On the Choice of Subspace for Iterative Methods for Linear Discrete Ill-posed Problems

with a large matrix A of ill-determined rank. Thus, A has many “tiny” singular values of different orders of magnitude. In particular, A is severely ill-conditioned. Some of the singular values of A may be vanishing. We allow m ≥ n or m < n. The right-hand side vector b̃ is not required to be in the range of A. Linear systems of equations of the form (1) with a matrix of ill-determined rank are obtained when discretizing linear ill-posed problems, such as Fredholm integral equations of the first kind with a smooth kernel. They also arise in image restoration. Following Hansen (1998), we refer to linear systems of equations with a matrix of ill-determined rank as linear discrete ill-posed problems. In many linear discrete ill-posed problems that arise in the applied sciences and engineering, the right-hand side vector b̃ is contaminated by measurement and discretization errors, i.e.,