Enriched Krylov subspace methods for ill-posed problems Original Research Article

We describe a modification of the conjugate gradient method for the normal equations (CGNR) that allows us to enrich the Krylov subspaces, in which the iterates are determined, with vectors containing pertinent information about the desired solution. The enriched CGNR method easily can be adapted to the solution of linear systems arising from penalized least-squares problems and Tikhonov regularization. Applications to the solution of linear discrete ill-posed problems illustrate that enrichment of the Krylov subspaces can improve the quality of the computed approximate solutions and reduce the computational effort required for their determination.