Inverse problems – the process of recovering unknown parameters from indirect measurements – are encountered in various areas of science, technology and engineering including image processing, medical imaging, geosciences, astronomy, aeronautics engineering and machine learning. Statistical and probabilistic methods are promising approaches to solving such problems. Of these, the Bayesian methods provide a principled approach to incorporating our existing beliefs about the parameters (the prior model) and randomness in the data. These approaches are at the forefront of extensive current investigation. Overwhelmingly, Gaussian prior models are used in Bayesian inverse problems since they provide mathematically simple and computationally efficient formulations of important inverse problems. Unfortunately, these priors fail to capture a range of important properties including sparsity and natural constraints such as positivity, and so we are motivated to study non-Gaussian priors. In this thesis we provide a systematic study of the theory and applications of Bayesian approaches to inverse problems with non-Gaussian priors. We develop the theory of well-posedness of infinite-dimensional Bayesian inverse problems with convex, heavy-tailed or infinitely divisible prior measures. We also introduce new prior measures that aim to model compressible or sparse parameters. Next, we demonstrate the applications of Bayesian approaches to important inverse problems in industrial applications: the estimation of emission rates of particulate matter, and the estimation of acoustic aberrations in ultrasound treatment. We propose two Bayesian approaches for the problem of estimating the emission rates of particulate matter into the atmosphere from far field measurements of deposition. Next, we present a Bayesian method for estimation of acoustic aberrations in high intensity focused ultrasound treatment of tissue in the brain using magnetic resonance images. The final contribution of this thesis is a systematic construction and convergence analysis of regularizations of the Dirac delta distribution. Point sources arise naturally in many models and we discuss smooth regularizations of these.
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