Analysis of Discrete Ill-Posed Problems by Means of the L-Curve

TLDR

In discrete ill‑posed problems, plotting the solution norm against the residual norm—especially for Tikhonov regularization—provides a convenient visual summary of regularized solutions. The paper advocates using this solution‑vs‑residual norm graph to guide the numerical treatment of discrete ill‑posed problems. The authors quantitatively characterize the graph and derive key relationships between regularized solutions and its features. They show that various regularization‑parameter selection methods correspond to locating the L‑shaped corner of the graph.

Abstract

When discrete ill-posed problems are analyzed and solved by various numerical regularization techniques, a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector. In particular, the graph associated with Tikhonov regularization plays a central role. The main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill-posed problems. The graph is characterized quantitatively, and several important relations between regularized solutions and the graph are derived. It is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic L-shaped “corner” of the graph.