Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

To apply the Generalized Cross-Validation (GCV) as a stopping rule for an iterative method, we must estimate the trace of the so-called influence matrix which appears in the denominator of the GCV function. In the case of conjugate gradient, unlike what happens with stationary iterative methods, the regularized solution has a nonlinear dependence on the noise which affects the data of the problem. This fact is often pointed out as a cause of poor performance of GCV. To overcome this drawback, in this paper we propose a new method which linearizes the dependence by computing the derivatives through iterative formulas along the lines of Perry and Reeves (1994) and Bardsley (2008). We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation both on 1D and on 2D test problems.