One-norm regularized inversion: learning from the Pareto curve

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. First, we show how these curves provide an objective criterion to gauge how robust one-norm solvers are when they are limited by a maximum number of matrix-vector products that they can perform. Second, we use Pareto curves and their properties to define and compute one-norm compressibilities. We argue this notion is key to understand one-norm regularized inversion. Third, we illustrate the correlation between the one-norm compressibility and the perfor- mance of Fourier and curvelet reconstructions with sparsity promoting inversion.