In this paper we propose a new method to apply the Generalized Cross-Validation (GCV) as a stopping rule for the Conjugate Gradient (CG). In general, to apply GCV to an iterative method, one must estimate the trace of the so-called influence matrix which appears in the denominator of the GCV function. In the case of CG, 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, our proposal linearizes the dependence by computing the derivatives through iterative formulas. We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1D and 2D test problems.

Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems / P., Favati; Lotti, Grazia; O., Menchi; F., Romani. - In: APPLIED MATHEMATICS AND COMPUTATION. - ISSN 0096-3003. - 243:(2014), pp. 258-268. [10.1016/j.amc.2014.05.109]

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

#### Abstract

In this paper we propose a new method to apply the Generalized Cross-Validation (GCV) as a stopping rule for the Conjugate Gradient (CG). In general, to apply GCV to an iterative method, one must estimate the trace of the so-called influence matrix which appears in the denominator of the GCV function. In the case of CG, 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, our proposal linearizes the dependence by computing the derivatives through iterative formulas. We compare the proposed method with other methods suggested in the literature by an extensive numerical experimentation on both 1D and 2D test problems.
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2014
Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems / P., Favati; Lotti, Grazia; O., Menchi; F., Romani. - In: APPLIED MATHEMATICS AND COMPUTATION. - ISSN 0096-3003. - 243:(2014), pp. 258-268. [10.1016/j.amc.2014.05.109]
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11381/2752902`
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