Evaluation of cross-validation formulas for choosing the regularization parameter for inversion of Fredholm integral of the first kind with non-negativity constraints

The inversion of Fredholm integral equation of the first kind is encountered in many applications where measurements have to be inverted to obtain information of a distribution of property values. Such problems can be solved using regularization techniques. The central issue in regularization techniques is the choice of an effective regularization parameter and it is desirable that this choice is made in an objective and automatic manner so that it is practical for use in online monitoring applications. In this work, two formulations based on cross-validation which account for non-negativity constraints are considered to search for the optimal regularization parameter. These are evaluated by applying to simulations of two different problems viz. the estimation of the distribution of adsorption energies and the extraction of particle size distributions from turbidity measurements. It was found that of the two methods for computing the cross-validation scores, one was computationally intensive but robust in that the search range of the regularization parameter can be broad. The second formula while much simpler and computationally faster provided a reliable regularization parameter only when a sufficiently narrow search range is used. This study indicates that a two-step approach which combines the two formulations could provide a method that will strike a balance in terms of computational speed while at the same time providing reliable values of the regularization parameters.