Projected Newton method for noise constrained lp regularization

Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The l(p) norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of p and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding l(p) regularized solution such that the discrepancy principle is satisfied. We project the problem on a low-dimensional generalized Krylov subspace and compute the Newton direction for this much smaller problem. We illustrate some interesting properties of the algorithm and compare its performance with other state-of-the-art approaches using a number of numerical experiments, with a special focus of the sparsity inducing l(1) norm and edge-preserving total variation regularization.