An MCMC Method for Uncertainty Quantification in Nonnegativity Constrained Inverse Problems

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 oneand 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.