Learning regularization functionals - a supervised training approach

We consider the solution of a distributed parameter estimation problem where the data are contaminated by noise. A common approach to solve such a problem is to use Tikhonov style regularization; however, it is not always clear what type of regularization penalty should be used for a given problem as different regularization operators may yield very different solutions. Here we use supervised learning techniques to determine a regularization functional given a training set of feasible solutions. Our approach leads to a constraint optimization problem that we solve using inexact sequential quadratic programming type methods. We illustrate the methodology with two examples.