A Newton root-finding algorithm for estimating the regularization parameter for solving ill-conditioned least squares problems

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a priori based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a χ² distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of the system matrix, is found to be very efficient for parameter estimation, requiring on average about 10 Newton steps. Additionally, the theory and algorithm apply for generalized Tikhonov regularization using the generalized singular value decomposition. The performance of the Newton algorithm is contrasted to standard techniques, including the L-curve, generalized cross validation and unbiased predictive risk estimation. This χ²-curve Newton method of parameter estimation is seen to be robust and cost effective in comparison to other methods, when white or colored noise information on the measured data is incorporated.