Near-optimal parameters for Tikhonov and other regularization methods

Choosing the regularization parameter for an ill-posed problem is an art based on good heuristics and prior knowledge of the noise in the observations. In this work, we propose choosing the parameter, without a priori information, by approximately minimizing the distance between the true solution to the discrete problem and the family of regularized solutions. We demonstrate the usefulness of this approach for Tikhonov regularization and for an alternate family of solutions. Further, we prove convergence of the regularization parameter to zero as the standard deviation of the noise goes to zero.