The University of Montana
Department of Mathematical Sciences

Technical report #28/2010

An Efficient MCMC Method for Uncertainty Quantification in Inverse Problems

Johnatha M. Bardsley, Univ. of Montana
Department of Mathematical Sciences
University of Montana
Missoula, Montana 59812
USA

E-mail: bardsleyj@mso.umt.edu

Abstract

The connection between Bayesian statistics and the technique of regularization for inverse problems has been given significant attention in recent years. For example, Bayes’ law is frequently used as motivation for variational regularization methods of Tikhonov type. In this setting, the regularization function corresponds to the negative-log of the prior probability density; the fit-to-data function corresponds to the negative-log of the likelihood; and the regularized solution corresponds to the maximizer of the posterior density, known as the maximum a posteriori (MAP) estimator. While a great deal of attention has been focused on the development of techniques for efficient computation of MAP estimators (regularized solutions), less explored is the problem of uncertainty quantification, which corresponds to the problem of determining the shape, at least to some degree, of the posterior density in high probability regions. One way to do this is to sample from the posterior density using a Markov chain Monte Carlo (MCMC) method. In this paper, we present an MCMC method for use on linear inverse problems with independent and identically distributed Gaussian noise and Gaussian priors (quadratic regularization functions). From the MCMC samples, an estimator (regularized solution), and measures of variability in the estimator, are computed. Additionally, samples of the noise and prior precision parameters are computed, making regularization parameter selection unnecessary.

Keywords: inverse problems, regularization, image reconstruction, Bayesian inference, Markov chain Monte Carlo, uncertainty quantification.

MSC numbers: 15A29, 62F15, 65F22, 94A08

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