The Library

Sparse deterministic approximation of Bayesian inverse problems

Tools

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

Preview

Text
WRAP_Stuart_1103.4522.pdf - Submitted Version
Download (605Kb) | Preview

Official URL: http://dx.doi.org/10.1088/0266-5611/28/4/045003

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

Official Date:

April 2012

Dates:

Date	Event
April 2012	Published

Volume:

Vol.28

Number:

No.4

Number of Pages:

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Not Peer Reviewed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), 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.