Projected Newton Method for noise constrained Tikhonov regularization

Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy principle. However, most approaches require the solution of the Tikhonov problem for many different values of the regularization parameter, which is computationally demanding for large scale problems. We propose a new and efficient algorithm which simultaneously solves the Tikhonov problem and finds the corresponding regularization parameter such that the discrepancy principle is satisfied. We achieve this by formulating the problem as a nonlinear system of equations and solving this system using a line search method. We obtain a good search direction by projecting the problem onto a low dimensional Krylov subspace and computing the Newton direction for the projected problem. This projected Newton direction, which is significantly less computationally expensive to calculate than the true Newton direction, is then combined with a backtracking line search to obtain a globally convergent algorithm, which we refer to as the Projected Newton method. We prove convergence of the algorithm and illustrate the improved performance over current state-of-the-art solvers with some numerical experiments

Implicit iteration methods in Hilbert scales under general smoothness conditions

For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting operator monotonicity of certain functions and interpolation techniques in variable Hilbert scales, we study these methods under general smoothness conditions. Order optimal error bounds are given in case the regularization parameter is chosen either {\it a priori} or {\it a posteriori} by the discrepancy principle. For realizing the discrepancy principle, some fast algorithm is proposed which is based on Newton's method applied to some properly transformed equations