Estimation of the Regularization Parameter in Linear Discrete Ill-Posed Problems Using the Picard parameter

摘要

Accurate determination of the regularization parameter in inverse problemsstill represents an analytical challenge, owing mainly to the considerabledifficulty to separate the unknown noise from the signal. We present a newapproach for determining the parameter for the general-form Tikhonovregularization of linear ill-posed problems. In our approach the parameter isfound by approximate minimization of the distance between the unknown noiselessdata and the data reconstructed from the regularized solution. We approximatethis distance by employing the Picard parameter to separate the noise from thedata in the coordinate system of the generalized SVD. A simple and reliablealgorithm for the estimation of the Picard parameter enables accurateimplementation of the above procedure. We demonstrate the effectiveness of ourmethod on several numerical examples. A MATLAB-based implementation of theproposed algorithms can be found athttps://www.weizmann.ac.il/condmat/superc/software/