NDLI: Fast Algorithms for Bayesian Uncertainty Quantification in Large-Scale Linear Inverse Problems Based on Low-Rank Partial Hessian Approximations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	van Bloemen Waanders, B. Wilcox, L. C. Akelik, V. Flath, H. P. Hill, J. Ghattas, O.
Copyright Year	2011
Abstract	We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the low-rank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on large-scale three-dimensional convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.
Starting Page	407
Ending Page	432
Page Count	26
File Format	PDF
ISSN	10648275
DOI	10.1137/090780717
e-ISSN	10957197
Issue Number	1
Volume Number	33
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-02-24
Access Restriction	Subscribed
Subject Keyword	uncertainty quantification fast algorithms Computational problems in statistics Lanczos low-rank approximation convection-diffusion Inverse problems PDEs in connection with statistics large-scale statistical inverse problem Bayesian inference Hessian PDEs in connection with control and optimization contaminant transport
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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