934.Google Scholar
[59]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), 2150–2164.Google Scholar
[60]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), 3083–3099.Google Scholar
[61]Spall, J. C. (2003) Introduction to Stochastic Search and Optimization, John Wiley and Sons, Hoboken, New Jersey.Google Scholar
[62]Stefanou, G., Nouy, A. & Clément, A. (2009) Identification of random shapes from images through polynomial chaos expansion of random level set functions. Int. J. Numer. Methods Eng. 79 (2), 127–155.Google Scholar
[63]Stuart, A. M. (2010) Inverse problems: A bayesian perspective Acta Numerica 19, 451–559.CrossRefGoogle Scholar
[64]Ta, Q. A., Clouteau, D. & Cottereau, R. (2010) Modeling of random anisotropic elastic media and impact on wave propagation. Eur. J. Comput. Mech. 19 (1–3), 241–253.Google Scholar
[65]Todor, R. A. & Schwab, C. (2007) Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27 (2), 232–261.Google Scholar
[66]Walter, E. & Pronzato, L. (1997) Identification of Parametric Models from Experimental Data, Springer, Heidelberg.Google Scholar
[67]Wan, X. L. & Karniadakis, G. E. (2006) Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28 (3), 901–928.Google Scholar
[68]Wan, X. L. & Karniadakis, G. E. (2009) Solving elliptic problems with non-Gaussian spatially-dependent random coefficients. Comput. Methods Appl. Mech. Engrg. 198 (21–26)1985–1995.Google Scholar
[69]Wiener, N. (1938) The homogeneous chaos. Am. J. Math. 60 (1), 897–936.Google Scholar
[70]Xiu, D. B. & Karniadakis, G. E. (2002) Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2), 619–644.Google Scholar
[71]Zabaras, N. & Ganapathysubramanian, B. (2008) A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach. J. Comput. Phys. 227 (9), 4697–4735.Google Scholar
[72]Zabreyko, P. P., Koshelev, A. I., Krasnosel'skii, M. A., Mikhlin, S. G., Rakovshchik, L. S. & Ya. Stet'senko, V. (1975) Integral Equations – A Reference Text, Noordhoff International Publishing, Leyden.Google Scholar