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Effective New Methods for Automated Parameter Selection in Regularized Inverse Problems

The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting a good parameter value by maximizing the probability of the data, {with no prior knowledge of the noise variance}. These concepts are developed for $\ell_2$ and consequently $\ell_1$ regularization models by way of their Bayesian interpretations. Based on these concepts, an iterative scheme is proposed and demonstrated to converge accurately, and analytical convergence results are provided that substantiate these empirical observations. For some of the most common inverse problems, including MRI, SAR, denoising, and deconvolution, an extremely efficient algorithm is derived, making the iterative scheme very attractive for real case use. The computational concerns associated with the general case for any inverse problem are also carefully addressed. A robust set of 1D and 2D numerical simulations confirm the effectiveness of the proposed approach.