Dealing with Boundary Artifacts in MCMC-Based Deconvolution

Many numerical methods for deconvolution problems are designed to take advantage of the computational efficiency of spectral methods, but classical approaches to spectral techniques require particular conditions be applied uniformly across all boundaries of the signal. These boundary conditions– traditionally periodic, Dirichlet, Neumann, or related–are essentially methods for generating data values outside the domain of the signal, but they often lack physical motivation and can result in artifacts in the reconstruction near the boundary. In this work we present a data-driven technique for computing boundary values by solving a regularized and well-posed form of the deconvolution problem on an extended domain. Further, a Bayesian framework is constructed for the deconvolution, and we present a Markov chain Monte Carlo method for sampling from the posterior distribution. There are several advantages to this approach, including that still takes advantage of the efficiency of spectral methods, that it allows the boundaries of the signal to This work was done by National Security Technologies, LLC, under Contract No. DE-AC52-06NA25946 with the U.S. Department of Energy and supported by the Site Directed Research and Development program. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. DOE/NV/25946–???? Email addresses: [email protected] (Johnathan M. Bardsley), [email protected] (Aaron Luttman) Preprint submitted to Linear Algebra and Its Applications September 30, 2013 be treated in a non-uniform manner–thereby reducing artifacts–and that the sampling scheme gives a natural method for quantifying uncertainties in the reconstruction.