Generalized Cross Validation stopping rule for Iterated Tikhonov regularization

Buccini A.

2021-01-01

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Abstract

Ill-posed inverse problems arise in many fields of science and engineering. These problems are usually very sensitive to the presence of noise in the measured data. Regularization methods aim at reducing this sensitivity. Among these methods Iterated Tikhonov (IT), in both its standard and general form, has been widely investigated due to its ease of implementation and high level of accuracy. One of the most crucial point for its implementation is the stopping criterion. Possibly the most popular choice is the Discrepancy Principle (DP). Unfortunately, the DP requires additional knowledge of the problem, namely the norm of the noise that corrupts the data. In this paper we present a stopping criterion based on the Generalized Cross Validation (GCV) that does not require any additional knowledge of the problem. We propose it for both Iterated Tikhonov in standard and general form. One of the main drawback of the GCV is its computational cost. Thanks to the projection into Krylov subspace of fairly small dimension we are able to formulate our proposal for both small and large scale problems. The proposed method still requires the estimate of some parameters. Even though our approach is fairly stable with respect of the choice of these parameters, we develop a completely automatic method that does not require any hand-tuning and can be run in a truly "plug and play"fashion. We show the performances of the proposed methods on some selected numerical examples.