Discrete Ill-posed Least-squares Problems with a Solution Norm Constraint

Straightforward solution of discrete ill-posed least-squares problems with errorcontaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. Error propagation can be reduced by imposing constraints on the computed solution. A commonly used constraint is the discrepancy principle, which bounds the norm of the computed solution when applied in conjunction with Tikhonov regularization. Another approach, which recently has received considerable attention, is to explicitly impose a constraint on the norm of the computed solution. For instance, the computed solution may be required to have the same Euclidean norm as the unknown solution of the error-free least-squares problem. We compare these approaches and discuss numerical methods for their implementation, among them a new implementation of the Arnoldi–Tikhonov method. Also solution methods which use both the discrepancy principle and a solution norm constraint are considered.