# Designing Optimal Regularized Inverse Matrices for Inverse Problems

Matthias Chung^{*}, Julianne Chung and Dianne O'Leary

In this talk, a new framework for solving inverse problems is developed where training data is used to compute an optimal regularized inverse matrix. The forward model is not required, however, knowledge regarding the forward model can be incorporated. An optimal regularized inverse 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 the empirical Bayes risk minimization problem. Once computed, the optimal regularized inverse matrix can be used to solve inverse problems very efficiently.

Mathematics Subject Classification: 65F22 65F20 62C10 65K05

Keywords: regularization; inverse problems; Bayes risk; empirical risk

Minisymposion: Noise Estimation, Model Selection and Bilevel Optimization