Projected Newton Method for noise constrained Tikhonov regularization

TL;DR

This paper introduces a novel Projected Newton method that efficiently solves noise constrained Tikhonov regularization problems by simultaneously determining the regularization parameter and the solution, reducing computational costs.

Contribution

The paper presents a new algorithm that combines projection onto Krylov subspaces with a line search to efficiently solve Tikhonov regularization with discrepancy principle constraints.

Findings

The proposed method converges globally.

It outperforms existing solvers in computational efficiency.

Numerical experiments demonstrate improved performance.

Abstract

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.