The University of Montana
Department of Mathematical Sciences

Technical report #18/2011

An MCMC Method for Uncertainty Quantification in Nonnegativity Constrained Inverse Problems

Johnathan M. Bardsley

Department of Mathematical Sciences
University of Montana
Missoula, Montana 59812
USA
E-mail: bardsleyj@mso.umt.edu

Colin Fox
Department of Physics
University of Otago
Dunedin 9054
New Zealand
E-mail: fox@physics.otago.ac.nz

Abstract

The development of computational algorithms for solving inverse problems is, and has been, a primary focus of the inverse problems community. Less studied, but of increased interest, is uncertainty quantification for solutions of inverse problems obtained using computational methods. In this paper, we present a method of uncertainty quantification for linear inverse problems with nonnegativity constraints. Our approach utilizes a Bayesian statistical framework, and we present a simple Markov chain Monte Carlo (MCMC) method for sampling from a particular posterior distribution. From the posterior samples, estimation and uncertainty quantification for both the unknown (image in our case) and regularization parameter are performed. The primary challenge of the approach is that for each sample a large-scale nonnegativity constrained quadratic minimization problem must be solved. We perform numerical tests on both one- and two-dimensional image deconvolution problems, as well as on a computed tomography test case. Our results show that our nonnegativity constrained sampler is effective and computationally feasible.

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

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

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