New insights into one-norm solvers from the Pareto curve

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Abstract

Geophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems with two-norm priors where they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one norm, Pareto curves remain relatively unexplored. We show how these curves lead to new insights in one-norm regularization. First, we confirm the theoretical properties of smoothness and convexity of these curves from a stylized and a geophysical example. Second, we exploit these crucial properties to approximate the Pareto curve for a large-scale problem. Third, we show how Pareto curves provide an objective criterion to gauge how different one-norm solvers advance towards the solution.

BiBTeX

@article{HennBergFrieHerr:2008,
  Title =        {New insights into one-norm solvers from the Pareto
                  curve},
  Author =       {G. Hennenfent and E. van den Berg and M. P.
                  Friedlander and F. Herrmann},
  Volume =       {73},
  Number =       {4},
  Journal =      {Geophysics},
  Month =        {July--August},
  pages =        {A23--A26},
  Year =         2008,
  doi =          {10.1190/1.2944169}
}