New insights into one-norm solvers from the Pareto curve

Conclusions

The sheer size of seismic problems makes it a certainty that there will be significant constraints on the amount of computation that can be done when solving an inverse problem. Hence it is especially important to explore the nature of a solver's iterations in order to make an informed decision on how to best truncate the solution process. The Pareto curve serves as the optimal reference, which makes an unbiased comparison between different one-norm solvers possible.

Of course, in practice it is prohibitively expensive to compute the entire Pareto curve exactly. We observe, however, that the Pareto curves for many of the one-norm regularized problems are regular, as confirmed by the theoretical Result 1. This suggests that it is possible to approximate the Pareto curve by fitting a curve to a small set of sample points, taking into account derivative information at these points. As such, the insights from the Pareto curve can be leveraged to large-scale one-norm regularized problems, as we illustrate on a geophysical example. This prospect is particularly exciting given the current resurgence of this type of regularization in many different areas of research.

New insights into one-norm solvers from the Pareto curve