Kilmer, Misha E.O'Leary, Dianne P.Numerical solution of ill-posed problems is often accomplished by discretization (projection onto a finite dimensional subspace) followed by regularization. If the discrete problem has high dimension, though, typically we compute an approximate solution by projection onto an even smaller dimensional space, via iterative methods based on Krylov subspaces. In this work we present efficient algorithms that regularize after this second projection rather than before it. We prove some results on the approximate equivalence of this approach to other forms of regularization and we present numerical examples. (Also cross-referenced as UMIACS-TR-98-48)en-USTechnical Report