Direct analytic model of the L-curve for Tikhonov regularization parameter selection

We present an approximation to a direct, non-parametrized analytic expression for the L-curve used in the regularization of ill-conditioned linear systems which is constructed starting from the exact solution for the case of a linear operator with just one singular value. For the general case, this expression suggests one choice of Tikhonov regularization parameter per singular value at which to compute the curvature of the log–log L-curve plot. It suffices to test a small number of candidate values in the vicinity of the mean square residual of the unregularized solution. Monte Carlo simulations show that choosing the maximum curvature generated from within this set of candidate values selects an optimum value of the regularization parameter which yields a solution whose norm typically lies within a fraction 0.125/κ_L of the exact L-curve value, where κ_L is the maximum log–log curvature at the L-curve corner. For large scale ill-posed problems, a new regularization parameter selection method based on the construction of an L-curve from a single sequence of conjugate gradient iterates is presented. It includes a new stopping rule for terminating the iterative solution of large scale least squares problems. Numerical experiments show that the accuracy of the regularized solution obtained with the value of the regularization parameter at the corner of this curve is close to that of the exact L-curve solution.