Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. Regularization aims to reduce this sensitivity. Typically, regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term, and a q-norm to measure the regularization term, has received considerable attention. The relative importance of these terms is determined by a regularization parameter. This paper discussed how the latter parameter can be determined with the aid of the discrepancy principle. We primarily focus on the situation when p = 2 and 0 < q ≤ 2, where we note that when 0 < q < 1, the minimization problem may be non-convex.

Tikhonov regularization is one of the most popular methods for the solution of linear discrete ill-posed problems. In many applications the desired solution is known to lie in the nonnegative cone. It is then natural to require that the approximate solution determined by Tikhonov regularization also lies in this cone. The present paper describes two iterative methods, that employ modulus-based iterative methods, to compute approximate solutions in the nonnegative cone of large-scale Tikhonov regularization problems. The first method considered consists of two steps: first the given linear discrete ill-posed problem is reduced to a small problem by a Krylov subspace method, and then the reduced Tikhonov regularization problems so obtained is solved. The second method described explores the structure of certain image restoration problems. Computed examples illustrate the performances of these methods.

The nonstationary preconditioned iteration proposed in a recent work by Donatelli and Hanke appeared on IP can be seen as an approximated iterated Tikhonov method. Starting from this observation we extend the previous iteration in two directions: the introduction of a regularization operator different from the identity (e.g., a differential operator) and the projection into a convex set (e.g., the nonnegative cone). Depending on the application both generalizations can lead to an improvement in the quality of the computed approximations. Convergence results and regularization properties of the proposed iterations are proved. Finally, the new methods are applied to image deblurring problems and compared with the iteration in the original work and other methods with similar properties recently proposed in the literature

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method

The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions of these problems, if they exist, are very sensitive to perturbations in the available data. Regularization replaces the original problem by a nearby regularized problem, whose solution is less sensitive to the error in the data. The regularized problem contains a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term and a q-norm to measure the regularization term has received considerable attention. The balance between these terms is determined by a regularization parameter. In many applications, such as in image restoration, the desired solution is known to live in a convex set, such as the nonnegative orthant. It is natural to require the computed solution of the regularized problem to satisfy the same constraint(s). This paper shows that this procedure induces a regularization method and describes a modulus-based iterative method for computing a constrained approximate solution of a smoothed version of the regularized problem. Convergence of the iterative method is shown, and numerical examples that illustrate the performance of the proposed method are presented.