Some numerical aspects of Arnoldi-Tikhonov regularization

Large linear discrete ill-posed problems are commonly solved by first reducing them to small size by application of a few steps of a Krylov subspace method, and then applying Tikhonov regularization to the reduced problem. A regularization parameter determines how much the given problem is regularized before solution. If the matrix of the linear discrete ill-posed problem is nonsymmetric, then the reduction often is carried out with the aid of the Arnoldi process, while when the matrix is symmetric, the Lanczos process is used. This paper discusses several numerical aspects of these solution methods. We illustrate that it may be beneficial to apply the Arnoldi process also when the matrix is symmetric and discuss how certain user-chosen basis vectors can be added to the solution subspace. Finally, we compare the application of the discrepancy principle for the determination of the regularization parameter to another approach that is based on the solution of a cubic equation and has been proposed in the literature.(c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.