Non-negativity and iterative methods for ill-posed problems

Abstract

The Generalized Minimal RESidual (GMRES) method and conjugate gradient method applied to the normal equations (CGNR) are popular iterative schemes for the solution of large linear systems of equations. GMRES requires the matrix of the linear system to be square and nonsingular, while CGNR also can be applied to overdetermined or underdetermined linear systems of equations. When equipped with a suitable stopping rule, both GMRES and CGNR are regularization methods for the solution of linear ill-posed problems. Many linear ill-posed problems that arise in the sciences and engineering have non-negative solutions. This paper describes iterative schemes, based on the GMRES or CGNR methods, for the computation of non-negative solutions of linear ill-posed problems. The computations with these schemes are terminated as soon as a non-negative approximate solution which satisfies the discrepancy principle has been found. Several computed examples illustrate that the schemes of this paper are able to compute non-negative approximate solutions of higher quality with less computational effort than several available numerical methods.