New insights into one-norm solvers from the Pareto curve |

**Gilles Hennenfent ^{}, Ewout van den
Berg^{}, Michael P. Friedlander^{}, and
Felix J. Herrmann^{}**

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 into
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.

- Introduction
- Problem statement
- Pareto curve
- Comparison of one-norm solvers

- Geophysical example
- Conclusions
- Acknowledgments
- Bibliography
- About this document ...

New insights into one-norm solvers from the Pareto curve |

2008-03-27