Estimating Solutions of First Kind Integral Equations with Nonnegative Constraints and Optimal Smoothing

A method is presented for estimating solutions of Fredholm integral equations of the first kind, given noisy data. Regularization is effected by a smoothing term which is the $L^2 $-norm of the estimate. We propose a scheme by which an approximately optimal amount of smoothing may be computed, based only on the data and the assumed known noise variances. Numerical examples are given for estimating inverse Laplace transforms.