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

Geophysics, 73(4):A23–A26, 2008

Geophysics, 73(4):A23–A26, 2008

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.

```
@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}
}
```