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 (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.