A simplified L-curve method as error estimator

TL;DR

This paper introduces a simplified L-curve method that uses the derivative instead of curvature to select regularization parameters, providing a practical error estimator with proven convergence.

Contribution

The paper proposes a simplified L-curve approach that replaces curvature with derivative, enabling it to serve as an effective error estimator with theoretical convergence guarantees.

Findings

The simplified method behaves similarly to the original L-curve.

It can serve as an error estimator under typical conditions.

The authors prove convergence of the new method.

Abstract

The L-curve method is a well-known heuristic method for choosing the regularization parameter for ill-posed problems by selecting it according to the maximal curvature of the L-curve. In this article, we propose a simplified version that replaces the curvature essentially by the derivative of the parameterization on the $y$-axis. This method shows a similar behaviour to the original L-curve method, but unlike the latter, it may serve as an error estimator under typical conditions. Thus, we can accordingly prove convergence for the simplified L-curve method.