TY - JOUR
T1 - An MCMC method for uncertainty quantification in nonnegativity constrained inverse problems
AU - Bardsley, Johnathan M.
AU - Fox, Colin
N1 - Funding Information:
This work was supported by the National Science Foundation under grant DMS-0915107. J.M. Bradsley would like to thank the University of Montana and the Department of Physics at the University of Otago, New Zealand, and Dr Colin Fox, in particular, for their support during his 2010–11 sabbatical year.
PY - 2012/6
Y1 - 2012/6
N2 - 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 (UQ) for solutions of inverse problems obtained using computational methods. In this article, we present a method of UQ for linear inverse problems with nonnegativity constraints. We present a Markov chain Monte Carlo (MCMC) method for sampling from a particular probability distribution over the unknowns. From the samples, estimation and UQ 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.
AB - 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 (UQ) for solutions of inverse problems obtained using computational methods. In this article, we present a method of UQ for linear inverse problems with nonnegativity constraints. We present a Markov chain Monte Carlo (MCMC) method for sampling from a particular probability distribution over the unknowns. From the samples, estimation and UQ 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.
KW - Markov chain Monte Carlo
KW - bound constrained optimization
KW - image reconstruction
KW - inverse problems
KW - uncertainty quantification
UR - http://www.scopus.com/inward/record.url?scp=84861638139&partnerID=8YFLogxK
U2 - 10.1080/17415977.2011.637208
DO - 10.1080/17415977.2011.637208
M3 - Article
AN - SCOPUS:84861638139
SN - 1741-5977
VL - 20
SP - 477
EP - 498
JO - Inverse Problems in Science and Engineering
JF - Inverse Problems in Science and Engineering
IS - 4
ER -