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201606 https://hdl.handle.net/2286/R.2.N.201606 http://rightsstatements.org/vocab/InC/1.0/ All Rights Reserved 2025 49 pages Masters Thesis Academic theses en Pawloski III, Robert Michael Espanol, Malena Platte, Rodrigo Lan, Shiwei Arizona State University Partial requirement for: M.A., Arizona State University, 2025 Field of study: Mathematics Inverse problems are prevalent across numerous practical applications in which researchers seek to recover unknown information given some noisy information. Choosing to implement a Bayesian framework, rather than traditional regularization methods, overcomes the ill-posedness of such problems while allowing for uncertainty quantification in the computed results. Experiments are often designed to include numerous data collection methods, resulting in datasets of varying accuracy. This motivates the development of computational methods within the Bayesian paradigm that reconcile sets of observations of varying accuracy levels. This thesis considers the addition of sparse, high-accuracy data scattered across the domain. The goal is to use the limited high-fidelity data, along with lower-fidelity data spanning the entire domain, to develop improved reconstructions with lower uncertainty. With this objective in mind, a preexisting Gibbs sampling algorithm is modified to allow the incorporation of a secondary dataset. A partitioning of the domain during preprocessing is proposed, allowing for the construction of spline estimations using the higher-accuracy data. Combining this with a reasonably accurate initial reconstruction, a Markov chain Monte Carlo algorithm is applied to compute a correction vector to the initial solution. This technique is established within the context of the deblurring problem, where numerical results demonstrate that mean reconstructions can be computed with lower error and narrower credible intervals. In addition, applications of this method in radiography are explored and used to illustrate how the proposed method can be employed to estimate the density profiles of unknown axially symmetric objects. Applied Mathematics Mathematics Statistics Bayesian modeling Gibbs sampling image deblurring Inverse Problems Uncertainty Quantification Data Integration Methods for Bayesian Inverse Problems