Schwab, Ch. (Christoph) and Stuart, A. M.. (2012) Sparse deterministic approximation of Bayesian inverse problems. Inverse Problems, Vol.28 (No.4). 045003. ISSN 0266-5611

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Abstract

We present a parametric deterministic formulation of Bayesian inverse problems with an input parameter from infinite-dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. The first step in this process is to estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number N of unknowns appearing in the parametric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Bayesian statistical decision theory, Banach spaces, Differential equations, Partial
Journal or Publication Title:	Inverse Problems
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	0266-5611
Date:	April 2012
Volume:	Vol.28
Number:	No.4
Number of Pages:	33
Page Range:	045003
Identification Number:	10.1088/0266-5611/28/4/045003
Status:	Not Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung [Swiss National Science Foundation] (SNSF), Seventh Framework Programme (European Commission) (FP7/2007-2013), Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	AdG247277 (FP7)
References:	[1] R. Andreev, Ch. Schwab and A.M. Stuart. In preparation. [2] H.T. Banks and K. Kunisch. Estimation techniqiues for distributed parameter systems. Birkh¨auser, 1989. [3] M. Bieri, R. Andreev, and C. Schwab. Sparse tensor discretization of elliptic SPDEs. SIAM J. Sci. Comp., 2009. [4] Babuˇska I., Tempone R. and Zouraris G. E. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42 , no. 2, 800–825. 2004. [5] A. Chkifa, A. Cohen, R. DeVore, and Ch. Schwab. Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs Report 2011-44, Seminar for Applied Mathematics, ETH Z¨urich, Switzerland http : //www.sam.math.ethz.ch/reports/2011/44 (in review). [6] A. Cohen, R. DeVore, and Ch. Schwab. Convergence rates of best N-term Galerkin approximations for a class of elliptic SPDEs. Journ. Found. Comp.Math. Volume 10, Number 6, December 2010, pp. 615-646 [7] A. Cohen, R. DeVore, and Ch. Schwab. Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Analysis and Applications (2011). [8] Cotter S. L., Dashti M., Stuart A.M. 2010, Approximation of Bayesian inverse problems in differential equations. SIAM J. Numer. Anal. 48, No. 1, 322-345. [9] R. DeVore. Nonlinear approximation. Acta Numerica 7(1998), 51–150. [10] C.J. Gittelson. Adaptive wavelet methods for elliptic partial differential equations with random operators Report 2011-37, Seminar for Applied Mathematics, ETH Z¨urich, Switzerland http://www.sam.math.ethz.ch/reports/2011/37 (in review). [11] V. Ha Hoang, Ch. Schwab and A.M. Stuart, (in preparation) (2012). [12] M. Hairer, A. M. Stuart, and J. Voss. Analysis of SPDEs arising in path sampling, part II: The nonlinear case. Annals of Applied Probability, 17:1657–1706, 2007. [13] L. Hoermander. An Introduction to Complex Analysis in Several Variables (3rd. Ed.) North Holland Mathematical Library, North Holland Publ., (1990). [14] J. Kaipio and E. Somersalo. Statistical and computational inverse problems, volume 160 of Applied Mathematical Sciences. Springer, 2005. [15] J. Liu, Monte Carlo Strategies in Scientific Computing. Springer Texts in Statistics, Springer- Verlag, New York, 2001. [16] Y. M. Marzouk, H. N. Najm and L. A. Rahn, Stochastic spectral methods for efficient Bayesian solution of inverse problems Journ. Comp. Phys. 224 No. 10, June 2007, Pages 560-586. [17] Y. M. Marzouk and D. Xiu, A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems Communications in Computational Physics, 6, 826-847, 2009. [18] Y. M. Marzouk and H. N. Najm Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems Journal of Computational Physics 228, Issue 6, 1 April 2009, Pages 1862-1902 [19] D. McLaughlin and L.R. Townley. A reassessment of the groundwater inverse problem. Water Resour. Res., 32:1131–1161, 1996. [20] C.P. Robert and G.C. Casella, Monte Carlo Statistical Methods. Springer Texts in Statistics, Springer-Verlag, 1999. [21] G.O. Roberts and C. Sherlock, Optimal Scaling of Random Walk Metropolis algorithms with discontinuous target densities http://www.imstat.org/aap/future-papers.html (to appear in Ann. Appl. Proba. 2012). [22] Ch. Schwab and C.J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numerica 20 (2011), 291-467. [23] P.D. Spanos and R. Ghanem. Stochastic finite element expansion for random media. J. Eng. Mech., 115:1035–1053, 1989. [24] P.D. Spanos and R. Ghanem. Stochastic Finite Elements: A Spectral Approach. Dover, 2003. [25] A.M. Stuart. Inverse problems: a Bayesian approach. Acta Numerica, 19, 2010. [26] R.A. Todor and C. Schwab. Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Num. Anal., 27:232–261, 2007. [27] N. Wiener. The homogeneous chaos. American Journal of Mathematics, 1938.
URI:	http://wrap.warwick.ac.uk/id/eprint/48173

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