Evaluation of Parameter Choice Methods for Regularization of Ill-Posed Problems in Geomathematics

Many different parameter choice methods have been proposed for regularization in both deterministic and stochastic settings. The performance of a particular method in a specific setting and its comparison to other methods is sometimes hard to predict. This chapter reviews many of the existing parameter choice methods and evaluates and compares them in a large simulation study for spectral cutoff and Tikhonov regularization. The numerical tests deal with downward continuation, i.e., an exponentially ill-posed problem, which is found in many geoscientific applications, in particular those involving satellite measurements. A wide range of scenarios for these linear inverse problems are covered with regard to both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods.