On the choice of subspace for large-scale Tikhonov regularization problems in general form

Abstract

Many applications in science and engineering require the solution of large linear discrete ill-posed problems. The matrices that define these problems are very ill-conditioned and possibly numerically singular, and the right-hand sides, which represent the measured data, typically are contaminated by measurement error. Straightforward solution of these problems generally is not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to the error in the right-hand side. The penalty term is determined by a regularization matrix. A suitable choice of this matrix may result in a computed solution of higher quality than when the regularization matrix is the identity. Two iterative solution methods based on the global Arnoldi decomposition method have been proposed in the literature for the solution of large-scale penalized least-squares problems that stem from Tikhonov regularization. In one of these, the regularization matrix influences the choice of the solution subspace; in the other one, it does not. This paper compares these approaches both with respect to the quality of the computed solution and computing time.