Inverse problems arise in scientific applications such as imaging, geophysics, astronomy, and computational biology, and computing accurate solutions to inverse problems can be both mathematically and computationally challenging. In this talk, a new framework for solving inverse problems is developed by incorporating training data to compute an optimal regularized inverse matrix. This matrix is obtained by incorporating probabilistic information and solving a Bayes risk minimization problem. We present theoretical results for the Bayes problem and discuss efficient approaches for solving associated empirical Bayes risk minimization problems. Once computed, the optimal regularized inverse matrix can be used to solve inverse problems very efficiently.
This is joint work with Julianne Chung (Virginia Tech) and Dianne O'Leary (University of Maryland, College Park).